Valuation of Structured Products

Transcription

1 Valuation of Structured Products Pricing of Commodity Linked Notes Shahid Jamil, Stud nr: SJ80094 Academic Advisor: Jochen Dorn Department of Business Studies Aarhus School of Business, University of Aarhus February 2011

2 1 Abstract Structured products including commodity linked structured products have complex composition. These are suitable for those investors who want a complete or partial protection of their investment. A typical structured product is a combination of a risk free bond and an option. The bond part guarantees capital protection while the option part provides the possibility of payoff. The option pricing is the tricky part of these products and a wide range of theories are available to price them. In this thesis the well known Black & Scholes option pricing frame work is applied and the theoretical estimated price of the selected commodity linked structured notes are compared with their issue price to evaluate if these products are offered to the investors at fair price.

3 2 Table of Contents 1. Introduction Structured Products Structured products are suitable for investors who Disadvantage of structured Products Difference between a Conventional Bond and a Commodity Linked Bond Commodity Linked Bonds, a brief history Classification Classic Products Corridor Products Guarantee Products Turbo Products Products with exotic option components Barrier Products Rainbow Products Structure of structured products The Bond Component The Option Component Swaps Participation Rate Understanding Options Exotic Options Path dependent options Asian options Lookback options Ladder options Barrier options Time dependent options Multifactor options Payoff modified options Option Pricing Theory Assumptions Stochastic Process... 26

4 Properties of a stochastic process The Markov Property Wiener Process Generalized Wiener Process Geometric Brownian motion Ito s Lemma Risk Neutral Valuation The Black- Scholes Equation (BS) Options on dividend paying stock Commodity Options Options on many underlying Black- Scholes Pricing Formulas Upper and Lower bounds for the call option Forward Contract Futures contracts Futures Options Pricing of European futures options An overview of the selected products DB Råvarer 2013 Basel (the Notes ) Payoff Structure Risk Factors Issuance costs Embedded option The underlying asset Analysis of Råvarer Basis Payoff Structure Risk factors Issuance costs Embedded option Underlying Asset Pricing of the selected products Pricing of DB Råvarer 2013 Basel Deriving the Zero Coupons Term Structure for Råvare.r Basis

5 4 7.3 Option Pricing Estimating Volatility Monte Carlo Simulation Variance Reduction Technique Generating Random numbers DB Råvarer Basel Pricing of Råvarer Basel Evaluation of the Model Possible extensions to the thesis Conclusion References... 67

6 5 1. Introduction Structured products are drawing more and more investors now a day. Retail and institutional investors alike are piling into these products. Structured products are suitable for defensive or conservative investors because investments in structured products assure a complete or partial protection of their invested capital but at the same time, they can take advantage of the economic exposure to the growth potential of the selected underlying. The underlying asset can be a single or a basket of the underlying assets. The popular structured products offer exposure to the equities, foreign exchange, indices, volatility indices, commodities and commodity indexes such as S&P commodity index, the Dow Jones-AIG commodity index or the Rogers International commodities index. Commodity indices differ from the equity indices. The underlying investments are not shares, bonds or the commodities themselves but futures contracts on a single or a basket of commodities (a contract to buy or sell an asset at a given future date for a set price). Futures contracts generally expire after three months; therefore the so called rolling principal is applied for futures index where the index sponsor replaces near to expiry contracts with the longer maturity contracts. Structured products have become very advanced too in their structure. The complex mechanisms within their structures are sometimes difficult to understand by the investors and even sometimes by the financial managers too. So, the theme of this thesis is to present an in depth analysis of the selected structured products. The following will describe exactly what this thesis will be answering What are structured products and their composition? How the individual components of a structured product are valued. That is, how are bonds priced (coupon and zero coupon bonds), and how the underlying embedded options are priced (plain vanilla and exotic options)? What is the theoretical fair value of the selected structured products and whether these products are offered to the investors at fair value, overvalued or undervalued?

7 6 Structured products can be found in a wide variety of underlying assets. The underlying assets can be from equities to equity indices, foreign exchange, interest rates, commodities or commodity indices. This thesis will mainly focus on valuing commodity linked structured products. The contents of the thesis are split into two parts. The first section mainly focuses on the theory behind the structured products. This section consist of chapters 2, 3, 4 and 5. Section two consists of the valuation of the selected products and comparison of the theoretical price with the issuing price of the selected products. Chapters 6, 7, 8 and 9 will be part of the second part of this thesis. Chapter 2 begins with the introduction and definition of structured products and then defining the commodity linked structured products. It will also describe in general how structured products are engineered. This chapter will also discuss the advantages and disadvantages of commodity linked products, their brief history, the difference between a conventional bond and a structured bond and explanation of different concepts within structured products. Structured products normally consist of two components i.e. the bond and an embedded option. The option component is generally the most tricky and complex part of these products. Chapter three, four and five consist of the option theory including their classification and the underlying concepts involved in option pricing in particular stochastic process, geometric Brownian motion and generalized Wienner process. Black and Scholes option pricing framework along with underlying assumptions and the concept of risk neutral world will be discussed in chapter five. Chapter six and seven deal with the pricing issues of the selected products. Chapter six begins with the analysis of prospectus of the selected products. While chapter seven starts with pricing of the bond and option part of these structured products. Valuation of the option components start with an overview of the Monte Carlo simulation. The qualitative and quantitative comparison of theoretical and issue price of the products is performed. Chapter eight starts with a critical evaluation of the underlying assumptions and their effect on valuation model. This thesis will give an understanding of how the structured products are priced and therefore will trigger readers interest to find improvements in the pricing model and possibly to use other pricing models too. So, a

8 7 brief description of the alternative option pricing models is also discussed in chapter 8. Finally, the thesis ends up with a brief conclusion in chapter 9. Although, structured products are available with a wide variety of underlying assets, but this thesis will focus on commodity linked products. The products are selected from the Danish market, therefore, interest rates will also be considered from the Danish market. The valuation will be performed by applying the well known Black & Scholes option pricing theory because the main theme of this thesis is the valuation of the selected products and not to evaluate the performance of different option pricing models. Therefore, other advanced option pricing models are not considered in this thesis. In order to price the option component in Black & Scholes frame work, Monte Carlo simulations are applied which follows the principal of risk neutral random walk. Tax issues will not be considered in the valuation process. Default risk of the issuing firm will be also disregarded because the selected products in this thesis are from the issuers with very good credit ratings. The data for this thesis has been downloaded directly from the Dow Jones UBS commodity index web site, while the data for deriving zero coupon term structure was down loaded from data stream. Finally, I would like to thank my advisor Jochen Dorn for his use full guidance and patience during the thesis.

9 8 2. Structured Products Structured products have emerged as an important instrument in financial markets. A structured product can be defined as a security that combines the features of a fixed income security with the characteristics of a derivative transaction. Generally, a structured product contains two components i.e. a fixed income security (a zero coupon bond that guarantees full or part of the invested capital) and an option or forward like instrument which has a specific class of the asset as an underlying. The underlying assets can be equity, interest rate, an index, inflation, foreign exchange, commodities or credit. The underlying can be a single asset or a basket of multiple assets. The additional payoff of a structured product depends on the performance of the underlying asset. Structured products are also said to be a marriage of a fixed income security and an option like instrument 1. When the underlying in a structured product is a commodity or a basket of commodities or a commodity index then they are called commodity- linked structured products. The underlying commodities can be for example crude oil, gas oil, metal (gold, silver, copper, and precious metals), energy etc. Commodity indices are different from the other indices. The underlying investments are not bonds or shares or the commodities themselves but it is futures contracts on the commodities. A futures contract is defined as the contract which gives its holder the right but not the obligation to buy or sell an asset (commodities, equity, foreign exchange etc) at a given future date at an agreed price. Futures have normally three months expiry date. These expiry dates are normally standardized. The index sponsor is therefore required to replace the expiring futures contracts with the new ones traded on the futures market (every three months). This is called the rolling principle. When a futures contract expires, the index will treat it as sold and the proceeds are reinvested in the new futures contract that will again expire after the next three months. 1 BNP Paribas equities & Derivatives handbook

10 9 The index level takes care, the price movements in the underlying commodities and takes into account the price difference between the old and the new futures contract that are rolled. 2 Commodity linked structured products are available in a wide variety of range. One of them is Commodity- Linked bonds/ Notes, which is also the topic of this thesis. Commodity linked notes or bonds are classified into two classes, i.e. The Forward type and the option type. In a forward type bond, the coupon and or principle payment to the bearer are linearly related to the price of the stated reference commodity i.e. it allows the holder to receive either the nominal face value or the designated commodity amount at maturity. While, an option type bond, the coupon payments are similar to that of a conventional bond but at maturity, the bearer receives the face value plus an option to buy or sell a predetermined quantity of the commodity at specified price 3. In literature both the terms (bonds and note) are used interchangeably. 2.1 Structured products are suitable for investors who 1. Want protection of their invested capital by hedging the risk of existing investments. 2. Want to enhance the return from their investment while controlling risks, whereby the structure is designed to enhance equity return with leverage. 3. Want to diversify with the adjustable risk/ return profiles and market cycle optimization capabilities of structured products. 4. Want to exploit their market view with more freedom and flexibility. 5. Want a growth by capitalizing on the market upside while protecting the downside. 6. Want to benefit from periodic returns with limited risks. This income type of structure is built to deliver coupons while protecting capital 4 2 Barclays Wealth, Light Energy Commodity Plan 2,3 Joseph Atta- Mensah. Commodity- Linked Bonds 4 BNP Paribas equities & Derivatives handbook

11 Disadvantage of structured Products Despite the fact, that structured products including commodity linked products provide capital protection and a possible payoff from the option component, the investor still loses the payoff associated with a traditional risk free bond. In a structured product, the investor receives only the invested capital if the option expires out of the money, but in a traditional bond, the investor receives a risk free interest of his invested capital along with the invested capital. This lost risk free interest or profit is called the Opportunity cost and can be defined as the forgone risk-free rate of return that could have been achieved if the principal would have been invested in the safe fixed- income securities such as Treasury bills 5. For example, if an investor invests 100 DKK in treasury bonds for one year with a 5% interest rate. He will receive DKK105 at maturity while if he invests in a structured product, he will only get DKK 100 at maturity plus a possible payoff from the option embedded in it, because he will actually invest 100*1,05^-1 = DKK 95 in the risk free investment and DKK 5 will go to buy a call option plus administration fee. This DKK 5 from investment in risk free bond will be the opportunity cost that he will miss if he would invest in a structured product. The payoff of a traditional bond will exceed as long as the option component of a structured product is out of the money, at the money or if it is in the money but still below DKK Difference between a Conventional Bond and a Commodity Linked Bond6 Commodity linked bonds are different from conventional bonds in many aspects. Some of the key differences between are 1. In conventional bonds, the investor receives fixed coupon payments i.e. interest payments during the life cycle of the bond (annually or semiannually etc) and the face value at the maturity of the bond. While the holder of a commodity linked bond receives the physical units of the underlying commodity or equivalent of its monetary value. Similarly the coupon payments may or not be 5 Lehman Brothers, A guide to Equity_Linked Notes

12 11 in units of the underlying commodity (it depends upon the performance of the underlying). 2. The nominal return of a conventional bond held to maturity is known while the real return is not known because of inflation. On the other hand both the real and the nominal return of a commodity linked bond are unknown. 3. The results of Atta- Mensah study also show that the coupon rate for a conventional bonds are greater than that of the commodity linked bonds whose terminal payoff is greater of the face value and monetary value of a pre-specified unit of a commodity. 4. The coupon rate of a conventional bond is less than that of a commodity linked bond that pays its holders on maturity the minimum of the face value and the monetary value of a pre-specified unit of a commodity. 2.4 Commodity Linked Bonds, a brief history The concept of structured note is considered to be relatively new in the financial markets. In reality these products have been in existence for a considerable time. For example callable notes and equity linked securities i.e. convertibles and debt with equity warrants are the precursors to the structured note products that are common place today. Commodity linked bonds were introduced during 1970 s when the oil backed bonds were issued by the Mexican Government in the financial market. These bonds were called Petrobonds. Each 1000 Peso bond was linked to barrels of oil with a coupon payment of % annually and had three years to maturity. Later on the French Government issued gold backed bonds during They were known as Giscard in the financial markets. These bonds have 7% coupon rate and redemption value was indexed to the one kilogram bar of gold. In 1981 Eco Bay Mines Company of Canada also issued gold warrants. Commodity linked bond with sliver as an underlying was issued during 1983 and again in 1985 by the Sunshine Mining Company in USA. The objective was to hedge against the fluctuations in the price of silver. Later on, bonds indexed to other commodities like nickel, copper, silver, cobalt and platinum were issued during 1984 by Inco. (a leading producer of metals). Now a days, commodity linked bond are issued by many investment banks around the world. These bonds are linked to the performance of a basket of specific commodities or commodity

13 12 index for example Goldman Sachs Commodity Index (GSCI), London Metal Exchange (LME), S & P commodity index or Dow Jones UBS commodity index. 2.5 Classification Structured products including commodity linked structured products are available with a wide variety of product characteristics and heterogeneous characteristics in the market. However, Pavel A Stoimenov and Sascha Wilkens, in their article about the equity linked structured products in the German market, Are Structured Products Fairly Priced? have classified them according to the underlying option components embedded in the product. As is shown in the figure 1, structured products can be divided into two categories i.e. Plain vanilla Option- Component and exotic Optioncomponent. Plain vanilla products are further classified into Classic, Corridor, Guarantee and Turbo while Exotic components products are further classified into Barrier and rainbow. Barrier structured products are further divided into Knock- in, Partial Knock- in and knock out products. The brief definitions of these products are discussed below. Figure Classic Products A classic structured product has the basic characteristics of a bond except that the issuer has the right to redeem it at maturity by repayment of its nominal value or delivery of previously fixed number of specified shares. In general, structured products can be

14 13 categorized into with and without coupon payments. Products with coupon payments are called as reverse convertibles, while those without coupon payments are named as discount certificates Corridor Products The payout of a corridor structured product depends on whether the underlying at maturity is quoted within a certain range Guarantee Products A guarantee product is similar to that of a corridor product. The only difference is that in a guarantee product, fixed minimum repayments are guaranteed to the investor. So, if price of the underlying falls below the reference value, then the investor will always get the guaranteed amount Turbo Products The payout of a turbo product is doubled if the underlying is quoted within a certain price range at maturity. This is called turbo effect. But there are three possibilities at maturity. If, for example, L and K are lower and upper reference prices, then at maturity, if 1. S t fixing L, the product is redeemed in shares; 2. L < S t fixing < K, a cash settlement with s(2 S t fixing - L) occurs; 3. K S t fixing, the maximum amount s (2K - L) will be paid. 2.6 Products with exotic option components Barrier Products Barrier products are the most common type of structured products, where the embedded option is a barrier one. The redemption of a barrier product depends upon whether the underlying reaches a certain fixed price barrier during its lifetime. In a knock in product, if the underlying reaches or crosses a fixed pre specified lower price barrier, then the stocks are delivered at maturity. In such a case the product behaves like a classic product. A knock in pays the maximum amount if the underlying is always above this barrier regardless of the S t Fixing. In the case of a knock out product, if the underlying

15 14 reaches or crosses the pre specified upper price barrier, then the issuer loses his choice of redemption and the products behaves like a regular bond in this case. In a partial time knock in product, the barrier criterion is tested only within a certain time interval, generally a few months immediately before maturity Rainbow Products The rainbow products have more than one underlying. In a rainbow product, the issuer has the right to choose between the specified underlying on redemption. 2.7 Structure of structured products Commodity linked notes like other types of structured products provide partial or 100% capital protection depending upon the investor s specific needs. A typical commodity linked bond provides 100% capital protection at maturity independent of the performance of the underlying commodity or commodity index. The structure of a simple commodity linked note can be sketched as Option/ Swap Bond Commodity linked Note Payoff Principal This figure shows that when an investor buys a structured product (equity, commodity or any other asset as an underlying), he/ she actually has bought a package which consists of a bond and an option or a swap (forward or future) and the fee on top of it. The payoff of a structured product (note) is equal to the par amount of the note plus a commodity / equity etc linked coupon. The payoff is either 1. Zero, if the underlying has depreciated from the initial agreed upon strike level

16 15 2. Or the participation rate times the percentage change in the underlying commodity/ equity times the par amount of the note 7. In order to understand how a structured note works, we consider a simple example, where an investor wants to invest 100 DKK over five years with full capital protection and exposure to the S & P Commodity index. This means that the investor will get at least his/ her 100 DKK at the end of five years no matter if the index depreciates or appreciates. The investment bank will apply the appropriate interest rate (treasury bonds, LIBOR or REPO rate) and find out the present value of 100 DKK (future value). For example, if the five years interest rate was 5 %. Then the present value of this five years zero coupon bond will be DKK (100 * ) today. It means that the structure provider (investment bank) will have = DKK to purchase an option or a futures/ forwards contract. Now let s consider that a five years S & P call option costs and 2 is the administration and margin costs, then the investor will benefit from % (( ) / 23.65) participation in the S & P index s upside The Bond Component The bond component of a structured product is the most important part of it. It is also the major part of any structured product. The bond component ensures that the investor will receive the agreed amount of his/ her investment at maturity. The agreed amount can be a 100 % of the invested capital or it can also be partial protection depending upon the product. Structured products in general have the characteristics of a zero coupon bond but it can also have coupon payments (annual or a semi- annual). The main advantage of a zero coupon bond is that the investor gets all his investment back at the same time instead of coupon payments at the end each period (annual or semiannual). The risk free interest rate applied to a zero coupon bond ensures that the present value of the investment will grow continuously until maturity. The risk free interest rate is normally taken from the government bonds (the rate at which the state borrows money). If an amount A is invested for n years at an interest rate R per annum and if R is compounded once per annum i.e. m=1, then the terminal value of the investment will be 7 Lehman Brothers ( equity Linked Notes)

17 16 A(1 + R m )nm If m +, we compound more and more frequently, then we obtain the well known compounding frequency interest rates and the future value or the terminal value will be given as Ae Rn In the same way the present value of a future amount can be written as Ae Rn The pricing of a zero coupon bond or any other fixed income security can be derived if we know the zero coupon (ZC) yield curve. The term structure of ZC rates (also known as ZC yield curve) is the curve relating maturities t (time horizons) with the corresponding ZC interest rate R(t). F t (1+R(t)) t Price = T t=1 = T t=1 F t B(t) (1) Here B(t) means the discount factor at time t ( the prices of zero coupon rates with face value of 1) R(t) is the zero coupon rate derived from B(t) and F (t) is the known cash flow or also called the principal amount. When a structure note/ bond have the features of a coupon bond, then it can be considered as the portfolio of zero coupon bonds. The price of such a bond can be written as the present value of the sum of all cash flows (coupon payments) for each period plus the principal amount and can by the following expression Price = n CF t n. B t + F t B n = CF t. (1 + R t ) t + F t (1 + R t ) n (2) Where, CF t is the cash flow in time t

18 The Option Component The option component is the 2 nd part of a structured product. Option component provides the chances of payoff. Options are of two types i.e. a call and a put. A call option provides its holder the right to buy an asset at a certain date on a pre specified price while the put option gives its holder the right to sell an asset at a pre specified price on a certain date. Call options are normally embedded in structured products because it is easy to earn on something whose price is increasing rather than decreasing as in case of a put option. Option component is also the risky part of any structured, because the payoff depends on the performance of the underlying. If the option embedded in a structured product expires out of the money (i.e. the strike price of the call option is higher than the corresponding price of the underlying) than it will not be exercised and the holder will get no profit but instead loose the money to buy that option but he will still receive his invested capital. If on the other hand, an option expires in the money, then it will be exercised and the holder will earn profit along with the guaranteed capital. The expression at the money means that when the strike price of the call option is equal to the price of the underlying. There are two possibilities to exercise an option. In a European style option, the holder can exercise his right to buy or sell an asset only at the maturity of the option. In an American style, the holder can exercise his right to buy or sell the underlying before the maturity of the option too Swaps Commodity linked structured products can also be found with swaps in their structures. Forwards are an example of swap and commodity swaps are in fact a series of forward contracts on a commodity with different maturity dates and the same delivery prices. 8 The commodity linked products as mentioned by Schwartz 9 issued for example by Sunshine company during 80 s or by the Mexican Government during 1979 backed by silver and oil respectively are examples of the products with forward type component in the structured product, where the company promised to pay either the face value or market value of the underlying commodity. 8 John C Hull: P-173 Option, Futures and Other derivatives 9 Eduardo S. Schawartz: The Pricing of Commodity Linked bonds

19 Participation Rate Participation rate determines that how much the product will participate in the performance of the underlying. It can be defined as the exposure of a product to movements in the price of its underlying. A participation rate of 100 % means that the investor would receive the return that will be exactly equal to the rise in the price of the underlying. For example if the underlying has increased by 25% at maturity, then the investor will also receive 25% return. But if it is low as mentioned in the example on page 9 i.e % then the investor will get DKK 21(87.83% * 25). Participation rate depends on the value of the option embedded in the structured product, the administration and other issuing costs and the present value of the bond component of the product. The participation rate depends on many factors. For example if the issuing costs of the product are low then the participation rate can be higher. Similarly, if the value of embedded option is high/ low then the participation rate can lower/ high. Participation rate is generally not set prior to the expiry of the issuance period and it appears as estimate in the prospectus. The participation rate can be calculated by the following relation Participation rate = issue price costs PV of bond Price of option component 100 (3) Participation rate is also named as Gearing. The above equation also shows that there other factor which determines the participation rate. For example, the interest rate used to calculate the present value of the bond component, the life time of the product and volatility of the underlying asset. For example a low interest rate will result in high present value of the bond component and can reduce the participation rate and vice versa. Similarly volatility of the underlying asset can also affect participation rate. If the volatility of the underlying asset is lower, consequently the option will have lower value and ultimately a higher participation rate. Cheaper options embedded in the structured products also result in high participation rate. For example, exotic options are generally cheaper than plain vanilla options. Therefore, now a day exotic options are generally embedded in the structured product which increase the participation rate and can result in higher payoffs at the end.

20 19 3 Understanding Options Options are classified into plain vanilla and exotic options. Plain vanilla options are standard options while the exotic options are complex in nature. The complex options have low prices as compared to the standard options. Therefore, exotic options are generally embedded in structured products, which also make the products interesting for investor s point of view. Some examples of exotic options are barrier, chooser, look back, Asian, Himalayan and basket options. Their detail will be discussed later on. If the price of an embedded option/ options is lower, then the participation rate will be higher and more payoff for the investor. 3.1 Exotic Options An option whose characteristics, including strike price calculations/ determinations, payoff characteristics, premium payment terms or activation/expiration mechanisms vary from standard call and put options or where the underlying asset involves combined or multiple underlying assets are called exotic options (Das2001, p718). Exotic options are also called thirds generation risk management products. Although it is hard to classify all the options, but they can be roughly divided into five to six categories Path dependent options In path dependent options the final payoff depends on particular path that asset prices follow over their life rather than asset s value at expiration. The path of the underlying determines payoff and structure of the options. Path dependent options are further divided into weak and strong path dependent options. In strong path dependency, the payoff depends on some property of the asset price path along with the value of the underlying at present moment of time and some other extra variable (Wilmot 2007, p252). Examples of strong path dependent options are Asian options Asian options are examples of strong path dependent options. In Asian options the payoff is determined by comparing the strike/ spot price of the underlying with the 10 Das divided exotic options into five classes while Wilmott into six categories.

21 20 average value of strike/ spot price during a specific period of time. They are strongly path dependent because their value prior to expiry depends on the path taken and not just where they have reached. Asian options were originated from Tokyo office of the bankers trust in Asian options are normally cheaper than the plain vanilla options because averages are less volatile and therefore less risky. Average can be calculated by means of arithmetic, or geometric average of the prices. In Asian options There is a specific period over which the prices are taken. End of the averaging interval can be shorter than or equal to the options expiration date, the starting value can be any time before. In particular, after an average option is traded, the beginning of the averaging period typically lies in the past, so that parts of the values contributing to the average are already known. The market generally uses discrete sampling, like daily fixing. Weighting different weights may be assigned to the prices to account for a nonlinear, i.e. skewed, price distribution The wide range of variations covers also the possible right for early execution. Asian options are popular in risk management for currencies and commodities because they provide protection against rapid price movements or manipulation in thinly traded underlying at maturity, i.e. reduction of significance of the closing price through averaging. These options reduce hedging costs because they are cheaper than standard options. Average Price Options can be used to hedge a stream of (received) payments (e.g. a USD average call can be bought to hedge the ongoing EUR revenues of a US based company). Different amounts of the payments can be reflected in flexible weights, i.e. the prices related to higher payments are assigned a higher weight than those related to smaller cash flows when calculating the average. With Average Strike Options the strike price can be set at the average of the underlying price which is a helpful structure in volatile or hardly predictable markets. An average price call pays (A T K) +, where A T denotes the geometric or arithmetic average price of the underlyings ti. The geometric average of the underlying can be calculated as Geometric Mean = n n i=0 S ti (4)

22 21 And the arithmetic or the simple average can be calculated as A T = 1 n n i=0 S t i (5) Week path dependency means that the option depends only on the underlying price and the time. Barrier options are examples of week path dependent options. The payoff in these options depends on if the underlying hits a pre specified value at some time before expiry Lookback options In these options the purchaser has the right at expiration to set the strike price of the options at the most favorable price for the asset that has occurred during a specified time. In a lookback call option, the buyer can choose to buy the underlying asset at the lowest price that has occurred over a specified period, typically the life of the option. Details about lookback options can be found in Fx options and structured products by Uwe Wystrup Ladder options The strike price in these options is periodically reset based on the underlying evolution of the asset price. A ladder option can be identical to lookback when the amount of reset is set to infinity Barrier options Barrier options are weekly path dependent options. Das also classified them as limit dependent options because their payoff depends on the realized asset path via its level. Certain aspects of the contract are triggered if the asset price becomes too high or too low. For example, an up- and out call option pays off the usual max (S-K, 0) at expiry unless at any time previously the underlying asset has traded at a value Su or higher. It means if the asset reaches this level then it is said to knock out and become worthless. The option can also be knocked in instead of Knock out, where the payoff is received only if the level is reached (Wilmott 2007, P288). Barrier options can be divided into two types (out option and in option) i.e. up- and out, down- and- out, upand- in and down- and- in.

23 22 The out option pays off only if a specified level is not reached. Otherwise the option is said to have knocked out and becomes worthless. The in option pays off as long as the level is reached before expiry. If the barrier is reached then it is said to have knocked in. In options contracts starts worthless and only become active when the predetermined barrier is reached. If the barrier is set above the initial asset value then it is said to have an up option and if the barrier is set below the initial asset value then it is said to have down option Barrier options generally are of American style. It means that the barrier level is active during the entire duration of the option: any time between today and maturity the spot hits the barrier, the option becomes worthless. If the barrier level is only active at maturity the barrier option is of European style and can in fact be replicated by a vertical spread and a digital option. Apart from a lower or an upper barrier, double barrier options are also available. Double barrier options have both upper and lower barrier. In double out option the contract becomes worthless if either of the barriers is reached. In a double in option one of the barriers must be reached before expiry, otherwise the option expires worthless. In some cases a so called rebate is paid if for example in an out option the barrier level is reached. The rebate may be paid as soon as the barrier is triggered or not until expiry. The above mentioned barrier options are standard in nature. The barrier options can also have exotic type features for example resetting of barrier, outside barrier options, soft barriers and Parisian options. Detail discussion can be found in Wilmott 2007, p Time dependent options In time dependent options the buyer has the right to nominate a specific characteristic of the option as a function of time for example the expiration of the option. Preference or chooser option is an example of time dependent option. In a chooser option, at a predetermined date (normally after commencement and before expiry) the buyer can choose if the contract should be a call or a put option. Bermudan options are also example of time dependent options, where early exercise of the option is possible on certain dates or periods.

24 Multifactor options In multi factor options, the payoff depends on the relationship between multiple assets. It means there is second source of randomness such as a second underlying asset. Compound, basket, exchange, quanto, rainbow are the examples of multifactor options. In Compound options (options on options), the holder has the right but not an obligation to buy or sell another predetermined options at a pre agreed time. The compound options can be a call on a call, a put on a put, a call on a put and put on a call. Compound options have two strike prices and two exercise dates. For examples in a call on a call option, on the first exercise date T1, the holder of the compound option is entitled to pay the first strike price K1, and receive a call option. This call option gives him the right to buy the underlying asset for the second strike price K2 on the second exercise date. The compound option will be exercised on the first exercise date only if the value of the option on the date is greater than the first strike price. In basket options the payoff is based on the cumulative performance of the underlying assets and in exchange options the holder has the right to exchange one asset for another. The underlying assets can be individual stocks or stock indices, currencies or commodities etc. If the payoff is determined on performance of maximum or minimum of two more underlying assets, then these option are named as Rainbow option. A quanto option can be any cash-settled option, whose payoff is converted into another currency at maturity than that of the underlying asset at a pre-specified rate, called the quanto factor. There can be quanto plain vanilla, quanto barriers, quanto forward starts, quanto corridors, etc. 3.5 Payoff modified options These options entail adjustment to the linear and smooth payoffs that are associated with conventional options (Das 2001, p723). Examples include Digital options: Digital options have discontinuous payouts irrespective of the normal options whose payoffs are smooth. In a normal option, if it is further in the money the higher the payout to the purchaser. While in digital option the payout is normally fixed provided if certain conditioned are met. For examples, in the typical structure of digital option, if the strike price is reached the payouts

25 24 are fixed predetermined amounts no matter how much the option is in the money. Contingent premium: A contingent premium option is basically a European option. The premium will be paid to the writer if the contingent premium option finishes "in the money". Otherwise, if the option expires "at the money" then, no premium will be paid. It means no premium is paid in the beginning of the contract and is due at expiration of the options only if it expires in the money. In other words, the contingent premium structure is a combination of a conventional option and a digital option. Power options: A power option is a derivative with payoff depending on the asset price at expiry raised to some power α, where α is higher than 1.

26 25 4 Option Pricing Theory This section will discuss the underlying concepts in Black & Scholes option pricing theory. For example, the assumption of model, stochastic process, Markove property, generalized Wienner process, geometric Brownian motion, Ito s lemma, risk neutral evaluation and finally the Black & Scholes option pricing formulae are discussed. 4.1 Assumptions The famous Black and Scholes (B&S) model has several underlying assumptions like other option pricing models. The understanding of these assumptions will help to analyze the advantages and the drawbacks of the model. The underlying assumptions are discussed below 1. The markets are efficient i.e. the markets are assumed to be liquid. There is price continuity. The markets are fair and provide all information to all the players. It means no transaction costs in buying or selling stock or options. 2. The underlying is perfectly divisible and short selling is allowed. A seller who does not own a security will simply accept the price of the security from a buyer and will agree to settle with buyer on the same future date by paying him an amount equal to the price of the security on that date 3. There are no costs of carrying to the commodity (evaporation, obsolescence, insurance etc) and that the commodity is held for speculative purposes like a stock. 4. The commodity price, firm value and the interest rate follow continuous time diffusion processes. It means that the interest rate is known and it is constant (risk free) through time (Schwartz 1982). In other words there exists a risk free security that returns $1 at time T when 1e -r(t-t) is invested at time T. 5. The stock/commodity price follows a random walk in continuous time (geometric Brownian motion) with a variance rate proportional to the square of the price. Thus the distribution of the possible stock/ commodity price at the end of any infinite interval is log normal and the variance rate of the return on the stock/ commodity is constant 6. The model deals with European style options only that can be exercised at maturity only.

27 Stochastic Process We know that the prices of commodities, stocks and interest rates etc change over time in the financial markets. If the change in value is uncertain over time i.e. if the change in price of a commodity, equity or currency exchange is unpredictable over time then this kind of price behavior is called a stochastic process. In other words any variable whose value changes over time in an uncertain way is said to follow a stochastic process (Hull 2008, p 259) and it can be discrete time and continuous time stochastic process. In a discrete time process the value of a variable is assumed to change at fixed time intervals of time, while changes can take place at any time in a continuous time stochastic process Properties of a stochastic process The Markov Property A stochastic process is said to have the Markov property, when only the present value of a variable is relevant to predict its future value (Hull 2008, p 259) i.e. the process has no memory beyond where it is now. It means that the past history of that variable and pattern of changes in value would be irrelevant to predict future prices. So it means that to predict the future price of a commodity bundle, the only relevant price will be the today s price and it will be independent of its price during the last week or year Wiener Process Wiener process is a particular type of Markov process which has a mean change of zero and a variance rate of 1.0 per year. Wiener process is also called Brownian motion (named after a Scottish botanist Robert Brown). Brownian motion has been used in physics to describe the motion of the particle that is subject to a large number of small molecular shocks. It is among the simplest type of continuous stochastic process. In mathematical finance, this concept was first used by Louis Bachelier during the 1900 in his PHD thesis, where he presented the stochastic analysis of stock and option markets A variable Z follows a Wiener process if it has the following two properties (Hull 2008, p261) 1. The change Z during a small period of time t will be

28 27 Z = ε t (6) Here, ε has a standardized normal distribution with mean zero and standard deviation of 1, that is; N (0, 1). 2. The values of Z for any two different short interval of time t are independent. It means that Z has independent increments and Z 1 is independent of Z 2 if t 1 does not overlap with t 2. The first property shows that Z has a normal distribution with i.e. Mean of Z = Z (t) Z (0) = 0 Standard deviation of Z = t And variance of Z = t The Wiener process is both the Markov and Martingale process (zero drift stochastic process). By martingale process, it means that the expected value at any future time is equal to its value today. Martingale property is an important part of the risk neutral evaluation Generalized Wiener Process It is clear from the Wiener process that if we choose it as a model then the stock/ commodity price can take negative values at any point in time with a probability of 0.5 and it will have a constant zero mean and it is not an ideal model to price stock prices. So we have to consider a better model called Generalized Wiener Process. The basic Wiener Process also states that Z has a zero drift rate and a variance rate of 1.0. Zero drift means that the expected value of Z at any future time is equal to its current value and the variance rate of 1.0 means that the change in a time interval of length T equals T. Here we consider a discrete time random walk X 0 = x, X i = X i-1 + a t + b t ε i where ε i ~ N (0, 1) And the increments are given by X i = a t + b t ε i

29 28 Here a is the constant drift rate and b is the volatility rate. When we take smaller and smaller time steps t, then the above equation can be written as X (t) = x + a t + b Z (t) (7) The above equation is called the Generalized Wiener Process. The above equation can be written in the differential form as follows dx = a dt + b dz (8) So for a stock price we can conclude that Its expected percentage change in a short period of time remains constant, not it s expected absolute change in a short period of time. The size of the future stock price movements is proportional to the level of the stock price Geometric Brownian motion Generalized wiener process fails to capture a key aspect of the stock / commodity prices i.e. the percentage return required by the investors is independent of the stock price. It means that the investor will demand the same return, does not matter if the stock price is DKK 10 or DKK 100. So, the assumption of constant expected drift rate needs to be replaced by the assumption that expected return (i.e. expected drift divided by the stock price) is constant. So, if P is the price of a commodity bundle at time t, then the expected drift rate in P should be assumed to be µp for some constant parameter µ. It means that the expected increase in P (in a short period of time t) is µp t. The parameter µ is the expected rate of return on the price, expressed in decimal form. If the volatility of the commodity price is zero, then the model implies that P = µ P t And as t approaches to zero then dp = µp dt or dp P = μ dt (9) Integrating the equation (7) between 0 and time T, we get