Validation of New Pricing Model for Exotic Options Pioneering Again

Final Year Report Sept 2010 Validation of New Pricing Model for Exotic Options Pioneering Again Trainee: Dima HAMZE Tutor: Nadia ISRAEL Responsible: Alain CHATEAUNEUF

Acknowledgements I would like to take this opportunity to pay tribute and express my deepest appreciation to everyone who was by my side till the completion of my graduate studies. To Professor Alain CHATEAUNEUF, I attribute the level of my Masters degree to your encouragement, effort and advices. One simply could not wish for a better and friendlier responsible. I am heartily thankful for being by my side. In my daily work I have been blessed with helpful and cheerful group of colleagues whose contribution in assorted ways to the research and the making of this paper deserves special mention. It is a pleasure to convey my gratitude especially to Nadia ISRAEL without you this paper, would not have been completed on due time. Moreover, in this humble acknowledgment, collective and individual thanks go to you all FXO team members. Words fail me to express my appreciation to my family and their non-ending attention and prayers. I owe you my deepest gratitude for providing me with the moral support I required throughout all my studies. My Father, Ghassan HAMZE, in the first place is the person whose encouragement, guidance and support from the initial to the final step of my university studies helped in shaping the person I am now. My Mother, Hala TABSH, whose persistent confidence in me ever since I was a child always elevated my self-esteem during the hardest times. Nadine, Razan and Nour, you made your blessings available in a number of ways thanks for being supportive and caring sisters. Finally, I gratefully acknowledge all my professors who supported me in any aspect and contributed to the successful realization of my paper and the completion of my internship. Last but not least I am indebted to many of my friends who were a great source of encouragement and assistance along my academic journey. Page 2

Contents Acknowledgement Outline I. About Murex, FXD and Basic definition A. General Presentation B. Brief Description of FXD Market a. Quotations and conventions of the FOREX Market b. Main traded exotic products of the FOREX Market C. Volatility and Smile Construction and Particularity of the FX Market a. Volatility b. Some Sensitivity measure i.delta ii.vega iii.volga iv.vanna c. Smile i. Strike scale ii. Delta scale iii. Smile construction in FXO d. Pricing models and volatility i. Local ii. Stochastic Page 3

iii. Hybrid II. Pricing models for Exotic options A. Existing Pricing models and tools for FX Options a. Replication Models (Skew model) b. Diffusion Models (Heston Model) c. Hybrid Model (Tremor I) B. Need for a more accurate model to price the partial barriers (Tremor II) III. Validation of Tremor II Model (To be continued) IV. Conclusion (To be continued) Bibliographies Page 4

A. General Presentation Murex, established in 1986, is a software company specialized in developing integrated front to back office financial software for the majority of derivative and physical assets, including but not limited to Interest rates, Foreign Exchange, Commodities, Equities, and Credit derivatives. With London constituting the largest concentration of users, Murex systems equip trading rooms of approaching 100 major financial institutions worldwide. To mention, the leading clients include ABN-AMRO, HSBC, J.P. Morgan Chase, etc.... Every day, thousands of users in banks, asset managers, corporations and utilities rely on Murex people and Murex solutions to support their capital markets activities across asset classes. For many years Murex has consistently been recognised as a solid leader in software development for trading, risk management and processing. Murex mainly focuses on five segments of clients: 1) Global banks that are the leading market makers. 2) Processing centres that are financial institutions that implement multi-business and multientity horizontal processing platforms. 3) Integrated trading and processing solutions.those are financial institutions looking for capital markets solution. 4) Asset managers and hedge funds. 5) Enterprise risk management that are the risk departments. Besides, Murex serves other segments of clients that include treasury centres of corporations and utilities. More than 800 staffs work with dedication and passion in Paris, New York, Beirut, Tokyo, Singapore, Dublin, Luxembourg, London, Beijing and Sydney. The front line interaction with clients, design testing and quality assurance, support and implementation. Moreover, the technical/architectural teams converge to facilitate clients requirements at the cutting edge of financial markets. They are represented by the financial consultancy activity that includes highly skilled financial professionals. It covers working with clients through the life cycle of Murex solutions that goes as follows: Pre-sale analysis and presentation, design and testing, implementation and configuration, and system extension. Murex s motto Pioneering Again sums it all up. Murex has always proved itself as a pioneering company that escort capital markets revolutions by offering innovating software solutions to the financial industry. Page 5

B. Brief Description of FXD Market a. Conventions and Quotations of the FOREX Market Some symbols: The Foreign Exchange options market is one of the largest and most liquid financial markets. This market has some particularities that will be discussed in the context of this report. Page 6

Exchange rates quotation: The exchange rates are quoted as (XXX-YYY) = (FOREIGN-DOMESTIC) = (DEN- Numéraire) = (UNDERLYING-BASE) The exchange rate is the quantity needed of domestic currency to buy one single unit of foreign currency. For example, EUR/USD currency pair is quoted as EUR-USD. Based on the explanation above, USD is the domestic currency and EUR is the foreign currency. It is worth to mention here that the term domestic is not related to the country of trading and that the slash (/) does not mean a division. It just symbolizes the currency pairs. Currency pair Default quotation Sample quote GBP/USD GBP-USD 1.8000 GBP/CHF GBP-CHF 2.2500 EUR/USD EUR-USD 1.2392 EUR/GBP EUR-GBP 0.9600 EUR/JPY EUR-JPY 135.00 EUR/CHF EUR-CHF 1.5500 USD/JPY USD-JPY 108.25 USD/CHF USD-CHF 1.2800 Exchange rates are usually quoted with five figures; taking the EUR-USD example again; an observed quote can be 1.2392. The last digit 2 is called the pip, and the middle digit 3 is called the big figure. Consider an exchange rate of 108.25 for USD-JPY pairs so an increase by 20 pips will give us a rate of 108.45 and a rise by 2 big figures will give us a rate of 110.25. Quotation of option prices: There are six ways to quote the Values and prices of vanilla options The Black-Scholes formula quotes d pips. The others can be computed using the following formulas. Page 7

Table taken from FX Options and Structured Products book by Uwe Wystup b. Main Exotic Options First generation of exotics Second generation of Third generation of exotics exotics Barriers Asians Accumulators Touch Baskets Tarns Best of/worst of Compounds KIKO Choosers Double average rate option Volatility Products Multi barrier In this paper, we will mainly focus on barrier options. Barrier options are vanilla options that either come to existence or are terminated when a barrier level is reached. There are two types of barriers: 1) Simple barriers that have one barrier 2) Double barriers that have two barriers. Each kind has its own sub-types listed in the table below. Page 8

Simple barrier Up and In Up and Out Down and In Down and Out Double barrier Knock In Knock Out In Options: If the barrier level is reached (when we hit the barrier) a simple option is obtained with the same characteristics as the barrier. Out Options: If the barrier level is reached (when we hit the barrier) the option is terminated and a rebate is obtained if specified. In Options If the barrier is reached during the option s life, the option is exercised. The exercise creates a new simple option at barrier s knock-time. If the barrier is not reached up to maturity, the option is early terminated and a Rebate is paid if specified. Out Options If the barrier is reached during the option s life, the option is early terminated After early termination; the option is dead and a rebate is paid if specified. If up to maturity the barrier is not reached; the option is either exercised if it was ITM or expired if it was OTM. Page 9

Simple barrier Option behaviour throughout its life: Example of Up and Out call : 1) When the spot goes beyond the Strike at a level far from the barrier, the option s Market Value (MV) increases until it gets close to the barrier. 2) Near the barrier, the MV decreases rapidly. 3) The MV becomes zero when the Spot hits the barrier. This barrier feature can be applied over specified time of the option s life rather than over the whole life. This application changes the option s style in to a window barrier according to the following criteria: 1) European Option: The window start date =window end date=maturity date. 2) American Option: The window start date =trading date and The window end date =maturity date. 3) Partial barrier option that includes 3 subtypes: The forward/late start: window start date> trading date and window end date=maturity date. The early end: window start date =trading date and window end date < maturity date. The mixed: window start date> trading date and window end date< maturity dates. C. Volatility and Smile Construction and Particularity of the FX Market a. Volatility Volatility usually refers to the standard deviation of the continuously compounded returns of a financial instrument over a specific time horizon. It is often used to quantify the risk of the Page 10

instrument over that time period. It is a main input parameter to compute the value of the option. For Non liquid options(no market quotes available), historical volatility is used: The historical volatility for an asset relates to a past period of time where volatility is observed at. For example we look at the figures for the past week, for the past month, and so forth. How to calculate the Historical Volatility? Step 1: Create a sequence of log-returns Step 2: Compute the average: Step 3: Compute the Variance: Step 4: Compute the standard deviation: This historical volatility does not have any predictive capabilities. On the other hand, for Liquid options (Market quotes available and quoted by traders, brokers and real time data providers), implied volatility can be used: Implied volatility is the calculated value of volatility by inverting the relevant option-pricing model.the prices are the inputs, observed on the market and the volatilities are the implied outputs. Those inputs reflect the market participants view and their predictions of the future. Therefore, instead of using the model to solve for the option's price, it is used to solve for the Page 11

option's volatility curve. By having this curve implied from several market quotes, other nonquoted options can be priced. b. Main Sensitivity measures discussed in the scope of this report: i. Delta: Is the first order derivative of the option price for a 1-percentage point increase in the spot. In FX market the Moneyness of vanilla options is expressed in terms of delta. Generally speaking, a 50 delta corresponds to an ATM option (K=S).More details on this subject will be tackled in the smile section. ii. Vega Vega is the change in the value of an option for a 1-percentage point increase in the implied volatility of the underlying asset price. Page 12

Cases Long option position ( Buying) ATM Further ITM or OTM Time Vega Always positive Greatest Vega Decreases Vega is greater for long dated options Page 13

iii. Volga: Volga measures second order sensitivity to implied volatility (the rate of change to Vega as volatility changes) iv. Vanna: Vanna is a second order derivative of the option value, once to the underlying spot price and once to the volatility. It is mathematically equivalent to Dvega/Dspot or Ddelta/Dvol. Vanna can be a useful sensitivity measure to monitor when maintaining a delta- or a Vega-hedged portfolio, as it will help the trader to anticipate changes to the effectiveness of a delta-hedge as volatility changes or the effectiveness of a Vega-hedge against change in the underlying spot price. c. Smile The Black-Scholes model assumes a constant volatility throughout. However, market prices of traded options imply different volatilities for different maturities and different deltas or strikes. The cumulative normal density function underestimates the probability of extreme occurrences. The implied distribution has fatter tail thus more probability for extreme occurrences, thus higher volatilities on the tails that leads to the existence of the smile. The volatility of exchange rate is far from constant. Page 14

Smile is defined as the plot of the options volatility as a function of its strike price or delta. It is the markets estimation of where the volatility will be should the market move. There are two common views of smile behaviour: i. Fixed (Sticky Strike) in which the volatility depends on the option strike for a certain maturity. Whatever the spot level is, for the same strike, one gets the same volatility.based on this assumption, the smile quoted in strike terms is indifferent to spot changes. If 2 FX options have the same strike they are priced with the same vol. ii. Floating (Sticky Delta) in which the volatility depends on the option delta for a certain maturity. This effectively centres the smile around the current spot/fwd value or the ATM. The implied smile floats with the spot level. Based on this assumption, the smile quoted in terms if delta is indifferent to the spot changes. (ATM volatility or 50 delta volatility is constant whatever the spot level). Page 15

With sticky delta we know that as the market moves a different volatility is applied to each strike and if 2 FX options have the same delta they are priced with the same vol. In FX Market, the smile is quoted on a delta scale, taking into account the most liquid points (5 points 10D put, 25D put, ATM, 25D call and 10D call). This kind of quotation allows: - Stickiness to the moneyness - Introduction of a common convention ( common language) - Replication of market behaviour/logic (An option with an ATM strike should not have the same volatility as an option with the same strike being OTM (out of the money). To highlight, the delta scale can be translated into a strike scale. It is an iterative process since: Delta= dc/ds and c = f (S,K,t,r, rf, vol). So for every strike we have a delta and for each delta we have volatility. iii. Smile construction in FXO To price accurately and be able to manage the risk, looking at the volatility smile is a must. FX derivative s market participants are confronted with the indirect observation of the smile in the market since it is built with specific input parameters that decompose it into symmetric part that reflects convexity (flies) and skew part (riskies) using Page 16

Risk Reversal (RR)=call put Strangle or Butterfly strategies(bf) =(call + put)/2 ATM options with strike = forward i.e. value of call= value of put. This is done on a delta scale following the market conventions for FXD. In summary, three volatility quotes are extracted from the market data for a given delta (Ex. Delta =+ or 0.25). ATM volatility. Risk reversal volatility for 25 delta. Strangle volatility for 25 delta. In terms of ATM volatility of call and put (0), OTM volatility of call (+) and put (-) we get This gives: Page 17

Smile Curve: d.pricing models and volatility The Black-Scholes starting point is that all options are valued at the same volatility, regardless of strike level. In smile world, Black Scholes considers the probable distribution of prices centred around the forward price (F) on the expiry date. Page 18

i. Local volatility models The most well known local volatility model is the binomial tree. The binomial tree with returns following the log distribution are used to calculate the local volatility that is the standard deviation of the returns at the node (n, i) Consider ln(x) evolves to ln(y) with probability p and to ln(z) with probability (1-p) Then E[lnX] = p ln(y) + (1-p) ln(z). And, V[lnX] = (standard deviation) ² Local volatility models expect the underlying price to be variable and the level of volatility to be wholly determined by the level of the underlying price Page 19

A local volatility model sees the probable distribution of volatility is 100% correlated with the level of the Forward price (see graphic). Another Local volatility model based on the diffusion process is Dupire Model which widely used in equity market. ii. Stochastic volatility Stochastic volatility models assume both the volatility and the stock price to follow a Brownian motion. Stochastic volatility models expect both the underlying price and volatility to be random. A Stochastic volatility model sees the probable distribution of (V) proportionate to the volatility of the Brownian variance (VoVol) and T, with the central point determined given a starting volatility (V0) and a terminal volatility (Θ) Page 20

iii. Hybrid, Universal or Mixed models are a mixture of the latter two A Stochastic/Local volatility hybrid model sees the probable distribution of (V) proportionate to the volatility of the Brownian variance (VoVol) and T, with the central point determined given a starting volatility (V0) and a terminal volatility (Θ) combined with a correlation between F & V. Page 21

Some notes on local and stochastic volatilities: -We observe that steep skew indicates locality. -The Correlation describes the skewness of the smile so: o o o If the correlation =0 the smile is completely convex and can be explained by stochastic volatility. If the correlation is different than 0 the smile is has higher risk reversals. If the correlation =1 the vovol is probably equal to 0 and the smile can be explained by local volatility. -The more local volatility is the less stochastic it is. II. Pricing models for Exotic options A. Existing Pricing models and tools for FX Exotics. To price exotics consistently with smile several models have been introduced: a. Replication Models (Skew model) Barrier and Touch options can be priced in the Black-Scholes framework (assuming constant volatility and rates throughout the life of the option). The Black-Scholes price uses only the ATM volatility at the expiry date of the option and doesn't take into account the smile curve. Because Barrier and Touch options are path-dependent products (the final payoff depends on the path followed by the spot price throughout the life of the option, not only on the value of the spot on maturity date), we cannot simply plug the smile volatility into the Black-Scholes price. One solution to get a better price for these exotic options would be to use a model where the volatility is not assumed constant throughout the life of the option (stochastic volatility model for example). Page 22

The Skew methodology allows traders to compute quickly an adjustment to the Black Scholes price of an exotic option that takes into account the presence of a smile curve. In principle, we start by measuring the B-S Vega, Vanna and Volga of the exotic option and then find the weights of the 3 liquid vanillas (25 delta and ATM call and put) with a smile off mode that will perfectly hedge the Vega, Vanna and Volga of the exotic option. Then we proceed by computing the smile cost of this portfolio (price with smile price with smile off).finally, we multiply this over-hedge cost by the probability of survival of our exotic option and add this adjustment factor to our B-S price of the exotic option. The shortcomings of this model are: It doesn t include the stochastic nature of volatility. It doesn t factor in the smile term structure thus it is incapable of adjusting for complex volatility moves. Doesn t replicate the market prices Thus, better techniques should be used to overcome those limitations. b. Diffusion Models (Heston Model) Heston Model is a stochastic volatility model that assumes both the underlying price and the volatility to follow a Brownian motion. It is an extension of the B-S model taking in to consideration the stochastic characteristic of volatility. Heston typology Page 23

Heston Models and FXO market. We concluded that Heston model did not replicate the market prices of FX option. Moreover, it doesn t take into account the well known correlation between the spot and the volatility and the spot and the risk reversals (also called smile dynamic). Therefore modelling the spot as a mixture between two models local and stochastic and monitoring the dynamics so that it matches spot with the correlation seems reasonable to replicate the market. d. Hybrid Model (Tremor I) Hybrid model is a mixture of the local volatility and the stochastic volatility model. It takes in to account the smile dynamics. The volatility used in the market ended up being neither completely stochastic nor completely local thus a logical guess would be a mix. Murex confirmed the aforementioned hypothesis and resulted in Tremor model that is a hybrid model used to price FX exotic options. The proportion of stochastic volatility is determined by the Cursor and the corresponding proportion of local volatility is deduced from the smile. The cursor explains the skewness and the convexity of the smile. Page 24

Tremor I Topology Tremor is a two factors model: The first diffusion process is the forward: it is a function of a stochastic volatility a local volatility represented by the quadratic function of the forward g(f). and The second process is the variance which is the volatility diffusion. The Kappa, also called the rate of mean reversion (controls the variance level over time prohibiting it from getting far from the mean).since tremor is calibrated per maturity pillar and doesn t take the term structure in to account; the kappa is considered as an input and is set to 1. The theta also called the long term variance requires a term structure to be calibrated so same as kappa it is set to the initial variance (actual volatility level). The volatility of volatility (vovol) describes the convexity of the smile. The more the convexity of the smile, the higher the vovol is. (More probability to be at the wings). The Correlation is already describes above. To recapitulate, we have six parameters to determine in order to match the five points of the smile: a, b, c, correl, vovol, and initial volatility. Page 25

This leaves us with six outputs and 5 inputs (5 vanilla prices of our calibration basket).so we need an additional input in order to calibrate the model. An extra input is introduced which is a factor that represents the proportion of stochastic volatility. This factor is called the cursor If the cursor =0% the model is purely local volatility one. If the cursor =100% the model is purely stochastic volatility one. The calibration is done in 3 steps (It will not be detailed due to confidentiality). We calibrate a reduced number of parameters of Heston parameters that fits 3 vanilla prices (10D s and ATM) then we apply the cursor on the vovol by that Heston doesn t explain the whole smile. We add the rest of the vanillas to our calibration basket to calibrate the local part. The local volatility will explain the rest of the smile. B. Need for a more accurate model to price the partial barriers (Tremor II) Tremor I didn t serve as a good tool to price accurately partial barriers. As a result, ameliorations should be done by introducing term structure. This is Tremor II the gap filler of tremor I and the tool to price all the barriers. This introduced term structure takes in to account the state of the market in the window time where the option is alive and where the impacts on the price should be taken in to account. The calibrated parameters will be functions of time (3 dimensions). Test plan is being elaborated to validate Tremor II for partial barriers. TO BE CONTINUED. Page 26

Bibliography: - Wystup, Uwe. FX Options and Structured Products. Chichester West Sussex, U.K: John Wiley and Sons, 2006. - Hull, John. Options,Futures and Other Derivatives. 6 th edition. Upper Saddle River, New Jersey: Pearson prentice hall, 2006. - Murex Internal documentation. Page 27