Time Charter Contracts in the Shipping Industry

Transcription

1 Time Charter Contracts in the Shipping Industry A Fair Valuation Perspective Renathe Elven MSc Finance Supervisor: Peter Løchte Jørgensen Department of Economics and Business Aarhus University Business and Social Sciences August 2013

2 Abstract This thesis studies a specific type of asset lease namely time charter contracts with purchase options. Time charter contracts are common in the shipping industry and give the charterer the operational control of the vessel leased, whereas the option to purchase the vessel gives the charterer the right, but not the obligation, to purchase the vessel at the options expiration. The options embedded in such contracts are often complex in nature such that they are granted for free rather than for their fair value. The intention of this thesis is to introduce fair valuation of the total value of time charter contracts with embedded options by introducing two potential models for valuation purposes. Both models are one-factor models that are assumed to model the main source of risk in the shipping industry namely the freight rate. The two adopted models ensure the freight rate to evolve in continuous time, and one of them allows for the derivation of analytic solutions for some simple freight rate contingent claims. The other model values the vessel underlying the contract, as well as an embedded European option to buy the vessel by implementing Monte Carlo simulation. Acknowledgements I will like to thank my supervisor, Peter Løchte Jørgensen, for his help and valuable comments during this period. I

3 Table of Contents Abstract... I Acknowledgements... I List of Tables... V List of Figures... V 1 Introduction Motivation Aim Structure Theoretical Framework Introduction to Important Terms in the Shipping Industry Freight Rates Spot Freight Rates Time Charter Equivalent Spot Freight Rates The Shipping Industry Agents in the Shipping Industry The Different Shipping Segments Vessels in the Shipping Industry The Shipping Market Model Costs in the Shipping Industry Business Risks in Shipping Price Risk Credit Risk Pure Risk Summing Up - Analyzing and Managing Freight Rate Risk The Four Shipping Markets The Freight Market The Sale and Purchase Market The Newbuilding Market The Demolition Market Time Charter Contracts with Embedded Options Background for the Models Selected II

4 2.8 The Dynamics of Freight Rates Shipping Market Cycles Freight Rate Dynamics The Ornstein-Uhlenbeck Process The Solution to the Ornstein-Uhlenbeck Process The Geometric Mean Reversion Process The Solution to the Geometric Mean Reversion Process Analysis Section Two Models Different Characteristics: A Comparison Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation of Freight Rate Contingent Claims Derivation of the Fundamental Partial Differential Equation Valuation of some simple Freight Rate Contingent Claims Claim to Receive Spot Freight Rate Flow from Time to Time Fixed for Floating Freight Rate Swap The Value of a Vessel European Option to Buy a Vessel Applications of the Geometric Mean Reversion Process: Vessel and European Option Valuation Monte Carlo Simulation The Value of a Vessel European Option to Buy a Vessel The Valuation Results: Comparisons Limitations Limitations Caused by the Models Selected The Parametric Property of the Models One-Factor The Assumptions Constant Market Price of Freight Rate Risk Constant Risk-Free Interest Rate Summary and Conclusions List of References III

5 7 Appendix The Ornstein-Uhlenbeck Process Detailed Solution The Ornstein-Uhlenbeck process - Derivation of the Mean and the Variance The Time Conditional Mean The Time Conditional Variance The Geometric Mean Reversion Process Detailed Solution The Ornstein-Uhlenbeck Process Claim to Receive Spot Freight Rate Flow from Time to Time The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel The VBA Codes IV

6 List of Tables Table 1: The variables in the shipping market model...8 Table 2: Base case parameter values Table 3: The dependence of fair time charter rates Table 4: The value of a 5-year time charter contract Table 5: The dependence of vessel values Table 6: Value of European option to buy a vessel Table 7: The value of a 5-year time charter contract with European purchase option Table 8: The dependence of vessel values Table 9: Value of European option to buy a vessel List of Figures Figure 1: Simulated spot freight rate from the Ornstein-Uhlenbeck process Figure 2: Simulated spot freight rate from the Geometric Mean Reversion process Figure 3: Simulated and expected value of a vessel V

7 1 Introduction Through the last decade, the shipping industry has been subject to extreme volatility in freight rates. Over the period from 2003 to mid-2008, freight rates increased by almost 300 per cent to exceptional levels. This large increase in freight rates was followed by a corresponding drop of 95 per cent over the last quarter of Such high volatility in the market also implies a shipping industry that is extremely risky. Therefore, the last decade s market fluctuations have changed the way the shipping industry views and manages its risks, and accordingly the derivatives market for freight have accelerated and a commoditization of the freight market is present. Today, agents that may not be involved in the underlying physical market, such that investment banks, hedge funds and other traders, can be seen participating in the shipping industry. 1.1 Motivation Thus, the accelerated derivatives market for freight has opened up possibilities for hedging and managing risk stemming from large volatility in freight rates. This leads to the topic of this thesis, namely fair valuation of time charter contracts with embedded options. Time charter contracts are common contractual agreements in the shipping industry which will be described in detail later. Options embedded in time charter contracts are wildly used and can be considered as a tool used to hedge against freight rate risk, this will also be described in detail later. More specific, the options which will be examined and valued here are call options which enable the charterer to purchase the vessel either during the contract period, or at the end of the contract period. The options serve as an insurance against undesirable movements in freight rates. Time charter contracts with purchase options are interesting from both academic and practical business management perspectives as they can be very complex and of significant economic importance. Jørgensen and Giovanni (2010) mention that they are aware of several shipping companies having a total net asset value where more than half of it stems from an estimated value of their portfolio of time charter contracts with purchase options. Thus, properly valuation of these contracts is extremely important in order to both support the stock markets valuation of shipping companies, and in order to assist managers of such companies in the general process of operation and risk management of their companies. High volatility in freight rates may create either a significant decrease in a shipping company s total reported net asset value or a significant increase in its total net asset value. 1

8 The complex nature of time charter contracts with purchase options makes fair valuation a difficult task. The need for development and analysis of good valuation models are therefore increasingly important. According to Alizadeh and Nomikos (2009), embedded options in the shipping industry are very often granted for free or for a nominal fee without being properly valued. 1.2 Aim Therefore, this thesis aims to shed light on the importance of fair valuation of time charter contracts with embedded options, as well as valuations of such contracts and a comparison between models. Two models will be introduced for valuation purposes and their valuation abilities will be compared. The model that will be used to obtain total values of a time charter contracts with embedded purchase options is adopted from Jørgensen and Giovanni (2010). The other model is adopted from Tvedt (1997) and will be introduced as an alternative model for valuations to the one from Jørgensen and Giovanni (2010). 1.3 Structure This thesis is divided into three main parts; a theoretical framework, an analysis section and a discussion section where limitations are elucidated. First, a theoretical framework of the shipping industry will be established. A fundamental introduction of the comprehensiveness of the shipping industry will be given, where the shipping market model will be emphasized. This simplified model describes the mechanisms that make freight rates evolve in accordance with the market cycles. Further, the time charter contracts with embedded options in the shipping industry will be described, as well as the background for the model selection. Finally, the dynamics of freight rates will be described forming the basis for the two models selected before the theoretical framework ends by a detailed description of these models, and for that purpose also a derivation of the two processes solution will be done. As an introduction to the analysis section the two models characteristics will be compared and discussed. Evidences of one model being more appropriate in freight rate modeling will be presented. Further, derivations will be done in order to value time charter contracts with purchase options. To a greater or lesser extent, both models will be applied for valuations. At the end, some of the limitations regarding the two models will be presented before the thesis will be summarized and concluded. 2

9 2 Theoretical Framework This section will first give a basic understanding of the shipping industry before it continues with a description of time charter contracts with embedded options. Further, the background that clarifies the reasons for the two models chosen will be presented, as well as a description of the freight rate dynamics. Finally, the two models adopted for the valuations will be presented in detail in which the solutions to both of them also will be derived. 2.1 Introduction to Important Terms in the Shipping Industry To prevent confusions, some relevant terms in relation to the shipping industry will be described in this first section of the theoretical framework. This will serve as a soft introduction to the comprehensive shipping industry Freight Rates Freight rates represent the cost of providing the service of seaborne transportation (Alizadeh and Nomikos 2009). Hence, freight rates are not tangible assets and can therefore not be stored. Kavussanos and Visvikis (2006) do also stress this special feature of freight services where they describe the demand for freight services as a derived demand. This is due to the fact that the freight service provided by the vessel is gone if it is not utilized at the time it is available. Again, the freight service is non-storable and it cannot be carried forward in time. Freight rates evolve through time according to market cycles that are prevalent in the shipping industry. These will be described later Spot Freight Rates The spot freight rate represents the cost of providing seaborne transportation today. It reflects the continuous balance between supply and demand for shipping services (Alizadeh and Nomikos 2009). Factors determining supply and demand will be described in Section where the freight rate mechanism will be examined. 3

10 2.1.3 Time Charter Equivalent Spot Freight Rates The time charter equivalent spot freight rate represents the spot freight rate less the voyage costs (Tvedt 1997). During this thesis, it is actually the time charter equivalent spot freight rate that is applied in the derivations. For reasons explained later, applying the time charter equivalent spot freight rate is in fact beneficial when valuing time charter contracts with purchase options. What is already important to note is that market-quoted spot freight rates embed varying degree of costs, and thus the time charter equivalent spot freight rates will not be directly comparable with the market-quoted spot freight rates. This is important to have in mind when comparing with market data. 2.2 The Shipping Industry This section will give an introduction to the shipping industry that plays a central role in the global economy and has also been at the forefront of global development through times. The main assets in the shipping industry are the vessels that can transport cargo from one part of the world to another. Already, it is worth mentioning that competition is an important key word in relation to this intriguing industry, and as for the bulk shipping segment that this thesis aim to address, the condition of perfect competition is present. When nothing else is stated, the whole section will be based on Stopford (2009) Agents in the Shipping Industry In the following, a description of the most important agents participating in the shipping industry will be given. Hence, some agents will be left outside this explanation since they are less important according to the aim of this thesis. The groups that will be explained here are the shipowners, the charterers and the shipbrokers. The shipowner has vessels for hire and enters the market with a vessel available free of cargo. The vessel has particular characterizations and the vessel s availability will be described in the contractual agreements. The shipowner may be looking for a short charter for the vessel or a long charter, in which will be dependent on the shipowner s strategy. The charterer can be either an individual or an organization; common to both of them is that they have a volume of cargo they need to transport from one location to another. The vessel type needed will be determined by the physical characteristics of the cargo. 4

11 The shipbroker operates as an intermediary between shipowners and charterers. According to who have a need, the shipbroker will be contacted and his task is to discover what cargoes or vessels are available, what expectations the shipowners/charterers have about what they will be paid or pay, and what is reasonable given the state of the market. The deal for their client is negotiated, and often in tense competition with other brokers The Different Shipping Segments Three different segments constitute the shipping industry; these are the liner shipping segment, the bulk shipping segment and the specialized shipping segment. This clearly division of the industry is necessary in order to meet specific needs of different customers; anything from grain to cars can be transported by sea. Shipping companies characteristics differ depending on which segment they operate in, this is due to differences in both the transported cargo, and in the dynamics of how the agreement between the shipowner and the charterer is conducted. The characteristics of each shipping segment will be described below. Despite the differences in shipping companies characteristics across these three segments, one shipping company can often operate in more than one segment and thus contribute to intense competition for the same cargo. Therefore, it is not convenient to treat the shipping industry as a series of isolated segments, but rather as a single market. Investors in the shipping industry move their investments from one segment to another, and if there is a supply-demand imbalance in one of the segments, this will also move on to the other segments. As this thesis aims to address the bulk shipping segment, liner shipping and specialized shipping will only shortly be described, whereas the bulk shipping segment will be examined in more detail. The liner shipping segment offers transport for cargoes that are too small to fill a single vessel and thus need to be grouped with others for transportation. Such cargoes are often highly valued and can be delicate in nature; the shipper thus often requires a special shipping service with a fixed tariff rather than a fluctuating market rate. Cargoes transported in the liner shipping segment are called general cargo and include loose cargo, containers and pallets. This creates complex administrative tasks and makes this segment very different from the bulk shipping segment. The specialized shipping segment is recognized by properties lying in between the liner shipping segment and the bulk shipping segment. This leads to a somewhat indefinite distinction between the specialized shipping segment and the two other segments. Cargoes that are considered special include cars, forest products, chemicals and refrigerated products. 5

12 The bulk shipping segment supplies transport for cargoes that need to be transported in large homogeneous shiploads. Generally a whole vessel is hired for transportation of one type of cargo, but it is also possible to carry different bulk cargoes in a single vessel. If so, each cargo occupies a separate hold or possibly even part of a hold. Commodities that often need to be transported in bulks are raw materials and bulky semi-manufactures. The bulk shipping segment is divided into dry bulk and liquid bulk. Dry bulk cargo transported in shiploads is mainly raw materials such as iron ore, coal and grain. Liquid bulk cargo includes crude oil, oil products, and liquid chemicals. Cost minimization of providing safe transport, as well as effective management of vessel investments is the main focus within the bulk shipping business. Costs can be minimized due to the characteristics of the bulk segment in which few transactions are handled. Typically, a vessel completes about six voyages with a single cargo each year. As already mentioned, the bulk shipping segment is subject to conditions of perfect competition. This creates rather volatile freight rates and prices (Kavussanos and Visvikis 2006). Perfect competition arises due to the many buyers and sellers of freight services. They negotiate on a relatively homogenous product, the freight service, and have no barriers to entry or exit the market. In addition, the freight markets are well organized markets. The product can be assumed to be almost perfectly homogenous since, in a particular route-trade, there are no significant differences in the quality of the freight service offered. The fact that the product is almost perfectly homogenous relates to some minor differences in vessel characteristics and customer relations that may exist between shipping companies. Another contribution to the condition of perfect competition is the availability of information in the freight markets. The Baltic Exchange, among others, brings together participants wishing to buy or sell the freight service. Relevant information on fixtures 1, prices, and cargoes/vessels available are collected and disseminated to the market. Such information is used by shipowners and charterers to make informed decisions in the freight markets (Kavussanos and Visvikis 2006). The feature of perfect competition arising in the bulk shipping segment, resulting in volatile freight rates, makes it especially interesting to address this particular segment; investors and charterers are to a great extent exposed to risk, and proper risk management becomes extremely important. In summary, these three shipping segments face different tasks depending on the value and volume of cargo, the number of transactions handled, and the commercial systems employed. 1 Stopford (2009) explains a fixture as an agreement where a vessel is chartered or a freight rate is agreed on. Further, he explains that the arrangement happens in much the same way as any major international hiring or subcontracting operation. 6

13 2.2.3 Vessels in the Shipping Industry In order to give a comprehensive picture of the shipping industry a description of different vessel types is, at this point, appropriate. Due to the differences in cargoes transported by sea, both within each segment and also across the segments, vessels are built and adjusted to fit the cargoes they transport. The result is several different vessel types and sizes in the world fleet 2. However, this section will only give a presentation of the different vessel types in the bulk shipping segment. There exist four different types of vessels in the dry bulk sector where each of them are classified by their size measured in dead weight tons (dwt). Handy bulk carriers are the smallest ones, those vessels are of to dwt. Handymax bulk carriers are of to dwt, Panamax are of to dwt, and Capesize are the largest vessels of over dwt. As any other physical asset vessels do also have a limited lifetime. The shipping industry is in continuous technological progress and the vessels operating in the industry therefore suffer from obsoleteness rather fast. Logically, as a vessel grows old or gets obsolete its value will also decrease, this reduction in value continues until the vessel s age lies between 20 and 30 years which is when it is normally scrapped. According to Kavussanos and Visvikis (2006), a useful tool to understand why different vessel sizes are necessary is the Parcel Size Distribution (PSD) of each commodity. A parcel is an individual consignment of cargo for shipment, each commodity can be transported in different parcel sizes, and these different parcel sizes constitute the Parcel Size Distribution. Based on the observation that some commodities are typically moved in larger sizes than others, the PSD function describes the range of parcel sizes in which that commodity is transported. In addition to the parcel size distribution, port and seaway restrictions have created differences in types and sizes of vessels The Shipping Market Model To understand the freight market and how the freight market cycles are generated the mechanisms determining spot freight rates will be explained next. These are the economic mechanisms that the shipping industry uses to regulate supply and demand. The ten most important economic variables determining supply and demand are collected in a simplified model called The Shipping Market Model. Ten variables which have an influence on the shipping market are listed and the purpose is to leave out less relevant details in order to create a 2 All existing vessels. 7

14 picture of how the spot freight rates are determined. Five of the variables influence demand in the shipping market, and the other five influences supply. The variables are listed in Table 1 below. Demand Supply 1. The World Economy 1. The World Fleet 2. Seaborne Commodity Trades 2. Fleet Productivity 3. Average Haul 3. Shipbuilding Production 4. Random Shocks 4. Scrapping and Losses 5. Transport Costs 5. Freight Revenue Table 1. Ten variables in the shipping market model. Source: Stopford (2009), page 136. First, the demand for sea transport will be examined. According to Kavussanos and Visvikis (2006), demand for freight services is a derived demand; the charterer s demand is not for the vessel, but for the service the vessel provides The World Economy The world economy is considered as the most important single influence on vessel demand. This is due to the world economy generating most of the demand for sea transport. Events in the world economy that generate demand for sea transport is import of raw materials for the manufacturing industry, and the trade in manufactured products. The relationship between the world industry and the demand for sea transport is complex and consists of two aspects of the world economy; the business cycle and the trade development cycle. These two aspects may create changes in the demand for sea transport. The business cycle has an important influence on the demand for sea transport in the short-term. Fluctuations in the world economy are directly transferred to the shipping market. In this way, the foundation for the cyclical behavior of freight rates is set. The trade development cycle is related to the long-term relationship between sea transport and the world economy. It says something about the speed of industrial growth relative to the speed of sea trade growth. The sea trade growth of individual regions will change as time goes by due to both the 8

15 change in a country s economic structure, and due to the ability of local resources of food and raw materials to meet local demand Seaborne Commodity Trades This variable explains the relationship between sea trade and the industrial economy both in the short-term and in the long-term. Short-term volatility arises due to seasonality in some trades. An example of such a trade is grain, which is subject to seasonal variations caused by harvests. Due to the difficulties in planning transportation of seasonal agricultural commodities, shippers rely heavily on the spot charter market when demand for sea transport arises. Thus, fluctuations in the grain market have larger impact on the spot charter market than other trades, like for example iron ore. Tonnage requirements in the transportation of iron ore are almost always met through long-term contracts. Demand affected by long-term trends in commodity trade is best identified by studying the economic characteristics of the industries that produce and consume the traded commodities. Overall, there are four types of changes that affect the demand for seaborne transport in the long-run; changes in the demand for that particular commodity, changes in the source from which supplies of the commodity are obtained, changes due to a relocation of processing plant changing the trade pattern, and changes in the shipper s transport policy. Changes in demand for that particular commodity may have an effect on the tonnage requirements if this particular commodity is imported. If, for example, the country decides to replace the imported commodity by a domestic commodity, this will naturally affect the demand for seaborne transport. Changes in the source from which supplies of the commodity are obtained happen when new sources are discovered. These new sources may happen to be located near countries that earlier imported the same commodity, this may result in imports of this commodity being redundant. Thus, the demand for sea transport is changed. Changes due to relocation applies to industrial raw materials, and may affect both the volume of cargo transported by sea, and the type of vessel used to transport this cargo. Raw materials are often transformed several times before the final product is made. If the transformation is done before it is shipped rather than after, the volume and characteristics of the vessel may be changed. Changes in the shipper s transport policy relates to, for example, switching between using long-term contracts and using the spot charter market. This will again affect the demand for sea transport. 9

16 Average Haul Cargo shipped over larger distances generates more demand for sea transport than cargo shipped over shorter distances. Therefore, demand is affected by the length of where the cargo is shipped. The demand of sea transport is measured in ton miles making sure that the distance effect is taken account for. Ton miles are defined by the tonnage of cargo shipped, multiplied by the average distance over which it is transported Random Shocks Random shocks can have major impact on the economic system, and in turn affect the cyclical process. Random shocks of more or less severity include weather changes, wars, new resources and commodity price changes. Economic shocks do often have the most important influence on the shipping market, the reason why is that the timing is usually unpredictable and they bring about a sudden and unexpected change in vessel demand. Political events often have an indirect effect on vessel demand. Examples of such events are a localized war, a revolution or strikes Transport Costs Raw materials will only be transported from other destinations around the world if the transportation costs are at a relatively low level, or if the quality of a product can be increased to a level that gives major benefits. Improved efficiency, bigger vessels and more effective organization of the shipping operation result in reduced transport costs and higher quality of service. Thus, also the amount of seaborne transport increases. Even though these five factors influencing demand for sea transport are a simplified picture of reality, they give an indication of the complex nature of seaborne demand. On the other hand, the supply for sea transport is quite different in nature as also will be seen in the following. The supply for seaborne transport is characterized as being slow in its adaptation to changes in demand. This is due to the time-lag created in response to an increase in demand; several years are needed for the completion of a new vessel. Responding to a decrease in demand is also a slow 10

17 process, once a vessel is built it is estimated to have a lifetime of years (Stopford 2009). Therefore, if a large surplus is to be removed, that process can take several years resulting in a timelag between the decreased demand and the surplus reduction The World Fleet This variable measures the total amount of vessels existing all over the world. Scrapping old vessels and deliveries of new vessels determine the rate of fleet growth, which can be positive or negative. The world fleet contains all the different vessel types and sizes. Since a new vessel is estimated to have a lifetime of 25 years on average only a few vessels are scrapped each year. The pace of adjustments to changes in the market is therefore measured in years. When demand for seaborne transport does not turn out as expected, supply is adjusted. This is the key feature of the shipping market model (Stopford 2009) Fleet Productivity The productivity of the vessels that constitute the world fleet can vary. This creates a flexibility element since a vessel often has several days where it does not transport cargo. Such ineffective activities include ballast time, cargo handling, incidents, repair, lay-up, waiting, short-term storage and long-term storage. The fleet productivity is measured in ton miles per dead weight ton and depends upon four main factors; speed, port time, deadweight utilization and loaded days at sea. Speed is measured by the time a vessel uses on a voyage. New vessels are often designed to go faster, but this reduces the transport capacity of the vessel. Also, older vessels are often subject to hull fouling which will reduce the maximum operating speed. Port time relates to the time a vessel is at port. Factors that determine the efficiency at port include the physical performance of the vessels and terminals, and the organization of the transport operation. The deadweight utilization measures how much of the total cargo capacity that is lost due to bunkers, stores, etc. Loaded days at sea are the time where the vessel actually transports cargo at sea. It is desirable to increase loaded days at sea to improve the efficiency of the world fleet. 11

18 Shipbuilding Production Shipbuilding is an important adjustment factor to the world fleet. In times of increased demand, shipbuilding can increase the world fleet to meet the demand required. But building new vessels is a lengthy process, therefore, when deciding to build a new vessel it is important that the need for this vessel in the future is identified. It can be difficult to predict future demand, and if the prediction turns out to be wrong, this can lead to excess of vessels in relation to required demand Scrapping and Losses The balance between delivery of new vessels and scrapping of old ones (or losses) determines the growth rate of the world fleet. A new vessel is estimated to have a lifetime between years, indicating the difficulty in estimating exactly when the vessel is to be scrapped. The reason why is that scrapping depends on the balance of a number of factors; age, technical obsolescence, scrap prices, current earnings and market expectations. These factors create some flexibility to the shipowner in deciding when a vessel is to be scrapped Freight Revenue The freight rate is the most important regulator of the supply of sea transport. Freight rates are used by the market to motivate decision-makers to adjust capacity in the short-term, and to find ways of reducing their costs in the long-term. The shipping industry consists of two main pricing regimes; the freight market and the liner market. As explained above, the liner market can be thought of as a retail shipping business; transport of cargo in small quantities is offered to many customers. The freight market (bulk shipping) is totally different, this can be thought of as a wholesale operation; transport of cargo in shiploads is offered to few customers at individually negotiated prices. In the short-term, supply is adjusted in response to prices by changing the vessels operation speed and move to and from lay up. In the long-term, investment decisions such as ordering new vessels and scrapping old ones, are heavily influenced by the freight rate. The supply and demand adjustment mechanism will be explained in more detail in Section 2.8 where freight rate dynamics are examined. 12

19 Summing Up According to Tvedt (1997) it is usual to assume that demand is quite inelastic to freight rates. The shipping industry is characterized by large scale operations, and the cost of transportation at sea is a minor share of the total oil price. Therefore, only to a very small extent, demand is supposed to depend on freight rates. Further, he points out that the supply can be quite inelastic to freight rates in the short run when there are no vessels available. The reason is that speed and efficiency in loading and discharging only to a limited degree can be increased. On the other hand, when freight rates have been low for a while many vessels may have been laid up. This makes it possible to increase short run supply by re-entering vessels that are laid up. 2.3 Costs in the Shipping Industry In this section, costs in the shipping industry will shortly be described. They are divided into four categories; capital costs, operation costs, voyage costs, and cargo-handling costs. The whole section will be based on Alizadeh and Nomikos (2009). Capital costs are related to interest payments and capital repayments. These costs depend on how the shipowner or the shipping company has financed their vessel purchases, and on the interest rate level. Fleet financing can take several forms, some of which include full equity, bank loans, bonds, public offerings and private placements. Shipping companies with high operational and financial capabilities may enjoy better financial agreements than shipping companies with relatively lower levels of credit and collateral. Operating costs are costs arising from the day-to-day running of the vessel. These costs are generally in the responsibility of the shipowner, and incur whether the vessel is active or idle. Operating costs include, among others, crew wages, stores and provisions, maintenance, and insurance. Such costs do not vary over time, but they grow at a constant rate normally in line with inflation. Voyage costs are costs related to a specific voyage. These costs include fuel costs, port charges, pilotage and canal dues. The specific voyage undertaken, and the type and size of the vessel are factors that decide the level of costs. Cargo-handling costs arise from the loading, stowage, lightering and discharging of the cargo. 13

20 2.4 Business Risks in Shipping The shipping industry is considered as one of the most volatile industries where participants in the markets are exposed to substantial financial and business risks. Fluctuations in freight rates, bunker prices, vessel prices, and even from fluctuations in the level of interest rates and exchange rates are all reasons why this is an extremely risky industry (Alizadeh and Nomikos 2009). All these factors have an impact on the cash flows of shipping investment and operations, thus they also influence the profitability of shipping companies as well as their business viability. Alizadeh and Nomikos (2009) divide business risk in shipping into three categories; price risk, credit risk and pure risk. In the following, these will be described and risk stemming from fluctuations in freight rates will be relied most weight Price Risk Price risk refer to a shipping company s costs and earnings which is uncertain and outside of direct control of the shipping company. The first source of price risk is freight rate risk which refers to the variability in earnings of a shipping company due to changes in freight rates. Alizadeh and Nomikos (2009) argue that this may be the most important source of risk for a shipping company due to the direct impact volatility in freight rates have on the profitability of the company. The management of risk arising from freight rates will be described after the other types of business risk in shipping have been presented. The second source of price risk is the operating-costs risk which refer to volatility in a shipping company s costs. Sharp and unanticipated changes in, for example, bunker prices will have a major impact on the operating profitability of shipping companies and vessel operators. The third source of price risk is the risk arising from exposure to changes in interest-rates. Most vessels in the shipping industry are financed through term loans priced on a floating rate basis, thus unanticipated changes in interest rates may create cash flow and liquidity problems for companies which may no longer be able to service their debt obligations. The fourth and last source of price risk is asset-price risk arising from fluctuations in the price of the assets of the companies. In the shipping industry, vessels are the most important assets, they are often used as collateral in vessel-finance transactions and a reduction in vessel value may therefore affect the creditworthiness of a shipowner and its ability to service debt obligations. Volatility in vessel values will also affect a shipping company s balance sheet. 14

21 2.4.2 Credit Risk Credit risk refers to counter-parties to transactions and their ability to perform their financial obligations in full and on time. Therefore, credit risk is also known as counter-party risk. In the shipping industry most of the deals, trades and contracts are negotiated directly between the counterparties, their trust and commitment to honor the agreement therefore becomes extremely important Pure Risk Pure risk relates to a decrease in the value of the shipping company s assets due to physical damage, accidents and losses. Also, risk of loss due to physical risks, technical failure and human error in the operation of the assets of a company are covered. In addition, the risk of legal liability for damages as a result of actions of the company is covered Summing Up - Analyzing and Managing Freight Rate Risk For the shipowner to be able to analyze the freight rate risks which he is exposed to, it is convenient to consider the vessels as investments as assets in portfolios (Kavussanos and Visvikis 2006). Through the freight services that vessels offer to charterers a stream of income is generated and the level of this income is dependent on the freight rate level at each point in time. Also, capital gains/losses created by selling the vessels at a price higher/lower than what they were bought at is part of the shipowner s investment strategy. To manage the risks arising from freight rates Kavussanos and Visvikis (2006) point out the use of financial derivatives. They explain that financial derivatives have been used in the shipping industry since 1985, but also that the popularity of these derivatives are far less popular than those available in other sectors of the economy. The time charter contracts with purchase options, which this thesis aims to analyze and valuate, is an instrument used to protect shipowners and charterers from risk arising from fluctuations in freight rates. 15

22 2.5 The Four Shipping Markets Within the shipping industry, markets play an extremely important role in the operation of the international sea transport. Stopford (2009) mentions the nineteenth-century economist, Jevons, who provided a definition of a market where the basic principles is still very suitable to the shipping industry. The definition is quoted below. Originally a market was a public place in a town where provisions and other objects were exposed for sale; but the world has been generalized, so as to mean any body of persons who are in intimate business relations and carry on extensive transactions in any commodity. A great city may contain as many markets as there are important branches of trade, and these markets may or may not be localized. The central point of a market is the central exchange, mart or auction rooms where traders agree to meet and transact business But this distinction of locality is not necessary. The traders may be spread over a whole town, or region or country and yet make a market if they are in close communication with each other. (Jevons, 1871, Ch. IV) Within the shipping industry there exist four different markets trading in different commodities. The freight market trades in sea transport, the sale and purchase market trades second-hand vessels, the newbuilding market trades new vessels, and the demolition market deals in vessels for scrapping. Since this thesis aims to model the valuation of contracts on sea transport, it is the freight market that is considered, and thus the freight market will be relied most weight in the following. However, for completeness the main characteristics of the three other markets will also be described. The whole section is based on Stopford (2009) The Freight Market The freight market is the marketplace where sea transport is bought and sold. Within the freight market different sectors are developed in order to support the different vessel types. The freight rates within each sector often behave quite differently from each other in the short term, but since it is the same broad group of agents participating in the shipping industry, what happens in one sector eventually ripples through into the others. There exist two different types of transactions in the freight market; the freight contract and the time charter. The freight contract is a fixed type of contract where the shipper buys transport from the shipowner at a fixed price per ton of cargo, and is used by shippers who prefer to pay an agreed sum and leave the management of the transport to the shipowner. On the other hand, the time charter contract is based on the spot freight market, and 16

23 the vessel is hired by the day. Experienced vessel operators who prefer to manage the transport themselves are the users of time charter contracts. Four different contractual agreements are used in the freight market; the voyage charter, the contract of affreightment, the time charter, and the bare boat charter. A voyage charter agrees on a fixed cargo price, measured in price per ton. In a contract of affreightment, the shipowner agrees to transport a series of cargo parcels for a fixed price per ton. These series of cargo parcels are agreed to be transported within a fixed time interval, for example within two months. The details of each voyage are in the concern of the shipowner. The vessels are then used in an efficient manner by, among others, switching cargo between vessels and arrange backhaul cargoes. The time charter contract takes a step further and gives the charterer the operational control of the vessels that carry the cargo. When the charterer has the operational control he instructs the master where to go and what cargo to load and discharge. Responsibilities left for the shipowner are ownership and management of the vessel. The length of the charter can vary from the time taken to complete a single voyage, to a period of months or years. When a vessel is chartered, the shipowner continues to pay the operating costs of the vessel. Operating costs include crew, maintenance and repair. Commercial operations, voyage expenses 3 and cargo handling costs are left to the charterer. Since time charters hand over the voyage costs to the charterer it is convenient to apply the time charter equivalent spot freight rate when agreeing upon a time charter contract. By subtracting the voyage costs from the spot freight rate time charter rates will reflect the net freight earnings through shipping operations (Alizadeh and Nomikos 2009). Therefore, when valuing time charter contracts with purchase options later on, the time charter equivalent spot freight rate will be applied. A bare boat charter can be arranged if the charterer wishes to have full operational control of the vessel without owning it. The owner of the vessel does not need to be a professional shipowner, it can also be an investor buying a vessel, and then entering into a bare boat charter. The charter period usually spans from ten to twenty years. Management of the vessel and operating and voyage costs is in the charterer s responsibility. 3 Voyage expenses include bunkers, port charges and canal dues (Stopford 2009). 17

24 2.5.2 The Sale and Purchase Market In the sale and purchase market second-hand vessels are traded with high intensity. Generally, the sale and purchase transactions are carried out through shipbrokers who have been instructed by the shipowner to find a buyer for the vessel. Most commonly, competition is created by offering the vessel through several broking companies. The sale and purchase market generates price volatility, and asset play 4 can result in high profits being an important source of income for shipping investors The Newbuilding Market As its name indicates, the newbuilding market trades vessels that are not yet built. For the building process, specifications of the vessel must be decided. The shipyard generally has their own standard designs, and it is therefore desirable that the buyer choose one of those. This will ease the negotiation process, the pressure on design and estimating resources will be reduced, and the shipyards standard designs are normally cheaper to build than a customized design The Demolition Market The demolition market has similarities to the second-hand market, but now it is the scrap yards that are the customers instead of the shipowners. When a shipowner is not able to sell his vessel in the second-hand market, he offers it on the demolition market. Also here, a broker generally handles the sale. Scrap values are determined by negotiation and depend on the availability of vessels for scrap and the demand for scrap metal. With the four shipping markets described, and with the different contractual agreements in the freight market in hand, it is time to move on to the primary theme of this thesis; namely time charter contracts and their embedded options. 4 Described by Stopford (2009) as well-timed buying and selling the vessels. 18

25 2.6 Time Charter Contracts with Embedded Options This section will give an understanding of the various options that often are embedded in time charter contracts in the shipping industry. Some of them are quite complex in nature, and for valuation purposes advanced numerical methods are necessary. Therefore, this section serves as an introduction to both simple options and more complex options, whereas simple European options will be valued later on. Time charter contracts with embedded options are common in the shipping industry. When agreeing upon a time charter, options to extend the lease period and options to buy the vessel are often embedded in the lease contract. The option to extend the lease period makes it possible for the charterer to lengthen the life of the contract period (Hull 2012). The option to buy the vessel is a so called purchase option (call-option) and gives the charterer the opportunity to buy the vessel at a predetermined price. These options serve as an insurance for the charterer against undesirable movements in the freight rate level over the contract period as he can terminate the contract by purchasing the vessel. Options embedded in contracts in the shipping industry are real options. This is options on physical assets (Hull 2012), such as the vessels in the shipping industry. According to Alizadeh and Nomikos (2009) the underlying assets of real options are cash flows affected by managerial decisions. In the shipping industry, owning a vessel results in cash flows when operating it in the freight market; by offering seaborne transport the shipowner captures the freight earnings. Jørgensen and Giovanni (2010) point out that these embedded options can have more or less complex properties, in addition to being of significant economic value. This makes such contracts interesting from both academic and practical business management perspectives. Due to the economic significance of such contracts, the need for development and analysis of good valuation models is increasing. Good developed valuation models will support the stock market s valuation of shipping companies and assist managers in the general process of operation and risk management of their companies. Embedded options can have different styles, the most common types of options in the shipping industry is European options, American options and Bermudan options. European options can only be exercised at a predetermined date (the expiration date) in the future (Hull 2012). If the embedded options in a time charter contract are of European style, the lease period can only be extended at one specific predetermined date. If the option to extend is exercised, the option to purchase the vessel normally also is extended until the end of the lease period. When 19

26 the option to purchase the vessel is of European style, the vessel can only be bought at one predetermined date in the future. If the option is not exercised at that point in time, the purchase option ceases to exist. Normally, the expiration date of the purchase option is at the end of the contract period. American options can be exercised at any point in time until the end of the contract period (Hull 2012). If the embedded options in the time charter contract are of American style, the charterer can extend the lease period or buy the vessel at any point in time in the contract period. Also here, if the option to extend is exercised, the purchase option is normally also extended. If the purchase option is exercised, the time charter contract ceases to exist. Bermudan options are American options with non-standard features. The option holder have the possibility to exercise the option at several predetermined dates in the contract period (Hull 2012). If the embedded options in the time charter contract are of Bermudan style, the charterer can choose to extend the lease period or purchase the vessel at several dates in the lease period. These dates are often set to once a year. As before, if the option to extend is used, the option to purchase the vessel is normally also extended. Since options are financial derivatives, the availability of reliable price information on the underlying asset is a necessary condition. Alizadeh and Nomikos (2009) emphasizes that available price information on the underlying freight market is necessary in order to trade derivatives on freight. Further, they explain that the price information on the underlying freight market needs to be continuous, measurable and fully transparent. The Baltic Exchange is the leading provider of freight market information (Alizadeh and Nomikos 2009). When derivative transactions are priced and settled in the freight market, they generally rely on freight indices provided by the Baltic Exchange. Several different freight indices exist due to both the differences between segments in the shipping industry, but also due to different vessel types and sizes. Since this thesis focus on the dry bulk segment, the Baltic Dry Index (BDI) is the index of interest. The BDI is an index calculated as the equally weighted average of the indices related to the different vessel sizes in the dry bulk segment (Alizadeh and Nomikos 2009). This index is used as a general market indicator reflecting the movements in the dry bulk segment. In addition to being of European, American or Bermudan style, options embedded in time charter contracts are written on the underlying average spot freight rate over the defined contract period; 20

27 they are path dependent 5 and therefore of Asian style (Koekebakker, Adland et al. 2007). As mentioned before, freight rates cannot be delivered in their physical form, they represent a cost of a freight service that cannot be stored or carried forward in time (Kavussanos and Visvikis 2006). Their non-storable feature is one reason why it is convenient to write freight rate claims on an average of spot freight rates over a defined period of time. Further, Koekebakker, Adland et al. (2007) explain that a charterer operating in the spot freight market is exposed to freight rate fluctuations during some period of time. In order to capture freight rate fluctuations over a defined period of time, it is convenient to treat freight rate contingent claims as path-dependent derivatives. Another aspect worth mentioning is the freight revenue process when operating a vessel in the spot freight market. The duration of a voyage can differ from a few weeks to several months; the expected freight revenue from each voyage is given by an estimated average of the forecasted fluctuations of the freight rates over the voyage s duration. Therefore, the spot freight rate is itself implicitly average based since it refers to fluctuations over a specific time period (Koekebakker, Adland et al. 2007). 2.7 Background for the Models Selected This section will give a presentation of previous empirical findings that support the two models adopted for valuations. Different arguments are presented that together form an empirical foundation for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process for valuation purposes. Due to the similarities between the two models, the following arguments will apply to both processes. But first, clarifying arguments for the two models adopted will be presented. The main intention of this thesis is to introduce the concept of fair valuation of time charter contracts with purchase options. For this purpose, the Ornstein-Uhlenbeck process is adopted and analytic solutions for the valuations will be derived. The Ornstein-Uhlenbeck process is a one-factor model with mean reversion properties which, as will be argumented for in the following, serves as well suited when modeling freight rates. However, two models will be introduced for valuation purposes and the second one is the Geometric Mean Reversion process. As its name indicates, mean reversion is also a property of this process and one factor is present also here. The Geometric Mean Reversion process is considered as more 5 According to Hull (2012) path-dependent options are options where the payoff depends on the path followed by the price of the underlying asset, not just its final value. 21

28 realistic in freight rate modeling compared to the Ornstein-Uhlenbeck process, arguments pointing in that direction will be examined later. Nevertheless, the Ornstein-Uhlenbeck process will be applied for the valuation of time charter contracts with purchase options due to the nice analytic solutions from Jørgensen and Giovanni (2010). Due to the lack of analytic solutions from the Geometric Mean Reversion process, valuation of time charter contracts with embedded options requires numerical routines such as the Finite Difference method 6. Such routines are complex and time intensive to implement, but Monte Carlo simulation 7 can be used to value vessels and European options from the Geometric Mean Reversion process. This will be done later on as an introduction to further studies of the use of this model. Now that the model choices have been clarified it is time to move on to empirical arguments for why these models are specially suited for freight rate modeling. First, both models are driven by one factor which is assumed to be the spot freight rate. Stopford (2009) points out four factors that are influential on the vessel value; freight rates, age, inflation and shipowners expectations for the future. Further he says that freight rates are the one factor that primarily influences vessel prices. As the freight market goes up and down, so will this continue to the sale and purchase market. Also, Jørgensen and Giovanni (2010) argue that the spot freight rate is the major source of uncertainty in the shipping industry, which results in the spot freight rate representing the main source of business risk in shipping. By adopting the Ornstein-Uhlenbeck process and choose the spot freight rate as the stochastic factor evolving through time, simple and more complex freight rate contingent claims in the shipping industry can be fairly valued, in addition to valuation of the vessels involved in these contingent claims. The Ornstein-Uhlenbeck process ensures the spot freight rate to evolve through time according to a mean reverting stochastic process. Empirical arguments according to the mean reversion property will be presented below, whereas an economic reasoning for mean reversion in freight rate movements will be presented in the next section where freight rate dynamics are examined. Fair valuation of time charter contracts with purchase options include valuation of the vessels underlying the contract. Therefore, another important justification for the models adopted and the spot freight rate as the uncertain factor in the models is the evidence of high correlation between spot freight rates and vessel prices. Adland and Koekebakker (2007) introduce their research by mentioning that freight rates are relatively highly correlated with vessel values since peaks and 6 Finite Difference methods value a derivative by solving the differential equation that the derivative satisfies (Hull 2012). 7 A procedure for randomly sampling changes in market variables in order to value a derivative (Hull 2012). This procedure will be described in detail in Section

29 troughs in the freight market have a tendency to quickly work their way into the sales and purchase market. They propose that the most important factor determining the price of a vessel is the vessel s age; this is due to the depreciation of the vessel s value during its lifetime. Further, they expect the one-year time charter rate to be the second important factor to the price of a vessel. In their research, they find strong evidence of correlation between freight rates and vessel prices. Also, they find that the vessel value is an increasing function of the freight rate level (Adland and Koekebakker 2007). Another argument for adopting both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process for valuation purposes is the evidence of mean reversion in spot freight rates. In his work, Adland (2000) finds only very slight mean reversion for low and medium values of the spot freight rate. But when the freight rate increases beyond per day, the mean reversion is stronger. This is the same as saying that the drift decreases since stronger mean reversion imply stronger attraction towards the long-term mean reversion level. Further, he explains that the decline in drift at high freight rates prevents the freight rate from exploding towards infinity. The evidence of mean reversion in freight rates is also acknowledged by Koekebakker, Adland et al. (2006) who state that the freight rate is expected to be mean reverting in one sense or another. They apply basic maritime theory to show why the spot freight rate process must be mean reverting (Koekebakker, Adland et al. 2006). The same theory is also to be found in Stopford (2009). In Section 2.8 below, where the freight rate dynamics are described, the economic reasoning will be presented. To sum up, the presented arguments point in the direction of a model that takes account of one variable evolving in a stochastic manner through time (the spot freight rate), and which also have mean reversion properties. Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process have these properties, and it is therefore reasonable to assume these models to be appropriate models for the valuations. 23

30 2.8 The Dynamics of Freight Rates The following section will describe the dynamics of spot freight rates and how they evolve in a stochastic manner through time. To understand the evolvements of spot freight rates a short description of the predominant market cycles in the shipping industry is essential. Thereafter, the freight rate dynamics will be described by first introducing the elasticity of freight rates in different market conditions, and then the behavior of freight rates will be described Shipping Market Cycles The shipping industry is pervaded by market cycles. Stopford (2009) identifies three components of a typical cyclical time series. The first is the long-term cycles driven by technical, economic or regional change. A long-term cycle moving upwards is good for business, whereas a long-term cycle moving downwards is bad for business. The second is the short-term cycles characterized by short-lived movements and are important drivers of the shipping market cycle. Short-term cycles are often referred to as business cycles, they fluctuate up and down and a complete cycle can last anything from 3 to 12 years from peak to peak. The third and last component is the seasonal cycles which are regular fluctuations within the year. As for the dry bulk shipping segment, weak markets appear often during July and August due to relatively little grain being shipped. Thus, seasonal cycles occur in response to seasonal patterns of demand for sea transport. Now that the shipping market cycles are shortly described it is time to move on to the dynamics of freight rates, which in fact is the factor that fluctuates according to the market cycles Freight Rate Dynamics According to Alizadeh and Nomikos (2009), the balance between supply and demand for shipping services is at any point in time reflected by the spot freight rates. The crucial factors for demand and supply are the ones listed in Table 1. When the shipping market is in recession and spot freight rates are at very low levels, an overcapacity is developed and many vessels cannot find employment, are laid up, slow steam or even carry part cargo. When the market is in such a recession, changes in demand due to external factors 8 can be absorbed by the extra available capacity. Thus, the impact on spot freight rates would be relatively small and they are considered as being inelastic to changes in demand. Gradually, market conditions will improve and freight rates will increase. Available vessels 8 Nomikos and Alizadeh (2009) propose seasonal changes in trade and random shocks (events) as external factors. 24

31 will be employed again until the world fleet is fully utilized and any increase in supply is only possible by increasing productivity through increasing speed and shortening port stays and ballast legs. When market conditions are good and spot freight rates are at high levels, any changes in demand due to external factors would create a relatively large movement in spot freight rates; spot freight rates are considered elastic in relation to changes in demand. Good market conditions imply high spot freight rates. Tvedt (1997) explains that very high spot freight rates may appear in shorter periods of time due to the demand for seaborne transport being inelastic to changes in spot freight rates, and that there is a short-term upper limit to supply. Further, he explains that good market conditions with high levels of spot freight rates will not be a persistent situation over time. The potential for supply adjustment, both newbuilding and demolition, will guarantee that very high or very low spot freight rates will not be a persistent situation over time (Adland and Cullinane 2006). High spot freight rates will tempt shipowners to order new vessels in order to capture high freight earnings. Due to the time lag from ordering a new vessel until delivery, a gambling position is created for the shipowner. Often, the market clears at a rate that is not high enough to cover the investment costs of a new vessel. If the shipowner manages to order a new vessel in time such that the rates are still high when the vessel is delivered, he may catch a high reward. But ordering a vessel when the spot freight rates are high is often too late because the rates will probably be back to normal low levels when the vessel is ready to be delivered. Summing up, the shipping market cycles and the continuous stochastic evolution of spot freight rates go hand in hand; spot freight rates will evolve in accordance with the current state of the shipping market and its cycles. This serves as an economic reasoning for the mean reverting behavior of spot freight rates which reinforces the empirical findings described in the Section 2.7 above. 2.9 The Ornstein-Uhlenbeck Process In the following, the model adopted to value time charter contracts with purchase options, namely the Ornstein-Uhlenbeck process, will be described in detail, along with the derivation of the solution to the process. The Ornstein-Uhlenbeck process has been applied to the shipping industry by, among others, Bjerksund and Ekern (1995), Tvedt (1997) and Jørgensen and Giovanni (2010). The basic underlying assumption is that the stochastic component of the instantaneous cash flow from owning a vessel, the spot freight rate, is characterized by an Ornstein-Uhlenbeck process (Bjerksund and Ekern 1995). When the spot freight rate is said to be stochastic, its value will change over time in an uncertain way (Hull 2012). Spot freight rates evolve in a stochastic manner and their 25

32 values can change at any point in time, they are therefore defined to be continuous-time stochastic variables. When spot freight rates are assumed to follow an Ornstein-Uhlenbeck process, they are ensured to behave like continuous-time stochastic variables due to the special feature of the model being an Itô process. An Itô process consist of two terms; a drift term which is a function of the value of the underlying variable 9 and time, and a variance term which also is a function of the value of the underlying variable and time. The variance term contains a standard Wiener process which is a particular type of a stochastic process with mean of zero and variance of one per year, it is denoted (Hull 2012). Both the drift term and the variance term are liable to change over time. Bjerksund and Ekern (1995) assume that the instantaneous cash flow generated by an operating vessel,, may be described as follows: (1) where represents the size of the cargo, represents the operating cost-flow rate, and represent the uncertain spot freight rate (annualized) at time per unit of cargo. For ease of notification, it is assumed that the spot freight rate is prevailing for the vessel as a whole and net of all costs. Thus, and (Jørgensen and Giovanni 2010), and is left as the instantaneous cash flow from operating a vessel. As described in Section where time charter equivalent spot freight rates are introduced, this assumption causes lack of the ability for direct market comparisons as marketquoted freight rates embed varying degree of costs. 9 The freight rate in this case. 26

33 As argumented for in Section 2.7,where the background for the model selected is examined, and in Section 2.8, where the dynamics of freight rates are examined, in addition to following Bjerksund and Ekern (1995), it is assumed that the spot freight rate can be modeled by the following stochastic differential equation: (2) where is the speed of mean reversion, is the constant long-term mean, is the instantaneous volatility of spot freight rates, and is a standard Wiener process defined on some probability space (Jørgensen and Giovanni 2010). The model is an Itô process where the first term is considered as the drift term, whereas the last term is considered as the variability term. This process is identical to the one Vasicek (1977) proposed for modeling interest rate dynamics. In line with Bjerksund and Ekern (1995), technical descriptions of the model, the drift term, ensures that the process always are pushed back to its long-term mean. When, the drift term is negative and the process will be pushed up to its long-term mean. On the other hand, when, the drift term is positive and the process will be pushed down to its long-term mean. Further, they explain that higher values of will create a stronger tendency of the stochastic process to move back towards its long-term mean. Thus, the higher the, the higher the degree of mean reversion. Further, they explain that the second term characterizes the volatility of the process with an increment of a standard Wiener process with characteristics as explained above. being Important to remember is that the model measures time in years, whereas actual spot freight rates are normally quoted on a daily basis. This is important to have in mind when comparing with market data where have to be considered instead of. is one day measured in years, where one year is assumed to contain 360 days (Jørgensen and Giovanni 2010). Due to the properties of the Wiener process being standard normal distributed, Equation (2) implies that future spot freight rates are normally distributed (Jørgensen and Giovanni 2010). The density function 10 will therefore have the classic bell shaped curve. In order to derive the mean and the variance of the distribution of future spot rates the solution to Equation (2) have to be derived, this will be done in the following section. 10 Plots the shape of the distribution curve of the random variable that is considered (Skovmand 2012), which in this case is the spot freight rate. 27

34 2.9.1 The Solution to the Ornstein-Uhlenbeck Process The solution to the process can be derived from the stochastic differential equation in Equation (2). This is desirable both in order to calculate the freight rate at time, and in order to derive the mean and the variance of the distribution of future spot freight rates. From the mean of the Ornstein-Uhlenbeck process the expected return from operating a vessel in the spot freight market over a defined period, can be calculated. The variance of the Ornstein-Uhlenbeck process calculates the volatility in spot freight rates over the same period. Both measures are useful for agents in the shipping industry as they give an indication of how the market will develop in the future. First, consider how to obtain the freight rate at time which is given by today s freight rate plus the sum of the dynamics of the spot freight rate evolving from time to time : (3) Equation (3) has to be solved explicitly in order to ensure the availability of analytic solutions. In order to do this, a temporary variable have to be introduced (Hammer, Hafsaas et al. 2011): (4) It is desirable to obtain the dynamics of this temporary variable since the dynamics of the spot freight rate are considered in Equation (2). The dynamics of Equation (4) are given by Ito s lemma 11 : (5) (6) 11 Describe the behavior of functions of stochastic variables. Such a function can be the price of a derivative which is dependent on the underlying stochastic variable and time (Hull 2012). 28

35 Now, the next step is to insert Equation (2) for in the equation above: (7) Manipulations 12 of Equation (7) give: (8) Continuing by integrating from time zero to time : (9) Calculating : Continuing with Equation (9): (10) 12 A detailed derivation is attached in the Appendix. 29

36 (11) (12) Calculating : Finally, the solution to Equation (2) becomes: (13) From Equation (13) the time conditional mean and variance of the normal-distributed future spot freight rate can be stated as follows: (14) (15) where the detailed derivations of Equations (14) and (15) are attached to the Appendix. 30

37 2.10 The Geometric Mean Reversion Process The Geometric Mean Reversion process will now be presented and the solution will be derived. As mentioned earlier, this process is more realistic in modeling freight rates, but to the author s knowledge analytic solutions do not exist. Therefore, when calculating vessel values and European option values numerical procedures need to be implemented. Those will be explained in more detail below. Following Tvedt (1997), the increment of the process is given by the following stochastic differential equation: (16) where is the speed of mean reversion, is the long-term mean reversion level, is the (annualized) spot freight rate at time, is the instantaneous volatility of spot freight rates, and is the increment of a standard Wiener process. As Tvedt (1997) describes, the Geometric Mean Reversion process is mean reverting and is also downwards restricted since it is not possible to take the natural logarithm of spot freight rates that equals zero or are negative - zero is an absorbing level. Another important feature of the Geometric Mean Reversion process is that it ensures the volatility to be progressively increasing in the freight rate level. This occurs in the last term of Equation (16) where the volatility parameter is multiplied by the time spot freight rate level. In his research, Adland (2000) finds evidence that the volatility is increasing in the freight rate level. More specific, he finds that the diffusion function,, is close to linear for low and medium freight rates, while it is increasing progressively for very high freight rates (Adland 2000). Assuming that the spot freight rate follows the Geometric Mean Reversion process is thus appropriate in relation to his findings. In the following, the derivation of the solution to the Geometric Mean Reversion process in Equation (16) will be done, whereas the time conditional mean and variance only will be presented. The same applications for the solution, the mean and the variance as for the Ornstein-Uhlenbeck process are applicable also here. 31

38 The Solution to the Geometric Mean Reversion Process The solution to Equation (16) is derived following Tvedt (1997). In order to simplify later calculations he suggests starting by dividing the whole process by and multiplying it with the integrating factor : (17) Further, he continues by defining a function this temporary function are desirable. They are given by Ito s lemma:. Again, the increments of (18) (19) Rearranging: (20) In order to proceed towards the solution two things have to be done. First, it is desirable to substitute the Geometric Mean Reversion process for. In order to ease the notification, and are defined such that. This gives: 32

39 where the first and the second term tend to zero, and where the last term tends to. such that Then, by rearranging Equation (17) to and substituting the process for Equation (20) becomes: (21) The next step is to rearrange this equation and integrate it from time zero to time : (22) (23) Finally, the solution is obtained by rearranging Equation (23) such that is alone, the rearrangement step by step is attached to the Appendix. The freight rate level at time,, can be expressed as: (24) with mean and variance: (25) 33

40 (26) where: 34

41 3 Analysis Section This section starts by an examination of the differences between the two considered models; the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process. Evidences of the Geometric Mean Reversion process being more appropriate in freight rate modeling will be presented and discussed. Further, the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process will be examined separately. When investigating the Ornstein-Uhlenbeck process, valuation of simple freight rate contingent claims will be introduced and for that purpose the partial differential equation will be established. Analytic solutions will be derived before they are used to the valuation of simple freight rate contingent claims. Finally, European option values to buy the vessel will be calculated, which in turn is used to obtain the total value of time charter contracts with purchase options. Next, the Geometric Mean Reversion process will be investigated and the purpose is to obtain vessel values and European option values to buy the vessel by Monte Carlo simulation. Finally, the section ends by a comparison of vessel values and option values obtained by these two models. As for the calculations throughout this section base case parameter values will be used. They are adopted from Tvedt (1997) where he has estimated parameter values in relation to both processes. The base case parameter values are presented in Table 2 below. However, when tables are presented for different volatilities and different spot freight rates, those specific varying parameter values will be adopted from Jørgensen and Giovanni (2010). Parameter Ornstein-Uhlenbeck Process Geometric Mean Reversion Process , ,1184 0, ,0033 1,5% 1,5% Table 2. Base case parameter values for both processes. 35

42 Freight Rate Value 3.1 Two Models Different Characteristics: A Comparison This section will give an examination of the differences between the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process. At a first glance they look quite similar, and they do in fact have some properties in common. But due to some important dissimilarities they behave quite differently from each other. In real life, negative values of the spot freight rate are impossible. This is due to the shipowner s option to lay up his vessel if the operational costs are not covered. In fact, due to the option to lay up, the spot freight rate will generally never fall below the given lay up level (Tvedt 1997). The Ornstein-Uhlenbeck process fails to take account of the impossibility of spot freight rates becoming negative since the process is not downwards restricted. Large volatility will often give negative spot freight rates because the process is normally distributed around a given mean. This is demonstrated in Figure 1 below where the process is simulated over a period of almost six years with time steps equal to one observation of the spot freight rate each day. The base case parameter values in Table 2 are applied and presented in the figure description. However, the volatility is adopted from Jørgensen and Giovanni (2010) in order to capture the high volatility effect. From Figure 1 it is clearly that the spot freight rates over the horizon can take on both negative and positive values. This is somewhat unrealistic compared to a real life situation ,00 Simulated Freigt Rate Following an Ornstein-Uhlenbeck Process , , ,00 Freight Rate , ,00 Time Figure 1. Simulated spot freight rate from the Ornstein-Uhlenbeck process. =0,00247, =14 371, =14 500, =1/360 and =

43 Freight Rate Value In relation to negative freight rates, what actually can happen in reality during short intervals is that the spot freight rates may become so low that the estimated time charter equivalent spot rate will become negative. This happens when the voyage income is less than the total of fuel consumption and harbor and channel costs, but in these cases the shipowner will probably rather lay up his vessel than keeping it in operation. Therefore, the market almost always clears at a positive time charter equivalent spot freight rate and the Geometric Mean Reversion process, which is downwards restricted, may give a more appropriate and realistic description of the spot freight rate. As Figure 2 below clearly indicate, the simulated Geometric Mean Reversion process only give positive values of the spot freight rate and is therefore also more realistic in relation to a real life situation. Again, the base case parameter values from Table 2 are applied and given in the figure description. However, in order to capture the high volatility effect the volatility is set equal to one Simulated Freight Rate Following a Geometric Mean Reversion Process Freight Rate Time Figure 2. Simulated spot freight rate from the Geometric Mean Reversion Process. =0,0033, =10,45, =14 500, =1/360 and =1, Inspired by Tvedt (1997) where he at one point has set the volatility parameter equal to 0,93. 37

44 3.2 Applications of the Ornstein-Uhlenbeck Process: Introducing Valuation of Freight Rate Contingent Claims In the following, some simple freight rate contingent claims whose value,, depend only on the spot freight rate factor process,, and time will be valued. Before this can be done, the fundamental partial differential equation that must satisfy in order to give the correct value of such contingent claims will be established. According to Hull (2012), a key property of the fundamental partial differential equation is that no variables affected by the risk preferences of investors are involved. This opens for risk-neutral valuation of claims dependent on only the spot freight rate factor process and time; the risk-neutral valuation result 14 can be applied. When no risk preferences enter the partial differential equation a simple assumption that all investors are riskneutral can be made. Thus, the risk-free rate of interest,, is assumed to be the expected rate of return on all investments. Also, over the contract period the risk-free rate of interest is assumed to be constant. The assumption of a risk-neutral world does therefore simplify the computations of the claims which are dependent on only the spot freight rate factor process and time. The spot freight rate essential in the partial differential equation is representing the price of a service, and is therefore not a physical asset. In order to price freight rate contingent claims by the partial differential equation, an assumption of the existence of a physical asset whose price is perfectly correlated with the spot freight rate need to be done (Jørgensen and Giovanni 2010). They explain that this is necessary to the application of standard no arbitrage arguments (Hull 2012). Further they explain that the vessels underlying the option contracts could be used, but hedging strategies based on trading vessels are very unpractical. Instead, they rely on the assumption of the existence of a liquid market for the relevant futures. Such markets do in fact exist in the real world; the Oslo-based International Maritime Exchange (IMAREX) 15 is an example. When both the assumption of a risk-neutral world and the assumption of the existence of a liquid market for the relevant futures are done, Ito s lemma and a risk-neutralizing hedge argument 16 can be used to derive the fundamental partial differential equation that give the correct value of the option contract. must satisfy in order to 14 The present value of any cash flow in a risk-neutral world can be obtained by discounting its expected value at the risk-free rate (Hull 2012). The mathematical expression will be presented later. 15 Writes freight rate derivatives upon the freight indices obtained from the Baltic Exchange and from Platts. 16 Transformation of the true probability measure to the risk-neutral probability measure which will be derived later. 38

45 3.2.1 Derivation of the Fundamental Partial Differential Equation From Equation (2), the single factor spot freight rate process is given as: where is the increment of a standard Wiener process under the true probability measure,. It is argumented that derivatives prices,, influenced by the factor process,, and time must have dynamics described by (Vasicek 1977). (27) where: and where is the market price of freight rate risk. In order to discount by the risk-free interest rate,, and thus apply the risk-neutral valuation result, the true probability measure have to be changed to the risk-free probability measure,. In short, the risk-neutral valuation principle states that a derivative can be valued by (a) calculating the expected payoff on the assumption that the expected return from the underlying asset equals the risk-free interest rate and (b) discounting the expected payoff at the risk-free rate (Hull, 2012, page 630).The transformation from the true probability measure to the risk-free probability measure can be done by applying Girsanov s Theorem. This procedure will be done in the following. Define: (28) 39

46 where is now the increment of a standard Wiener process under the risk-free probability measure,. is again the market price of freight rate risk. Then: (29) (30) (31) In the following, an assumption of a constant market price of freight rate risk is done such that reduces to. Therefore, the factor process under is given as: (32) (33) (34) (35) 40

47 Define. Then: (36) This is the dynamics of under the risk neutral probability measure,. The mean and the variance are given as in Equations (14) and (15). The only difference is that is replaced by. Finally, Ito s lemma can be used to characterize the dynamics of the -process under : (37) The next step is to substitute the Itô process for and in Equation (37). The Itô process under the risk-neutral probability measure is given in Equation (36): To ease the calculations, and are defined such that. Calculating : 41

48 In the limit, tends to zero, tends to, and tends to (Campbell, Lo et al. 1997). Hence, the first and the second term tend to zero, and the last term tends to. This reduces equation to. Inserting both and into Equation (37): (38) This is equal to: (39) (40) (41) 42

49 According to Jørgensen and Giovanni (2010), absence of arbitrage requires that: (42) (43) where is the dividend rate received by the claim. By introducing as the absolute cash flow from the claim the following partial differential equation is the one satisfy: must (44) Or, equivalently: (45) This is the partial differential equation that all claims depending on the spot freight rate process,, and time need to satisfy in order to be priced arbitrage free. Solving for will give the price of a particular freight rate contingent claim. Another way of solving Equation (45) is to manipulate the probabilistic Feynman-Kac representation of the solution to the partial differential equation. A partial differential equation can then be solved by an expectation of the discounted payoff of the derivative,, modified replacing the true probability measure,, by a risk-free probability measure, (Duffie 2001). By doing this, the risk-free interest rate,, can be used as the expected rate of return when discounting backwards in time. The probabilistic Feynman-Kac representation of the solution to the partial differential equation takes the following form (Jørgensen and Giovanni 2010): 43

50 (46) where would typically be the expiration date at which the value of the claim would be given as a known function of. Equation (46) is quite intuitive, it indicates that the time value of the claim is the expected value of the continuous flow of net dividends plus the value of the payoff at the maturity date, both discounted back to time. Finally, when the partial differential equation is established, with its solution, the valuation of some simple freight rate contingent claims can be done. 3.3 Valuation of some simple Freight Rate Contingent Claims The following section will present derivations of analytic solutions for evaluating simple freight rate contingent claims, as well as valuation results for each simple contingent claim Claim to Receive Spot Freight Rate Flow from Time to Time The first claim that will be valued is the claim to receive the spot freight rate on a continuous basis from time, when the current spot freight rate is, to time. To do this, the probabilistic Feynman- Kac representation from Equation (46) is used. In the following, the main steps of how Equation (46) is used to derive an analytic solution of the value of such a claim are shown, whereas a detailed step by step representation is attached to the Appendix. Defining the current value of receiving the spot freight rate on a continuous basis as: (47) 44

51 The freight rate process, : (48) Inserting in Equation (47) gives: ( ) (47) Since the expectation of the increment of a standard Wiener process equals zero (Hull 2012), the last term becomes equal to zero and only the two first terms are left. Further: (49) (50) (51) (52) where is an annuity factor of the present value using the discount rate of receiving a unitized continuous cash flow for years (Jørgensen and Giovanni 2010). 45

52 3.3.2 Fixed for Floating Freight Rate Swap When a shipowner and a charterer agrees on a time charter contract, the shipowner receives a fixed daily freight rate over the lifetime of the contract, and leaves the operation of the vessel in the charterers responsibility. If the shipowner had not chartered his vessel, he could have operated it in the spot freight market himself and received the floating spot freight rate over the same period of time. Therefore, when a shipowner enters into a time charter contract, he agrees to receive a fixed daily freight rate in exchange for a floating one. This is the same mechanism as an interest rate swap; an exchange of a fixed rate of interest on a certain notional principal for a floating rate of interest on the same notional principal (Hull 2012). It is in the shipowner s interest to determine the constant freight rate,, fixed at time that will be equivalent to receiving the variable spot freight rate over the period from time to time. In this way, the value of the freight rate swap will be equal to zero at initiation. This is also a property equal to an interest rate swap agreement (Hull 2012). To determine the fixed freight rate that will make the shipowner indifferent in his choice of whether to operate the vessel on his own, or charter it, Jørgensen and Giovanni (2010) apply Equation (52) and solve for : (53) Equation (53) indicate that the present value of the fixed freight rate have to be equal to the present value of receiving the spot freight rate on a continuous basis from time to. When ensuring this equality, the shipowner will be indifferent from receiving the fixed freight rate over the horizon, or receiving the floating spot freight rate over the same horizon. Solving for : (54) where the contract. will be the fair continuously paid time charter rate (fixed freight rate) during the life of 46

53 In the following, some fair valued time charter rates for different values of the speed of mean reversion and current spot freight rates will be presented. The calculation of these fair valued time charter rates is done by applying Equation (54) above. The base case parameter values from Table 2 are applied and given in the table description. As for this example, both the spot freight rates and the fair time charter rates are presented in daily values. It is worth mentioning that when the analytic solutions are used for valuations the parameter values have to be scaled up into yearly values so that the time units are proportionate to each other. This is the case for all remaining calculations during the next sections. Spot Freight Rate, , , , , , , , Table 3. The dependence of fair time charter rates (daily) on speed of mean reversion, spot freight rate,. =1,5% and = (daily). and current The results in Table 3 indicate that when the spot freight rate is per day, which is close to the long-term mean of per day, the fair time charter rate will also lie close to the long-term mean for all values of the speed of mean-reversion parameter. This is in line with intuition; when the spot freight rate is close to the long-term mean it is not expected to increase or decrease wildly the next five years, thus also the fair time charter rates should lay close to the spot freight rate. The story is different when the current spot freight rate differs from the long-term mean. For current spot freight rates that are below the long-term mean, the fair time charter rates are far from the long-term mean when the speed of mean reversion-parameter takes on low values, but are closer to the long-term mean when the speed of mean reversion-parameter takes on high values. This makes sense since when the speed of mean reversion is high, it is expected that the spot freight rate will 47

54 quickly return to the long-term mean. Therefore, the fair time charter rate should be close to the long-term mean. Opposite, when the speed of mean reversion is low, it is expected that the spot freight rate will return slowly to its long-term mean. The fair time charter rate should in that case be set closer to the current spot freight rate to secure as low difference between the two rates as possible during the contract period. For current spot freight rates lying above the long-term mean the opposite is present; when the speed of mean reversion is high the fair time charter rate is closer to the long-term mean than is the case when the speed of mean-reversion is low. Also this makes perfectly sense; low speed of mean reversion creates expectations of the spot rate to slowly return to the long-term mean. And again, high speed of mean reversion creates expectations of the spot freight rate to quickly return to the long-term mean, implying a fair time charter rate closer to the long term mean to reduce the possibility of large differences between these two rates. At the time a time charter contract is agreed on and the fair time charter rate is fixed, the value of the swap is, as mentioned before, equal to zero. However, Equation (47) clearly indicates that the value of receiving the current spot freight rate on a continuous basis during the contract period is based on an expectation. This implies uncertainty that the spot freight rate actually will evolve as expected. If the spot freight rate moves in other directions than expected, the fair time charter rate will also be different than the one that is fixed in the contract. This difference between the new fair time charter rate and the one that was set at initiation gives the swap contract a value - positive or negative. It is possible to calculate the value of a time charter contract entered into at an earlier point in time. Jørgensen and Giovanni (2010) present this valuation formula for a contract that receives the floating spot rate and pays a fixed time charter rate: (55) where time, is the prevailing fair time charter rate and is the contracted fixed time charter rate. This equation indicate that if the prevailing fair rate is higher than the contracted fixed time charter rate the swap has a positive value, and otherwise if the contracted fixed time charter rate is higher than the prevailing fair time charter rate. The annuity factor,, ensures that the value of the swap equals the discounted value of the continuous flow of the spread difference. 48

55 Valuation of a swap for different spot freight rates, fixed contracted time charter rates, and fair time charter rates will be done next. The base case parameter values from Table 2 are applied and presented in the table description. Table 4 below show the value of a 5-year time charter contract which is dependent on the current fair time charter rate, and the previously contracted time charter rate. In the table, spot freight rates, current fair time charter rates and previously contracted rates are presented in daily values, whereas the values for the 5-year contract are yearly 17 values. The Value of a 5-Year Time Charter Contract Spot and Current Time Charter Rate, Previously Contracted Rate Table 4. The value of a 5-year time charter contract. r = 1,5%, = (daily), = 0, As expected, when the previously contracted time charter rate differ from the current fair time charter rate, the swap contract does either have a positive value or a negative value. A current fair time charter rate higher than the previously contracted time charter rate gives the swap contract a positive value. Otherwise, a current fair time charter rate lower than the previously contracted time charter rate results in a negative value of the swap contract. Also, when the difference between the two rates is large, the value of the swap is also large - positive or negative The Value of a Vessel According to Jørgensen and Giovanni (2010), a vessel is a physical asset that earns rents to its owner through the flow of net spot freight rates during the vessel s lifetime. Therefore, they recommend using Equation (47) to value a vessel. But since the vessel will have a scrap value 18 when it reaches the end of its useful economic life, an extension of this formula has to be applied. Assuming that the 17 One year is assumed to contain 360 days. 18 When the vessel is scrapped the remaining steel can be sold to the steel industry (Stopford 2009). 49

56 final service date,, of a vessel as well as its scrap value,, are known with certainty, the vessel value can be calculated using this extension of Equation (52): (56) Equation (56) signify that the valuation formula for a vessel follows a Gaussian process with deterministically time-varying drift and diffusion coefficients (Jørgensen and Giovanni 2010). Therefore, future vessel values at any time are normally distributed under both the true probability measure and the risk-neutral probability measure. The time conditional mean and variance under both probability measures are given below (Jørgensen and Giovanni 2010): (57) (58) (59) In Table 5 below, vessel values as a function of different spot freight rates and varying remaining vessel lifetimes is calculated using Equation (56). The parameter values are again the ones from Table 2 and are given in the table description. The spot freight rates are given in daily values whereas the vessel values are given in yearly values. 50

57 Remaining Vessel Life ( ) Table 5. The dependence of vessel values on remaining vessel value ( ( ). =0,00247, = per day, =1,5% and = ) and the spot freight rate In line with intuition, Table 5 clearly indicates that vessel values are an increasing function of both the spot freight rate and the remaining vessel life. Higher spot freight rate gives higher cash inflow to the shipowner and therefore also the vessel value is increased. Longer remaining lifetime of a vessel also imply higher vessel value compared to a vessel with shorter remaining lifetime. In Figure 3 below, Equation (56) is used to calculate the evolution of a vessel s value over a lifetime of 25 years. This is the jagged line where the simulated vessel value is calculated over its lifetime. The expected vessel value is also calculated under the risk-neutral probability measure using Equation (57) above. For this purpose, all parameter values are adopted from Jørgensen and Giovanni (2010) and are presented in the figure description. In line with intuition, we can see that the value of a vessel is expected to decrease as it ages. 51

58 Value Simulated and Expected Vessel Value Simulated Vessel Value Expected Vessel Value Time Figure 3. Simulated and expected value of a vessel. =0,25, =0,05, =25, = , =20 000, σ=5 000 and Δt=1/ European Option to Buy a Vessel As described earlier, a European option to buy a vessel gives the charterer the right, but not the obligation, to buy the vessel at expiration. If the option is not exercised, the contract with the purchase option will expire and the vessel is handed over to the shipowner. Purchase options in shipping contracts are common practice, but simple European options are generally not used. As mentioned earlier, complex options happen to be used more often in shipping contracts. Although European purchase options are somewhat unrealistic in relation to real life, such options will be valued in order to demonstrate the interpretations of the nice analytic solutions derived through the sections above. Following Jørgensen and Giovanni (2010), the current date is denoted by and the expiration date of a European option to buy a vessel is denoted by. Also, the vessel must be scrapped at date for a value of. Further, will denote the exercise price 19 of the purchase option. The payoff function at expiry is then given as: (60) 19 The price at which the vessel may be bought at in the time charter contract (Hull 2012). 52

59 Due to the analytic solution for the vessel value in Equation (56), an analytic solution for the time value of this European call option can be derived (Jørgensen and Giovanni 2010). In the following, only the results will be presented, whereas a detailed step by step derivation will be attached to the Appendix. Thus, from the Appendix the time value of the European call option in Equation (60) is given as: (61) where: (62) (63) (64) (65) and where and denote the standard normal cumulative probability and density functions, respectively. Now, when the framework for valuing a European option to buy a vessel is established, purchase options for varying spot freight rates and freight rate volatilities will be valued using Equation (61). The parameter values are again taken from Table 2 and are given in the table description. Also, the spot freight rates and the freight rate volatilities are daily values, whereas the option values are annualized values. 53

60 Freight Rate Volatility, Spot Freight Rate, Table 6. Value of European option to buy a vessel. =1,5%, =0,00247, = , =0, =5, =25 and = = (daily), The option values in Table 6 confirm well-known option theory; as the freight rate volatility increase, so does the option value. Higher volatility imply higher probability of upside gains, thus the option value also increases. In addition, increased spot freight rate will also increase the option value since the cash flow from owning the vessel is increased. Since the option is European, the total value of the time charter contract with a purchase option can safely be decomposed into its leasing contract component and its option contract component (Jørgensen and Giovanni 2010). The total value of the 5-year time charter contract with an embedded European purchase option can be found by adding the value of the 5-year contract to the value of the embedded purchase option. The value of the 5-year contract is calculated by using Equation (55). Table 7 below show total fair contract value for varying spot freight rates and fixed time charter rates. Again, the base case parameter values from Table 2 are applied and given in the table description. 54

61 Fixed Time Charter Rate Spot Freight Rate, Table 7. The value of a 5-year time charter contract with European purchase option. r=1,5%, =0,00247, = (daily), =5 000 (daily), = , =0, =5, =25, = Table 7 indicates that the 5-year time charter contract with an embedded European option to buy the vessel can take on a negative value. Generally, this is the case when the spot freight rate is sufficiently low compared to the fixed time charter rate. This is a logical result since the spot freight rate represents the cash inflow to the charterer, and the fixed time charter rate has to be paid; cash inflow that is lower than the cash outflow creates a loss to the charterer, and the contract value therefore becomes negative and unfavorable. This is also the reason why the contract value decreases as the fixed time charter rate increases; higher rate which have to be paid will reduce the contract value. 3.5 Applications of the Geometric Mean Reversion Process: Vessel and European Option Valuation As described in Section 2.6 where time charter contracts with embedded options were examined, the options embedded in time charter contracts are path-dependent which imply that the payoff depends on the path followed by the price of the underlying asset, not just its final value. In this case, the payoff will depend on the path in which the freight rate follows over the contract period; the freight rate represents the price of operating a vessel in the spot freight market. To the author s knowledge there exist no analytic solutions to the Geometric Mean Reversion process. Thus, in order to value both the vessel and the option to buy the vessel, numerical procedures must be applied. For this purpose Monte Carlo simulation is implemented. 55

62 3.5.1 Monte Carlo Simulation Monte Carlo simulation is a numerical procedure which is applied when analytic results do not exist (Hull 2012). Especially, when the derivatives are path-dependent, as is the case for the options embedded in the time charter contracts, Monte Carlo simulation is a highly popular tool. The idea underlying Monte Carlo simulation is the feature of random sampling. When Monte Carlo simulation is applied to the valuation of an option the risk-neutral valuation result is used; several paths are sampled to obtain the expected payoff in a risk-neutral world and then the average payoff is discounted at the risk-free rate in order to achieve the option value. An assumption of a constant risk-free interest rate is done when applying Monte Carlo simulation, which fit perfectly in this case since the risk-free interest rate was assumed to be constant in Section 3.2 where valuation of simple freight rate contingent claims was introduced. The main drawback with Monte Carlo simulation is that it is extremely time consuming in the achievement of the required level of accuracy. This will be exemplified later The Value of a Vessel This section will introduce the use of Monte Carlo simulation when valuing a vessel applying the Geometric Mean Reversion process. The model adopted to the valuation is obtained from Tvedt (1997). However, some simplifying assumptions will be done for consistency compared to the ones done when investigating the Ornstein-Uhlenbeck process. The simplifying assumptions done will be explained when proceeding. According to Tvedt (1997) the instantaneous cash flow from operation and lay up until the vessel is scrapped, is given by: (66) where is the time charter equivalent spot freight rate, is the operation costs except for voyage related costs 20, is the costs of keeping the vessel mothballed and is an indicator function of the event, where. Keeping the vessel in operation is the optimal policy for the shipowner when the spot freight rate plus lay up costs are above the operation costs. When this is the case, is equal to one and. Otherwise, is equal to zero and. Again, for ease of notation it is assumed that the spot freight rate is quoted for the entire vessel and net of all 20 Those are subtracted from the spot freight rate in order to obtain the time charter equivalent spot freight rate. 56

63 costs. This results in the spot freight rate being the instantaneous net profits from an operating vessel: (67) Further, Tvedt (1997) explains that when the vessel reaches its maximum age at time, its value must be equal to the value of the vessel as scrap,. But if the value of the vessel as a going concern is less than the demolition value, the vessel may be scrapped before the estimated end of its lifetime. Formally, the termination date is equal to the stopping time given by: (68) where is as defined below. Equation (68) indicates that the termination date is reached the first time that the value of the vessel as a going concern is equal to or below the scrap value. Finally, the value of a vessel at time can be presented. It is represented by the market value of the cash flow generated from time to, and is given by: (69) where is the risk-neutral probability measure. Equation (69) indicates that the vessel value equal the discounted sum of the spot freight rates from time to, plus the discounted scrap value. Again, to the author s knowledge, no analytic solutions exist to Equation (69). To calculate the vessel value Monte Carlo simulation is therefore implemented in a Visual Basic for Applications (VBA) function, the codes will be attached in the Appendix. In the following, the procedure will shortly be explained. 57

64 First, vessel values for several points in time have to be calculated. Random numbers are therefore generated from the standard normal distribution for the increment of the standard Wiener process,, in the Geometric Mean Reversion process. The Geometric Mean Reversion process is then applied to obtain thousand different values for the freight rate evolving from time to time by Monte Carlo simulation. Each simulated freight rate is in turn used in Equation (68) to calculate several different vessel values in time. Finally, to obtain the vessel value in time, the vessel values are averaged over the thousand simulated paths. The VBA code will be attached to the Appendix. Vessel values as a function of different spot freight rates and varying remaining vessel life are presented in the table below. Also here, the base case parameter values given in Table 2 are applied and presented in the table description below. The number of yearly subdivisions 21 is set to 200, and the number of simulated paths is set to Remaining Vessel Life (T-t) X(t) Table 8. The dependence of vessel values for different freight rates and different remaining vessel lifetimes. =0,0033, =10,45, =1,5%, =0,1184 and = Table 8 clearly indicates that vessel values are an increasing function of both the spot freight rate and the remaining vessel life. As is in line with intuition, the longer the remaining vessel lifetime, the higher the vessel value. Also, higher spot freight rates indicate higher vessel value which is also in line 21 One year is divided into 200 equally time steps. 58

65 with intuition; higher spot freight rates generate higher cash inflow from owning a vessel, which in turn will increase the value of owning this vessel European Option to Buy a Vessel When an analytic solution does not exist, a concept called Monte Carlo on Monte Carlo simulation can be used to calculate the value of a European option to buy a vessel. This is done by a new simulation of freight rates using the Geometric Mean Reversion process. For each simulated spot freight rate from time to time vessel values are calculated using Equation (69) which is done by the same procedure as the calculation of vessel values above. In time, the payoff function is given by: (70) where is the exercise price of the purchase option. For each vessel value in time the payoff function is calculated. To find the option value in time, the discounted average of the payoff functions obtained in time is calculated using the risk-neutral valuation result: (71) where is the risk-free interest rate which is assumed to be constant over the defined period. Also this is done in VBA and the codes will be attached in the Appendix. Option values for different spot freight rates and different volatilities are presented in Table 9 below. The number of yearly subdivisions is again sat to 200, and the number of simulated paths has been sat to 500. The parameter values are again obtained from Table 2 and are presented in the table description. The exercise price is obtained from Jørgensen and Giovanni (2010). 59

66 Freight Rate Volatility, σ 0,1 0,3 0,5 0,7 0,9 Spot Freight Rate, X(0) Table 9. Value of European purchase option. Dependence on spot freight rate and freight rate volatility. =0,0033, =10,45, =1,5%, Maturity Option=15 years, Maturity Vessel=25 years, = , Scrap Value= The European option value is expected to be an increasing function of both the spot freight rate and the freight rate volatility. As the spot freight rate increases, the vessel will become more valuable due to higher cash inflow, and thus the option to buy such a valuable vessel will also increase. When the freight rate volatility increases the probability of large upside gains will increase, thus, the option value will also increase. Most of the values in Table 9 are acting in line with what is expected, but some values are somewhat strange. There can be several reasons for these strange results, but an obvious reason is that extremely few paths are Monte Carlo simulated. A modest number of 500 simulations are done due to the limited abilities of Microsoft Office Excel 22. The efficiency of Monte Carlo simulation can be increased by implementing variance-reduction techniques. The antithetic variates method is an example of such a technique; correlation is created across simulated paths in order to reduce the variance of the sum of random variables (Campbell, Lo et al. 1997). This technique ensures a doubled amount of simulated paths by utilizing the symmetry present in the normal distribution. Each simulated path can be reflected through its mean to produce a mirror-image with the same statistical properties (Campbell, Lo et al. 1997); a mirrored random variable is created with exactly the same statistical properties, but with the opposite sign. 22 A simulation of 500 paths ran a whole night before it was done. 60

67 3.6 The Valuation Results: Comparisons This section will shortly comment the vessel values and the European option values obtained from both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process. Important to note is that estimation procedures lie outside the scope of this thesis. This creates difficulties when comparing the valuation results. For correct comparison the parameter values to both models should have been estimated from the exact same dataset in order to ensure that they matches relatively to each other. This may be the reason why the vessel valuations give ambiguous results; it is expected that the Geometric Mean Reversion process will give higher vessel values as negative values are impossible. Also comparisons of the option values becomes difficult. By comparing the vessel values from the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process in Table 5 and Table 8 respectively, they clearly indicate quite similar results. Generally, the vessel values obtained from the Ornstein-Uhlenbeck process lie above the values obtained from the Geometric Mean Reversion process, which is ambiguous as the Geometric Mean Reversion process is expected to give higher values. Also, as both the spot freight rate increases and the remaining lifetime increases, so does the difference between the values also increase. As for the European option values, comparison becomes difficult due to both the estimation issue, and due to the reduced accuracy when obtaining values from the Geometric Mean Reversion process. A modest amount of 500 paths are Monte Carlo simulated, which in fact is too few when accurate results are desirable. However, the valuations serve as an introduction to further studies where Finite Difference methods could have been implemented for more accurate results. 61

68 4 Limitations When choosing a mathematical model for valuation purposes there will always be a trade-off between analytical tractability and goodness of fit to observations. This is important to have in mind when considering the valuation results during this thesis. This section will examine some of the simplifications done when modeling freight rates which create a gap between the valuation results and reality. 4.1 Limitations Caused by the Models Selected By choosing the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process for valuation purposes, important considerations going beyond the properties of the model will automatically not be taken into account. Some of the considerations left behind when choosing these two models will be discussed in this section The Parametric Property of the Models Both the Ornstein-Uhlenbeck process and the Geometric Mean Reversion process are parametric in nature; their distributions are assumed to belong to a parametric 23 family (Jorion 2007), which in this case is the normal distribution. In his article, Adland (2000) proposes a non-parametric model to model the time charter equivalent spot freight rate by using monthly data over a period of ten years. By doing this, misspecification of the density function s shape of the time charter equivalent spot freight rates is avoided, and estimations may appear more accurate compared to estimations based on a misspecified model. By estimating the marginal density of the time charter equivalent spot freight rate he finds that the distribution is slightly skewed to the right. This result does not correspond to the assumption of the time charter equivalent spot freight rates being normally distributed in both the Ornstein-Uhlenbeck process and in the Geometric Mean Reversion process. Adland (2000) s findings imply misspecification when assuming normally distributed time charter equivalent spot freight rates. Subsequent to this, the results appearing in this thesis will need to be considered with this possible misspecification in mind. But also Adland (2000) s results have shortcomings; a non-parametric model opens for greater estimation error than parametric models do, in addition he stresses that too few observations are obtained in order to get a statistically confident estimate. 23 To obtain parameter values in parametric models estimation procedures are necessary (Jorion 2007). 62

69 4.1.2 One-Factor The model adopted to the valuation of time charter contracts with purchase options is a one-factor model with an assumption of the spot freight rate being the factor influencing the value of a vessel. This assumption will disregard all information that is not embedded in the current spot freight rate level and the dynamics of the spot freight rate process observed in the past. Important fundamental market information that is expected to influence future freight rate dynamics is thus not taken into account at all (Adland and Strandenes 2007). A gap is then created between theory and reality due to the existence of several factors influencing vessel values. According to Alizadeh and Nomikos (2009), factors influencing vessel values can be classified into two groups; vessel-specific factors and market-specific factors. Vessel-specific factors generate a more intuitive understanding of why vessel values vary as those are related to the particulars and condition of each vessel. Examples of such factors are size, type, age and general condition. Marketspecific factors relates to the general state of the freight market. Those factors are more complex than the vessel-specific factors. Current and expected freight rate levels and market conditions are among the most important market-specific factors. Stopford (2009) argues that the vessel s age is the second most important factor in relation to vessel values. As also the vessel valuation results from both models indicate, a newer vessel is worth more than an older vessel. As a vessel gains age, it may lose performance and also higher maintenance costs may appear. Thus, this will lead to a lower vessel value in total, and when its market value falls below the scrap value the vessel is likely to be sold for scrapping. Due to the importance of the vessel s age, a multifactor model, where both the freight rate and the vessel s age are taken into account, could have been considered when valuing vessels. This would have included more realism to the model. 63

70 4.2 The Assumptions This section will address some of the simplifying assumptions done through this thesis, and describe how more reality would have been included if they were not taken Constant Market Price of Freight Rate Risk Since the risk-free interest rate have been applied as the discount factor the true probability measure,, had to be transformed to the risk-neutral probability measure,. To do this transformation Girsanov s Theorem was used, and the market price of freight rate risk,, was introduced and integrated in the Ornstein-Uhlenbeck process. Further, the market price of interest rate risk was assumed to be constant. The market price of freight rate risk can be thought of as the difference between implied forward freight rate 24 and the expected spot freight rate (Adland 2003). Therefore, the value of the market price of risk is closely related to the expectations of the term structure of freight rates in the future. The assumption of this being constant has been disproved by Adland (2003). He presents qualitative arguments and suggests that the market price of risk in the freight market in bulk shipping must be time varying and depend on the state of the spot freight market and the duration of the time charter in a systematic fashion. He argues for the existence of other factors than volatility in freight rates that support the hypothesis of a non-constant market price of risk. Finally, he finds it reasonable to consider the market price of risk as an increasing function of the spot freight rate level. An earlier research also done by Adland (2000) supports his conclusions discussed above. This is a quantitative based research which contribute with two separate results; that shipowners are not compensated for the risk associated with trading in the spot freight market when freight rates are low, and that the market price of risk is an increasing function of the freight rate level (Adland 2000). The first result may appear due to the lack of considering other risk factors than the volatility of freight rates when determining the market price of risk; when freight rates are low, the volatility of freight rates is also low, and the risk associated with trading in the spot market may be considered as small. The second result support his qualitative arguments discussed above that the market price of risk is increasing in the freight rate level. This indicate that shipowners are compensated for bearing freight rate risk when freight rates are at high levels; operating in the spot freight rate market is more risky in this case since high freight rates implies high volatilities. 24 The freight rate evolvements in the future. 64

71 4.2.2 Constant Risk-Free Interest Rate Throughout this thesis the risk-free interest rate is assumed to be constant. The risk-free interest rate do in fact vary through time, thus this is another assumption made to simplify the calculations and for the ability to apply the risk-neutral valuation result. To accommodate the shortcoming of constant risk-free interest rate, one approach could be to model the time structure of the risk-free interest rate outside the models. This could be done by using the Vasicek (1977) model which is identical to the Ornstein-Uhlenbeck process. The estimated time structure could then be implemented in the model as a simple variable. 65

72 5 Summary and Conclusions The aim of this thesis was to shed light on the importance of fair valuation of embedded options in time charter contracts, with options to purchase the vessel underlying the contract as the primary area of research. As these options can be very complex in nature, they are often granted for free rather than for their fair value. This creates misleading information of a shipping company s total net asset value as the embedded options are of highly economic significance to the company. Also, large volatility in freight rates can lead to large decreases or increases in the option values, which in turn can affect the shipping company s viability. Thus, important to stress is also the proper management of the risks a shipping company faces. Also, this thesis aimed to introduce two possible models for the valuations where each of them was examined and their properties were compared. This gave an indication of one model being more appropriate in freight rate modeling than the other. First, the comprehensive shipping industry was introduced. The shipping market model that describes the mechanisms controlling the shipping market cycles shortly described in Section 2.8.1, was examined. Further, various costs and risks occurring in this industry was described, as well as a description of the four existing markets. With the theories of the shipping industry in hand, embedded options in time charter contracts were examined before a comprehensive description of the two models could be established. The first model introduced for valuation purposes was the Ornstein-Uhlenbeck process which has been applied for valuation of time charter contracts with embedded purchase options. The solution to this process was derived before the derivation of the partial differential equation that all freight rate dependent claims must satisfy in order to be priced arbitrage free. Further, the partial differential equation enabled for derivations of analytic solutions to simple freight rate contingent claims - equal to the approach seen in Jørgensen and Giovanni (2010). The analytic solutions were then applied to valuations of simple freight rate contingent claims, as well as vessel valuation, European options to buy the vessel and finally, the total fair value of the time charter contract with purchase options. The second model introduced for valuation purposes was the Geometric Mean Reversion process. To the author s knowledge no analytic solutions exist and numerical routines were therefore needed for valuations. Vessel values and European options to purchase a vessel were valued by the application of Monte Carlo simulation. The Geometric Mean Reversion process was introduced as an alternative and more realistic valuation model. However, for valuation of time charter contracts with purchase options, implementation of Finite Difference methods would have been necessary. This may be an interesting area of further research. 66

73 The Ornstein-Uhlenbeck process gave analytic solutions such that time charter contracts with purchase options for different fixed time charter rates and spot freight rates could be valued. The results imply that the value of the time charter contract increase in the spot freight rate level, but decrease in the fixed time charter rate level. This was also expected as higher spot freight rates imply higher cash inflow and thus an increase in the value of the option to purchase the vessel. Whereas higher fixed time charter rates creates a larger gap from the long-term mean of , and thus also increasing the gap between the fixed time charter rate that would have been fair from the agreed fixed time charter rate. The Geometric Mean Reversion process facilitated for the use of Monte Carlo simulation. Vessel values and European option values for purchasing the vessel were obtained. However, accuracy in the results was reduced at the expense of an illustration of how Monte Carlo on Monte Carlo simulation can be applied to obtain option values when analytic solutions do not exist. A larger amount 25 of simulated paths would have been beneficial in relation to more accurate results, but this would have been very time consuming due to the shortcomings of Microsoft Office Excel. The intention behind the introduction of both models was the opportunity for comparisons both the model characteristics and the valuation results. The comparison of the valuation results highlighted the weakness when lack of estimation is present. In order to obtain correct comparisons between the valuation results obtained from both models, the parameter values need to match relatively to each other. This could have been ensured by using the exact same dataset when conducting the estimation procedure to both models, and at least the ambiguous vessel values obtained could have been avoided. The introduction of two models opened for an examination of model characteristics and a thoroughly review of which model that would have been best fitted to freight rate modeling. The discussion points in the direction of the Geometric Mean Reversion process being most appropriate, which is due to the fact that the process is downwards restricted with zero as an absorbing level. With the impossibility of freight rates being negative, this property becomes valuable compared to the Ornstein-Uhlenbeck process which is not downwards restricted. Thus, this thesis ends by concluding that the Geometric Mean Reversion process is more realistic in freight rate modeling compared to the Ornstein-Uhlenbeck process. 25 A suggestion of to simulated paths would have been beneficial. 67

74 6 List of References Adland, R. and K. Cullinane (2006). "The Non-Linear Dynamics of Spot Freight Rates in Tanker Markets." Transportation Research: Part E 42(3): Adland, R. and S. Koekebakker (2007). "Ship Valuation Using Cross-Sectional Sales Data: A Multivariate Non-Parametric Approach." Palgrave Macmillan Journals: Adland, R. and S. P. Strandenes (2007). "A Discrete-Time Stochastic Partial Equilibrium Model of the Spot Freight Market." Journal of Transport Economics & Policy 41(2): Adland, R. O. (2000). "A Non-Parametric Model of the Timecharter-Equivalent Spot Freight Rate in the Very Large Crude Oil Carrier Market." Foundation for Research in Economics and Business Administration. Adland, R. O. (2003). The Stochastic Behavior of Spot Freight Rates and the Risk Premium in Bulk Shipping. The Department of Ocean Engineering, Massachusetts Institute of Technology. Ph.D in Ocean Systems Management Alizadeh, A. H. and N. K. Nomikos (2009). Shipping Derivatives and Risk Management. Faculty of Finance, Cass Business School, City University, London, Pargrave Macmillan. Bjerksund, P. and S. Ekern (1995). Contingent Claims Evaluation of Mean-Reverting Cash Flows in Shipping. Real Options in Capital Investment: Models, Strategies, and Applications. L. Trigeorgis. London, Preager: Campbell, J. Y., et al. (1997). The Econometrics of Financial Markets. Princeton, New Jersey 08540, Princeton University Press. Duffie, D. (2001). Dynamic Asset Pricing Theory. Princeton and Oxford, Princeton University Press. Hammer, H., et al. (2011). Valuing Time Charter Contracts with Purchase and Extension Options. Bergen, Norges Handelshøyskole. Hull, J. C. (2012). Options, Futures, And Other Derivatives, Pearson Education Limited. Jorion, P. (2007). Value at Risk - The New Benchmark for Managing Financial Risk, The McGraw-Hill Companies. Jørgensen, P. L. and D. D. Giovanni (2010). "Time Charters with Purchase Options in Shipping: Valuation and Risk Management." Applied Mathematical Finance 17(5):

75 Kavussanos, M. G. and I. D. Visvikis (2006). Derivatives and Risk Management in Shipping, Witherby Shipping Business. Koekebakker, S., et al. (2006). "Are Spot Freight Rates Stationary?" Journal of Transport Economics & Policy 40(3): Koekebakker, S., et al. (2007). "Pricing freight rate options." Transportation Research: Part E 43(5): Skovmand, D. (2012). Supplementary Notes on: Linear Algebra, Probability and Statistics for Empirical Finance. Stopford, M. (2009). Maritime Economics, Routledge, Taylor & Francis Group, London and New York. Tvedt, J. (1997). "Valuation of VLCCs Under Income Uncertainty." Maritime Policy & Management: Vasicek, O. A. (1977). "An Equilibrium Characterization of the Term Structure." Journal of Financial & Quantitative Analysis 12(4):

76 7 Appendix The Excel spreadsheets where the tables and figures have been obtained are saved in a USB stick enclosed to this paper. 7.1 The Ornstein-Uhlenbeck Process Detailed Solution The freight rate at time is given by today s freight rate plus the sum of the dynamics of the spot freight rate evolving from time to time : A temporary variable is introduced in order to solve the equation above explicitly: The dynamics of this temporary variable are given by Ito s lemma: The Ornstein-Uhlenbeck process is substituted for above: 70

77 Manipulations of the equation above give: Continuing by integrating from time zero to time : Calculating : Using this result in the continuation: 71

78 Calculating : Finally, the solution to the Ornstein-Uhlenbeck process becomes: 7.2 The Ornstein-Uhlenbeck process - Derivation of the Mean and the Variance The Time Conditional Mean Taking the expectation of each term in the solution to the Ornstein-Uhlenbeck process: Since the expectation of a Wiener process equals zero the mean of the process becomes: 72

79 7.2.2 The Time Conditional Variance Taking the variance of each term in the solution to the Ornstein-Uhlenbeck process: Since the variance of the constants in the two first terms equals zero we are left with the following expression: 73

80 7.3 The Geometric Mean Reversion Process Detailed Solution To simplify later calculations: A temporary variable is introduced in order to solve the equation above explicitly: The increments of this temporary variable are given by Ito s lemma: Rearranging: 74

81 Now, it is desirable to substitute the process for. In order to ease the notifications and are defined such that. This gives: where the first and the second term tend to zero, and where the last term tends to such that: Then, by rearranging the manipulated Geometric Mean Reversion process to and substituting the process for : The next step is to rearrange this equation and integrating it from time zero to time : 75

82 What is left now is to rearrange this equation such that is alone: 76

83 Finally, the freight rate level at time,, can be expressed as: 7.4 The Ornstein-Uhlenbeck Process Claim to Receive Spot Freight Rate Flow from Time to Time Defining the current value of receiving the spot freight rate on a continuous basis as: The freight rate process, : Inserting in the expression for : 77

84 Since the expectation of a standard Wiener process equals zero (Hull 2012), the last term equals zero and only the two first terms are left. Further: 78

85 where is an annuity factor. 79

86 7.5 The Ornstein-Uhlenbeck Process - European Option to Buy the Vessel Define and the value of a vessel is then given by: The next step in the derivation of an analytic formula for the option value is to evaluate +. Again, by following Jørgensen and Giovanni (2010) consider: 80

87 where is the density function that describes freight rates (Skovmand 2012). The density function plots the shape of the distribution curve of the random variable that is considered, which is the freight rate in this case. Since the freight rates are assumed to be normally distributed by their density function plots the classic Bell Curve which takes the form (Skovmand 2012): which in this case will look like this (Jørgensen and Giovanni 2010): Continuing by inserting the expression for the density function gives: Further, define. Then: 81

88 Finally, the time t value of the European call option is given by premultiplying the above result with times the discount factor: 82

89 where: and where and denote the standard normal cumulative probability and density functions, respectively. 83

90 7.6 The VBA Codes 84

91 85