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Regression-Based Monte Carlo for Pricing High-Dimensional American-Style Options

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Regression-Based Monte Carlo for Pricing High-Dimensional American-Style Options Regression-Based Monte Carlo For Pricing High-Dimensional American-Style Options Niklas Andersson [email protected] Ume˚aUniversity Department of Physics April 7, 2016 Master's Thesis in Engineering Physics, 30 hp. Supervisor: Oskar Janson ([email protected]) Examiner: Markus Adahl˚ ([email protected]) Abstract Pricing different financial derivatives is an essential part of the financial industry. For some derivatives there exists a closed form solution, however the pricing of high-dimensional American-style derivatives is still today a challenging problem. This project focuses on the derivative called option and especially pricing of American-style basket options, i.e. options with both an early exercise feature and multiple underlying assets. In high-dimensional prob- lems, which is definitely the case for American-style options, Monte Carlo methods is advan- tageous. Therefore, in this thesis, regression-based Monte Carlo has been used to determine early exercise strategies for the option. The well known Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) has been implemented and compared to Robust Regression Monte Carlo (RRM) by C.Jonen (2011). The difference between these methods is that robust regression is used instead of least square regression to calculate continuation values of American style options. Since robust regression is more stable against outliers the result using this approach is claimed by C.Jonen to give better estimations of the option price. It was hard to compare the techniques without the duality approach of Andersen and Broadie (2004) therefore this method was added. The numerical tests then indicate that the exercise strategy determined using RRM produces a higher lower bound and a tighter upper bound compared to LSM. The difference between upper and lower bound could be up to 4 times smaller using RRM. Importance sampling and Quasi Monte Carlo have also been used to reduce the variance in the estimation of the option price and to speed up the convergence rate. 1 Sammanfattning Priss¨attning av olika finansiella derivat ¨ar en viktig del av den finansiella sektorn. F¨or vissa derivat existerar en sluten l¨osning, men priss¨attningen av derivat med h¨og dimensionalitet och av amerikansk stil ¨ar fortfarande ett utmanande problem. Detta projekt fokuserar p˚aderivatet som kallas option och s¨arskilt priss¨attningen av amerikanska korg optioner, dvs optioner som b˚adekan avslutas i f¨ortid och som bygger p˚aflera underliggande tillg˚angar.F¨or problem med h¨og dimensionalitet, vilket definitivt ¨ar fallet f¨or optioner av amerikansk stil, ¨ar Monte Carlo metoder f¨ordelaktiga. I detta examensarbete har d¨arf¨or regressions baserad Monte Carlo anv¨ants f¨or att best¨amma avslutningsstrategier f¨or optionen. Den v¨alk¨anda minsta kvadrat Monte Carlo (LSM) algoritmen av Longstaff och Schwartz (2001) har implementerats och j¨amf¨orts med Robust Regression Monte Carlo (RRM) av C.Jonen (2011). Skillnaden mellan metoderna ¨ar att robust regression anv¨ands ist¨allet f¨or minsta kvadratmetoden f¨or att ber¨akna forts¨attningsv¨arden f¨or optioner av amerikansk stil. Eftersom robust regression ¨ar mer stabil mot avvikande v¨arden p˚ast˚arC.Jonen att denna metod ger b¨attre skattingar av optionspriset. Det var sv˚artatt j¨amf¨ora teknikerna utan tillv¨agag˚angss¨attet med dualitet av Andersen och Broadie (2004) d¨arf¨or lades denna metod till. De numeriska testerna indikerar d˚aatt avslutningsstrategin som best¨amts med RRM producerar en h¨ogre undre gr¨ans och en sn¨avare ¨ovre gr¨ans j¨amf¨ort med LSM. Skillnaden mellan ¨ovre och undre gr¨ansen kunde vara upp till 4 g˚angermindre med RRM. Importance sampling och Quasi Monte Carlo har ocks˚aanv¨ants f¨or att reducera variansen i skattningen av optionspriset och f¨or att p˚askyndakonvergenshastigheten. 2 Niklas Andersson April 7, 2016 Contents 1 Introduction 5 1.1 Options . .5 1.2 Pricing Options . .6 2 Theory 7 2.1 Monte Carlo . .7 2.1.1 Kolmogorov's strong law of large numbers . .7 2.1.2 Monte Carlo simulation . .7 2.1.3 Central Limit Theorem . .8 2.1.4 Error estimation . .8 2.1.5 Advantages of Monte Carlo . .9 2.2 Dynamics of the stock price . .9 2.3 Monte Carlo for pricing financial derivatives . 10 2.4 Robust Regression . 13 2.5 Duality approach . 14 2.6 Variance reduction . 15 2.6.1 Importance sampling . 16 2.6.2 Importance sampling in finance . 16 2.7 Quasi Monte Carlo . 18 2.7.1 Discrepancy and error estimation . 18 2.7.2 Sobol sequence . 19 2.7.3 Dimensionality Reduction . 20 3 Method 21 3.1 Algorithms . 22 3.1.1 LSM and RRM . 22 3 Niklas Andersson April 7, 2016 3.1.2 Duality approach . 23 4 Results 24 4.1 LSM vs RRM . 24 4.1.1 Duality approach . 29 4.2 Quasi MC . 31 4.3 Importance Sampling . 34 4.4 Combinations . 36 5 Discussion 37 6 Appendix A1 39 6.1 Importance sampling example . 39 6.2 Not only in-the-money paths . 40 7 References 40 4 Niklas Andersson April 7, 2016 1 Introduction After the financial crisis that started in 2007, the derivatives markets have been much criticized and many governments have introduced rules requiring some over-the-counter (OTC) derivatives to be cleared by clearing houses[1]. This thesis has been performed at Cinnober Financial Technology. Cinnober is an independent supplier of financial technology to marketplaces and clearinghouses. When more and more complex derivatives are brought to the market the requirements of the clearing houses to clear and evaluate them increases. Therefore, the pricing of different derivatives is an important part of the financial industry today. A financial derivative is a financial instrument that is built upon a more basic underlying variable like a bond, interest rate or stock. This project focuses on the derivative called option and specifically the pricing of American-style multi-asset options using regression-based Monte Carlo methods. 1.1 Options There are two different kinds of options, call options and put options. Buying a call option means buying a contract that gives the right to buy the underlying asset at a specified date, expiration date T, for a specified price, strike price K. This is also referred to as taking a long position in a call option and for this you pay a premium. If one instead takes a short position in a call option you have the obligation to sell the underlying asset at the expiration date T for the strike price K and for this obligation you receive the premium. On the other hand, a long position in a put option gives the right to sell the underlying asset whereas the short position has the obligation to buy the underlying asset, see Table 1. Table 1: Explanation of the states Long/Short in a Call/Put option. Long: Gives the right to buy the underlying asset. Call # premium Short: Obligation to sell if long position choose to exercise. Long: Gives the right to sell the underlying asset. Put # premium Short: Obligation to buy if long position choose to exercise. The difference between an option and future/forward contracts is that the holder of a option has the right, but not the obligation, to do something whereas the holder of a future/forward contract is bound by the contract. Let's take a call option to buy a stock S(t) with strike prize K and expiration date T as an example. If the stock price S(T ) exceeds the strike price K in the future time T > t the holder can exercise the option with a profit of S(T ) − K. If instead the stock price S(T ) is less or equal to the strike price K the option is worthless for the holder. The payoff of a call option, C with strike price K and expiration date T , is thus, C(S(T ); K; T ) = maxf0;S(T ) − Kg: (1) Options that only can be exercised at the expiration date T are called European options and options that can be exercised at any time are called American options. There are also options whose value depend not only on the value of the underlying asset at expiration but on the whole path of the asset. The barrier option is an example of this kind of option. It either becomes active or stop being active if the underlying asset hits a pre-determined barrier during the lifetime of the option. A knock- in barrier option becomes active only if the barrier is crossed and knock-out barrier option pays nothing if the barrier is crossed. The payoff of a down-and-in barrier call option with strike 5 Niklas Andersson April 7, 2016 price K, maturity date T and barrier H is, C(S(T ); K; H; T ) = 1S(t)≤H · maxf0;S(T ) − Kg; (2) where 1S(t)≤H is 1 if S(t) ≤ H at any time in the interval t=[0,T] and 0 otherwise. There are also options on multiple underlying assets, these are called basket options. The payoff of a basket option is typically a function of either the maximum, minimum, arithmetic average or geometric average of the assets prices, see Table 2. Table 2: Different payoffs of multi-asset Call and Put options with strike price K and D number of assets. Type Call Put Maximum: maxf0; max(s1; :::; sD) − Kg maxf0;K − max(s1; :::; sD)g Minimum: maxf0; min(s1; :::; sD) − Kg maxf0;K − min(s1; :::; sD)g QD 1=D QD 1=D Geometric average: maxf0; ( i=1 si) − Kg maxf0;K − ( i=1 si) g PD PD Arithmetic average: maxf0; ( i=1 si)=D − Kg maxf0;K − ( i=1 si)=Dg 1.2 Pricing Options Pricing European options is often done using the well known Black-Scholes-Merton model.
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