Sequential Monte Carlo Methods in Options Pricing

By: Jacinth Grace Leong Mei En

Supervisor: Dr Jasra Ajay

ST4199 Honours Project in Statistics

Department of Statistics and Applied Probability

National University of Singapore 2018/2019

Content Page

Acknowledgements

Summary

Preliminaries & Literature Review

Options and Financial Markets

European Call Options

Barrier Options

Black Scholes Model

Monte Carlo Methods

Conventions and Overview

Importance Sampling

Sequential Monte Carlo Methods

Monte Carlo Methods

Derivation

Properties of Estimate

European Call Option Approximation

Evaluation of Results

Barrier Option Approximation

Importance Sampling

Properties of Estimate

Importance Sampling for Barrier Options

Monte Carlo and Importance Sampling – A comparison

Sequential Monte Carlo Methods

Sequential Importance Sampling (SIS)

Formulation

Application to Barrier Options

Sequential Importance Resampling (SIR)

Formulation

Application to Barrier Options

Algorithm

Importance Sampling and SMC – A comparison

Conclusion

References

Acknowledgements

Firstly, I would like to express my sincerest gratitude to my supervisor, Dr Jasra Ajay, for having me as his honours student and for his instrumental guidance throughout the period of this project. This project has been a good reminder of why I chose to major in Statistics.

Secondly, I would like to thank my family for their patience and the sacrifices made throughout my course of study.

Thirdly, my friends – for pushing me and motivating me. It was tiring (to say the least) to juggle the many commitments, but they kept me going on days I felt I hit a wall, whether they know it or not.

Last but not least, the faculty members in NUS’ Department of Statistics and Applied Probability. For helping to build the foundation necessary to complete this paper, and for the challenging yet rewarding experience.

Summary

This thesis follows the approximation of option prices in finance using Monte Carlo methods (MC). The aim of this paper is to justify the use of Monte Carlo simulation in option pricing. Variance reduction techniques are employed as well, which proves useful in reducing the variance of more complicated derivatives. Substantial work has been done in pricing derivatives in the field of finance. However, problems arise when the specifications used to model the underlying asset is high dimensional. In such instances, numerical methods such as Monte Carlo methods are useful. Moreover, with the proliferation of computational power, Bayesian computation has become increasingly popular. This paper explores the various MC methods, evaluating the efficiency and practicality of some of these methods. Algorithm and simulation results in this paper were produced in R.

Chapter 1 serves as an overview as well as a literature review of past work done on this topic, as well as the theoretical framework behind the methods used.

Chapter 2 looks at European call options, and serves to convince readers of the accuracy of MC methods in pricing these options.

Chapter 3 delves into barrier options – a form of exotic options, and applies standard MC methods, followed by importance sampling (IS), then sequential MC (SMC). We also compare the variance of these methods and analyse the trade-offs between using the different methods.

Chapter 4 contains the concluding remarks of this paper and proposes possible future direction in this field.

Preliminaries ; Literature Review

Options and Financial Markets

Options are a form of financial derivatives that gives the buyer the right, but not the obligation to buy or sell an underlying asset for a strike price at a specified date. The underlying asset can be a variety of financial instruments – foreign exchange, equities, fixed income and even commodities. Options are seen as a powerful hedging tool, and is also used as speculation. Hedging helps to reduce risk exposure as holders can exploit negative correlations in payoffs to protect their positions. Speculation through options gives investors a less costly option when placing bets on the direction they believe the market is likely to move. Options are traded on either organized exchanges, such as Chicago Board Options Exchange (CBOE), or over-the-counter markets.

Options exist in many different forms (Calls and Puts, Vanillas and Exotics, American and European, etc.) – but this paper does not serve to shed light on the numerous types there are. Consequently, we look at the application of Monte Carlo methods in (1) European Call options, and (2) Barrier Options.

Hull, John C: options, Futures and other derivatives (2018) gives a good overview of options as well as some of the basic concepts behind the derivation of option prices. This paper applies Monte Carlo methods to European call options and European Barrier options, specifically down and out barriers.

European Call Options

European call options are a form of vanilla options that have a maturity T in which the holder has the right to exercise the option and buy the underlying asset, henceforth called stock, at a price K, the strike price. The payoff to the holder would then be

(ST-K)+=max?(0, ST-K)(4)

Intuitively, this means the payoff to the holder would either be 0 or ST-K . If the stock’s price at maturity exceeds the strike price, the holder’s payoff would be positive. If ST;K, the holder would choose not to exercise the option, resulting in a 0 payoff.

In pricing European call options, the Black-Scholes model reduces to

C0= S0Nd1- Xe-rTNd2(5)

d1=lnS0X+(r+ ?22)T?T(6)

d2= d1-?T(7)

C0 the current call option value, N(d) the cumulative normal distribution, or ?(z?d). Following the Black Scholes model’s assumptions, all assets follow correlated geometric Brownian motion processes with constant volatilities over the life of the option, and the risk-free rate is constant.

Barrier Options

Zero pay-out if H is crossed before T (ceases to exist) Can only be exercised if stock price crosses H before T

H < S0 Down and Out Down and In

H > S0 Up and Out Up and In

Barrier options are a form of exotic options that is weakly path-dependent, and the payoff is determined by whether or not the price of the stock crosses a barrier before the expiration. Our analysis will focus on European down and out barrier options.

From the Black Scholes Model, we have the estimate

e-rTEST-K)+k=1mIa,bStk= e-rT(ST-K)+k=1mIa,bstkpstkstk-1dst1…dsTIntuitively, this formula gives us the present value of the expected barrier option price. Over the lifetime of the option, it is possible for the option to be knocked out. What does this mean from a financial perspective?

Black Scholes Model

The ubiquitous Black-Scholes Model is a mathematical model of financial derivative markets. Economists Black and Myron Scholes first first came up with the partial differential equation (PDE)

?V?t+12?2S2?2V?S2+rS?V?S-rV=0 (1)

where V is the price of the option as a function of the underlying asset’s price S at time t, r is the risk-free rate, and ? is the volatility of the underlying asset. There are underlying assumptions to this model, and work has been done to relax these assumptions over time. Most approaches use (1) to compute the expected value of the discounted terminal payoff under a risk-neutral measure. Financial asset prices can be defined as a stochastic process {St}t?0, T, defined on a probability space (?, F, P). S0 is the initial price of the underlying asset, while T is the terminal time (expiration date) of the option.

From the Black-Scholes PDE a stock’s risk-neutral dynamics is described using the following stochastic differential equation (SDE):

d StSt=r dt+ ? dWt,Wt~N(0, T)(2)

where Wt is a standard Brownian motion process.d StSt can be interpreted as the returns on the underlying asset. By Itô’s formula, (2) reduces to

ST=S0expr-12?2T+?Wt=S0expr-12?2T+?TZ(3)

whereby the Geometric Brownian motion follows a lognormal density. Let Z~N(0,1), then TZ~N(0, T).

Monte Carlo Methods

Monte Carlo methods have gained popularity as a computational tool to evaluate the volume of a set by interpreting its volume as a probability. The first work done on using Monte Carlo methods to price options was by Boyle BOYLE (1976). The literature is a good early review on using Monte Carlo methods to price derivatives. Numerous work has been done in the evaluation of options price using Monte Carlo following this. CITE BOOKS ON OPTIONS PRICING. MC methods branch out into three problem classes – optimization, integration and generating draws from a probability distribution.

Moreover, after a finite number of simulations, we are able to get information on the likely magnitude of the error in the estimate from the Central Limit Theorem. GLASSERMAN (2003) In this paper, in order to value the options, we determine stochastic processes for the underlying asset before simulating these paths. In the pricing of many financial derivatives, problems may arise when evaluating the expected prices of the underlying asset for complex financial instruments, making the calculation of their estimates intractable. Monte Carlo methods provide a powerful and flexible approach in dealing with complex problems and their advantages are more distinct in higher dimensions (increased complexity of underlying problem).

Let X be a random variable with mean ? and variance ?2. Simple Monte Carlo method seeks to estimate the expected value of the random variable ?=EX by sampling N independently and identically distributed (IID) samples Xi, i=1, …, N from a probability density function (pdf) of X and taking their average,

?= 1Ni=1NXiwhere ? has mean ? and variance ?2, and is a consistent and unbiased estimator of ?.

Conventions and Overview

Unbiasedness and consistency of estimate

Suppose we have an estimate of the option price as

CN=1ni=1Nf(STi)(4)

And suppose that the true value is denoted C. By the law of large numbers, CN?C as n? ?. From this result, CN-C ~ N(0, ?fn) by Central Limit Theorem.

Importance Sampling

In this paper, we look at importance sampling GLASSERMAN (1999) ASYMPTOTICALLY OPTIMAL IS, as well as Sequential Monte Carlo methods (SMC) Sequential monte carlo methods for options pricing, Jasra Ajay. More work has been done on utilizing importance sampling in financial engineering, though the work done on SMC methods in option pricing is limited.

In using standard Monte Carlo methods, the estimate can be subjected to high variability in certain cases, resulting in a need for us to look for us to look into variance reduction techniques. Importance sampling (IS) is a well-known Monte Carlo method that seeks to reduce the variance in the estimate by allowing more “important” outcomes to have greater sway on the estimate. Below is a short outline of importance sampling.

Suppose we start with the problem of estimating a parameter ?,

?= Ef?X= S ?x fx dx = S ?x fx dxg(x)gx dx = Eg?X fX g(X)= Eg?XwXwhere wx=f(x)g(x) is a weighting function and f and g are pdfs on domain S. Our importance sampling estimator is then

?IS= 1Ni=1N?Yi fYi g(Yi)=1Ni=1N?Yi wYiwhere Yi ? i=1,…, N are IID samples from g. It is noted that ?IS is an unbiased estimator of ?. Proof? Therefore, the key to the variance reduction when employing importance sampling is the selection of the proposal density g(x). Specifically, sampling from the proposal distribution should be easy and the the estimator should have a reduced variance. Why? Pg 34, variance reduction techniques in pricing financial derivatives.

The variance of ?X fX gX with respect to the density gx is given by

Varg?X fX gX=Eg?X fX gX2-Eg?X fX gX2

= ?2x f2x g2(x)gx dx- ?2= ?2x fxf(x) g(x) dx- ?2 =Ef?2x fx g(x)-?2

While the variance of ?X with respect to the density fX is given by

Varf?X =Ef?2X -Ef?X 2=Ef?2X -?2From the two equations above, it is clear that in order to reduce the variance of our original estimator, fx g(x)?1. However, as noted in ROSS (2013) SIMULATION page 202, it is possible for Varg?X fX gX to become infinite even if the above condition is fulfilled.

Sequential Monte Carlo Methods

Sequential Monte Carlo (SMC) methods, also known as particle filters, are a form of Bayesian filtering methods. Mathematically, SMC methods can be seen as a form of Feynman-Kac models. Traditionally, Kalman CITATION Gob09 l 1033 (Gobet, 2009) filters allow users to recursively generate a sequence of posterior distributions in order to get an analytical solution. However, a caveat to this is that the data have to follow a linear Gaussian state-space models. SMC methods are a class of sequentially calculated IS and is appropriate for dealing with non-linear non-Gaussian state-space models.

DOUCET (2001) provides a comprehensive overview of SMC and its developments through the years.

Monte Carlo Methods

Derivation

Properties of Estimate

European Call Options

The present value of the payoff presented in section 2.1 would then have to be multiplied by a discount factor e-rT, where r is the continuously compounded interest rate. The expected present value of payoff would then be

Ee-rT(ST-K)+ (8)

and the MC estimate of the call option is therefore

e-rTNi=1N(STi-K)+(9)

where N is the number of simulations and ST(1), …, ST(N) are independent and identically distributed samples.

In order to convince ourselves of the accuracy of Monte Carlo methods before commencing to more complex problems, we have performed the simulation of the estimated call option price for various values of N, K, S0, T, r and ?, comparing our estimate to the Black-Scholes estimate as calculated using equations (5)-(7).

Evaluation of Results

Figure 1: Variation of option price with varying S0. Other parameters are k=Barrier Options

In order to employ Monte Carlo methods for barrier options, we discretize the time interval 0, T into m uniform subintervals each of length ?t=T/m, where each grid point ti=i?t, i= 0, 1, …, m. For convenience, let Si=Sti. From equation (3), we get

Sn+1= Snexpr-12?2?t+??t Zn, n=0, 1,…, m-1(10)

where Zn~N(0,1).

In order to approximate a down and out barrier call option, we use the estimate

e-rTNi=1N(STi-K)+k=1TIa,b(Stki)to estimate the barrier option price based on the standard Monte Carlo method. The algorithm for this is as below.

Importance Sampling

From the results in section 3.2, it is clear that despite the ability to estimate the price of barrier options, the estimate suffers from high variability, especially when the probability of being knocked out is high. This is due to

Properties of Estimate

Importance Sampling for Barrier Options

Monte Carlo and Importance Sampling – A comparison

Sequential Monte Carlo Methods

Sequential Importance Sampling

Sequential importance sampling (SIS) is a sequential version of the classical Bayesian importance sampling. SIS considers a space of increasing dimensions.

qnx1:n= qn-1×1:n-1qn(xn|x1:n-1) = q1x1 q1x1… qnxn| x1:n-1So if X1:n-1(i)~qn-1×1:n-1 then sampling from Xn(i)|X1:n-1i~qnxnX1:n-1i would give X1:n(i)~qn(x1:n). The importance weights are updated according to

wnx1:n= ?nx1:nqnx1:n = wn-1×1:n-1?n(x1:n)?n-1(x1:n-1)qn(x1:n-1)Formulation

Application to Barrier Options

Sequential Importance Resampling

Sequential importance resampling (SIR) is built upon SIS, with an additional resampling scheme to decide whether or not to resample based on the weights.

SMC Samplers are similar to SIR

Formulation

Application to Barrier Options

Algorithm

Importance Sampling and SMC – A comparison

Conclusion

References

Metropolis-Hastings Sampling

In Metropolis-Hastings sampling, samples mostly move towards higher density regions, but sometimes also move downhill. In comparison to rejection sampling where we always throw away the rejected samples, here CITATION Hul18 l 1033 (Hull, 2018)we sometimes keep those samples as well. Its pseudo-code is given below. MH is a markov chain monte carlo method

Algorithm for MH sampling

1: Init x(0)2: for i=1 to N-1 do

3: u~U0,14:x*~q(x*|xi)5: if u;min px*q(xi|x*)px*q(x*|xi) then

6: x(i+1)?x*7: else

8: x(i+1)? x(i)9:end if

10: end for

Remark: in line 5, if q is symmetric then q(xi|x*)q(x*|xi)=1. This term was later introduced to the original Metropolis algorithm by Hastings.

Bibliography

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