Multilevel Monte Carlo Methods for American Options

Transcription

1 Multilevel Monte Carlo Methods for American Options Simon Gemmrich, PhD Kellog College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance November 19, 2012

2 To my wife Miriam and my son Jakob

3 Acknowledgements It is a pleasure to thank my supervisor Prof. Mike Giles for pointing me in the direction of this fun project and for his thorough help and guidance while I was working on this thesis. This thesis would not have been possible without him. I thank my former employer d-fine GmbH for the opportunity to participate in the Oxford Math Finance programme. Above all I thank my wife Miriam for her endless support. ii

4 Contents 1 Introduction and scope 1 2 American options Option basics Pricing models for American options Free boundary formulation Linear complementarity formulation Optimal stopping formulation Stopping rules and parametric approximations Finite Difference θ-scheme for American options The Longstaff-Schwartz algorithm Multilevel Monte Carlo method The setup Basic multilevel decomposition Monte Carlo estimators The multilevel algorithm Complexity and convergence theorem Numerical implementation for an American put option under a preset exercise strategy Discretizing the SDE Exact path simulation for geometric Brownian motion Estimators for a given exercise strategy Numerical example A brief aside on Brownian bridge construction and Brownian interpolation Equidistant time grids Time discretization in transformed coordinates iii

5 5 Optimizing exercise strategies Parametric forms for the exercise boundary of an American put option Stochastic approximation The optimization algorithm Numerical examples Conclusion 48 A MATLAB code 50 A.1 MLMCAmOp.m A.2 MLMCAmOpLevelL.m A.3 mlmc AmOp Test.m Bibliography 60 iv

6 Chapter 1 Introduction and scope The best mathematics is like literature - it brings a story to life before your eyes and involves you in it, intellectually and emotionally. Herbert Ellis Robbins The correct pricing of American style options is still an active field of research within the quantitative finance world. Unlike for their European counterparts there exist no general closed form solutions for the value of American put and call options. Various approaches exist in the literature on how to solve the valuation problem for American options. They differ greatly in style and spirit. Some are based on optimal stopping formulations. Others are based on partial differential equation and free boundary formulations or are phrased as a linear complementarity problem. Some are purely numerical in nature and employ finite difference schemes or binomial/trinomial tree methods. Others put more emphasis on the analytical work. Some approaches end up working with integral equations, in which the early exercise boundary features as an unknown. One of the most heavily used techniques is the famous Longstaff-Schwartz algorithm, which is based on Monte Carlo ideas and which tries to approximate the early exercise boundary by a least squares regression of the continuation value onto a certain set of basis functions. The aim of this thesis is to show how the ideas from multilevel Monte Carlo (MLMC) methods can be used in the valuation of American options. The core idea is to use and adapt the multilevel Monte Carlo method introduced in [Gil08] and [Gil] to find the value of an American option under a given exercise rule and then optimize over all exercise rules from a feasible set. In the spirit of the above quote by H. E. Robbins we hope to tell a consistent story and validate it through numerical tests. We work with simple test cases, for which 1

7 existing solution techniques will most likely outperform our new MLMC approach. This thesis should be understood as a first step towards a new competitive pricing algorithm for American options in cases where established finite difference and other lattice methods fail. Ultimately, it should stand the test against other Monte Carlo like methods such as the Longstaff-Schwartz approach. Heavy theorem proving and a full numerical analysis are beyond the scope of this thesis. The thesis is structured as follows. In Chapter 2 we review some of the known facts and pricing approaches for American options, thereby setting the stage for the underlying problem of this thesis. Chapter 3 then introduces the multilevel Monte Carlo method by Giles and reviews some of its properties. Chapter 4 continues with the numerical implementation and its intricacies. Finally, Chapter 5 first deals with stochastic approximation algorithms and then ties the knot around the whole pricing algorithm and applies the algorithm to several test cases. 2

8 Chapter 2 American options In our view, however, derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal. Warren Buffet, 2002 In this chapter we list some well-known facts about American options and set the stage for their numerical treatment by multilevel Monte Carlo methods. We start by briefly discussing the necessary options jargon. Afterwards, we mention the most common mathematical pricing frameworks for American options including the optimal-stopping formulation on which our numerical algorithm is based. Moreover, we discuss the finite difference θ-scheme and how it can be used to approximate the price of an American option written on a single underlying. We will later use the finite difference method to compute benchmark solutions against which we hold our multilevel method. The chapter ends with a very brief presentation of the famous Longstaff-Schwartz algorithm. This chapter is by no means intended as an exhaustive treatment of option intricacies. A nicely written account of option trading strategies is for example [Sin10]. 2.1 Option basics A financial option is a contract between two counterparties. On one side of the contract there is the option holder (also called the owner of the option) and on the other side of the contract there is the option seller (also called the writer of the option). The option contract gives the option holder the right, but not the obligation, to buy 3

9 or sell an underlying asset for a specified price at a specific time. There is a variety of option types with different properties. If the option gives the owner the right to buy the underlying asset it is referred to as a call option, if it gives the option holder the right to sell the underlying asset it is called a put option. The predetermined price for the underlying asset is called the strike price or simply the strike. The option holder buys the contract from the option seller in exchange for an upfront payment, usually called the option premium. The fair value of an option is derived from the value of the underlying asset and specific option characteristics. One of the main characteristics of each option is its exercise type. If the option can only be exercised on a single date the exercise type and the option itself are called European, if it can be exercised on several dates the exercise type and the option are classified as Bermudan and if the option can be exercised at any time up until its final maturity, the exercise type and the option are referred to as American. If an American and a European option are identical except for their exercise types, then the American option must be worth at least as much as its European counterpart. After all, the holder of an American option can always choose not to exercise before expiry, thereby limiting his rights to mimic the European option. A simple no-arbitrage argument then guarantees that the American option is at least as valuable as the European one. We note that the inequality here is not strict. The value of an American call option written on an underlying without dividends turns out to be equal to the value of the European counterpart. Assuming that no arbitrage opportunities exist, an American option must always be at least as valuable as its immediate payoff. Otherwise, an investor could simply buy the option and exercise it immediately in clear violation of the no-arbitrage assumption. Basic options without any special features are classified as plain vanilla. 2.2 Pricing models for American options The pricing and risk management of American options is still a field of ongoing research, especially when considering options which are written on more than one underlying asset. From a theoretical point of view, the pricing problem of an American option can be formulated in various ways. Some of the most common pricing approaches are based on an optimal-stopping formulation, 4

10 a dynamic programming formulation, a linear complementarity formulation or a moving boundary formulation. The linear complementarity and the moving boundary formulations make use of the associated partial differential equation and we briefly discuss them in sections and Longstaff and Schwartz introduced their well-known pricing algorithm for American options in [LS01] based on a dynamic programming formulation. The Longstaff- Schwartz algorithm uses a backward iteration where at every timestep the continuation value of the option is approximated with a least squares fit against a set of basis functions. We give a very brief introduction to the main ideas behind the Longstaff- Schwartz algorithm in section 2.4. Most tree based methods to approximate the value of an American option are also based on backward iterations. Such methods are simple yet powerful pricing tools. We will, however, not pursue any tree based methods in this thesis, but instead refer the reader to [Gla04, 8.3] for detailed information. With a Monte Carlo approach pricing and managing the risks of American derivatives is far more involved than pricing and managing the risks of European options. The added difficulty stems from the fact that it is a priori unclear when the option holder will choose to exercise the option. Also Monte Carlo methods usually work forward in time whereas most formulations for pricing American derivatives lend themselves better to methods working backwards on the time scale. The difference in value between an American derivative and its European equivalent is referred to as the early exercise premium. In contrast to European options there are generally no closed-form solutions for American option even for plain vanilla instruments. The pricing of an American option is based on the assumption that a rational option holder will choose to exercise his right optimally in some sense. Thus, finding an optimal exercise strategy is at the core of pricing an American option. Mathematically, this translates into the optimal stopping formulation discussed in section Free boundary formulation One way to phrase the American options pricing problem is as a free boundary problem. Here, we sketch this approach for a classical American put option. 5

11 At every point in time, depending on the current value of the underlying, it is either optimal to exercise the option or to continue to hold it. Mathematically, the (t, S) plane can be divided into a hold region (where it is optimal to hold the option) and an exercise region (where it is optimal to exercise it). The barrier between those two regions is called the optimal exercise boundary B (t). Denote the fair value of the American put option by P am and assume that the underlying price process is a geometric Brownian motion. In this situation the hold region lies above the exercise boundary B (t) in the (t, S) plane. Within the hold region the value function P am satisfies the Black-Scholes equation, i.e. L BS [P am ] := P am + 1 t 2 σ2 S 2 2 P am + r S P am S 2 S r P am = 0, (2.1) where r and σ are the usual drift and volatility parameters from the geometric Brownian motion SDE. At expiry T and along the optimal exercise boundary B (t) the option is worth its payoff, i.e. P am (T, S) = max (K S, 0) and P am (t, B (t)) = max (K B (t), 0). The transition between the two regions is smooth, in the sense of the so-called smooth pasting condition P am S (t, B (t)) = 1. Since the optimal exercise boundary is not known a priori but must be computed as part of the solution, the problem is much harder to solve than a simple boundary value problem. We note that the smooth pasting condition only holds for options whose payoff is sufficiently smooth. It does not hold in the case of an American digital put for example. Another point worth mentioning is that for more complicated American derivatives the hold region might not be simply connected at all times. free boundary problems are hard to solve numerically. formulation any further Linear complementarity formulation Overall, We will not consider this Another way to characterize the price of an American option is as the solution of a linear complementarity formulation. In this formulation we are looking for a function V (S, t) such that L BS (V ) 0, V (S, t) P (S, t) (V (S, t) P (S, t)) L BS [V ] = 0 6

12 and V (S, T ) = P (S, T ). Here, P (S, t) denotes the payoff function of the option and the Black-Scholes operator L BS is defined as in (2.1). The linear complementarity formulation is the clever combination of several features of an American option. Some of these can easily be argued heuristically. We had already argued before that based on the no-arbitrage assumption the option must be worth no less than its immediate payoff, i.e. V (S, t) P (S, t). Moreover, it is clear that the option value coincides with its payoff value whenever it is optimal to exercise. Also the terminal condition is clear from the definition of the option contract. It can be shown that within the hold region of the option its price function V (S, t) satisfies the Black Scholes equation. Combining these obeservations one ends up with the linear complementarity formulation above Optimal stopping formulation The last possible formulation of the pricing problem for American options which we want to discuss is the optimal stopping formulation. It is at the heart of our multilevel Monte Carlo approach. Finding the fair value V am of an American option contract is equivalent to solving the following optimal-stopping-time problem (see for example [Gla04, chapter 8]) V am = sup E [e ] τ 0 r(u)du P τ (S τ ). (2.2) τ T Here, the expectation is taken under the risk-neutral martingale measure. The instantaneous short rate process r(t) is used to discount the payoff P τ at time τ, and the supremum in (2.2) is taken over all admissible stopping times τ T between now and the final expiry of the option. For more exotic path-dependent options, e.g. lookback options, the payoff in (2.2) could be dependent on the whole path history, i.e. P τ ({S t : t [0, τ]}), but for the purpose of this thesis we restrict ourselves to (2.2). Also, for the sake of better readability, we will assume the short rate r to be constant from now on. To make (2.2) more explicit, we consider the special case of a classical American put option struck at K on a single underlying S with price process S t. If the underlying price process follows a geometric Brownian motion with interest rate r under the risk-neutral martingale measure, then (2.2) reads sup τ T E [ e τr max (K S τ, 0) ]. (2.3) 7

13 In case of a put option on a single underlying the hold and exercise regions are half planes divided by the optimal exercise boundary B (t) and the supremum in (2.3) is attained for an optimal stopping time τ of the form τ = inf {t 0 : S t B (t)}. (2.4) Figure 2.1 illustrates the situation for a sample path which hits the optimal exercise boundary exercise boundary underlying price process hold region exercise region t Figure 2.1: hold and exercise region under a given exercise rule Stopping rules and parametric approximations The connection in (2.4) allows us to interchangeably formulate the pricing problem either in terms of stopping times or in terms of exercise regions. Note that any choice of a stopping time ˆτ T will produce a (usually suboptimal) value in the expectation from (2.2), i.e. V (ˆτ) am = E [ e ˆτ r Pˆτ (Sˆτ ) ] V am. (2.5) 8

14 If the optimal exercise region E is known, then the optimal stopping time τ is the first time that S t enters the exercise region τ = inf τ T {t 0 : S t E }, (2.6) and the fair value of the option can be computed as a simple expected value according to V (τ ) am = E [ e τ r P τ (S τ ) ] = V am. (2.7) The mapping implicit in (2.6) can be used to assign a stopping time to any given exercise region. We can therefore think in terms of optimizing over exercise regions (or boundaries) in order to find the value of an American option. The appealing consequence is that given an exercise strategy (i.e. an exercise region), we can simply use a forward Monte Carlo simulation of the underlying path process, compute the payoff for each path and take an average to find the value of the option under the chosen exercise strategy. We want to emphasize that most numerical pricing algorithms for American options including the famous Longstaff- Schwartz algorithm are based on backward iterations and that forward algorithms are usually unsuited for the problem. Suppose now that we can parametrize all feasible exercise regions E θ by some parameter θ Θ and denote by τ(θ) the stopping time associated with E θ according to the mapping presented in (2.6). The option value is then approximately equal to Ṽ am = sup θ Θ E [ e τ(θ) r P τ(θ) ( Sτ(θ) )]. (2.8) Note however, that the value computed in (2.8) is in general biased low because we might not hit all feasible stopping times in our approximation. In other words, the set τ(θ) T will in general be a proper subset of T and hence Ṽ am = sup θ Θ E [ e τ(θ) r P τ(θ) ( Sτ(θ) )] sup τ T E [ e τ r P τ (S τ ) ] = V am. Nonetheless, Equation (2.8) is the foundation of our numerical approach. For an American put option a lot is known about the optimal exercise boundary. One of its main characteristics is its asymptotic behaviour close to the expiry date of the option, where it clearly hits the strike. In fact, the optimal exercise boundary of an American put option asymptotically approaches its strike level K displaying a square root behaviour (see e.g. [KK98]), i.e. B (t) K c T t for T t 1 (2.9) 9

15 with an appropriate constant c. This asymptotic feature determines the choice of our Monte Carlo time discretization in section Even more importantly, it influences our choice of a parametric family of feasible exercise boundaries in section 5.1, which we use in the optimization problem (2.8) and the numerical examples in chapter Finite Difference θ-scheme for American options In this section we recall the finite difference θ-scheme and how it can be used to approximate the value of an American put option on a single underlying. We follow the presentation in the Oxford lecture notes of Dr. Christoph Reisinger and will later use the finite difference method as a benchmark solution against which we compare our multilevel Monte Carlo approximations. This section is not supposed to be an exhaustive treatment of the finite difference method. First we consider a European put option on a single underlying with strike K and time to maturity T. Its fair value V (S, t) solves the Black-Scholes equation V + 1 t 2 σ2 S 2 2 V S + r S V 2 S r S = 0 on the domain (S, t) (0, ) (0, T ) and satisfies the terminal condition V (S, T ) = max (K S, 0). The finite difference scheme discretizes both the equation and its approximate solution. It starts with the terminal condition at time t = T and moves backwards in time towards t = 0 according to the discretized version of the partial differential equation. This only works on a bounded domain and hence we truncate the computational domain in the space direction, i.e. we only consider the equation for (S, t) (0, S max ) (0, T ) with some upper bound S max K and prescribe an appropriate boundary condition along S max (0, T ). For a put option this could for example be zero Dirichlet data. We subdivide the space direction into N + 1 equidistant grid points S i = i S for i = 0,..., N and S = S max /N and the time direction into M + 1 equidistant grid points t j = j t for j = 0,..., M and t = T/M, and use the notation Vn m to the approximation to V (S n, t m ) on these grid points. to refer Next, we discretize the partial differential equation. This is done using the following difference quotients for the spatial derivatives V S (S n, t m ) = V (S n+1, t m ) V (S n 1, t m ) + O ( ( S) 2) 2 S 10

16 and 2 V S 2 (S n, t m ) = V (S n+1, t m ) 2 V (S n, t m ) + V (S n 1, t m ) ( S) 2 + O ( ( S) 2). For the time derivative we choose between either a forward difference of the form or a backward difference of the form V t (S n, t m ) = V (S n, t m+1 ) V (S n, t m ) + O ( t) t V t (S n, t m ) = V (S n, t m ) V (S n, t m 1 ) + O ( t). t Choosing the backward difference leads to the so-called explicit Euler scheme V m n V m 1 n t σ2 S 2 n Vn+1 m 2 Vn m + Vn 1 m ( S) 2 + r S n V m n+1 V m n 1 2 S r S n = 0. The scheme is called explicit because the solution vector Vn m 1 explicitly in terms of the previous iteration step, i.e. at time t m 1 is given V m 1 n = A m n Vn 1 m + Bn m Vn m + Cn m Vn+1 m for the coefficients A m n B m n C m n = 1 2 n2 σ 2 t 1 2 nr t = 1 n 2 σ 2 t r t = 1 2 n2 σ 2 t nr t. Alternatively, if we choose to approximate the time derivative using the forward difference we end up with the so-called implicit Euler scheme V m+1 n t V m n σ2 S 2 n Vn+1 m 2 Vn m + Vn 1 m ( S) 2 + r S n V m n+1 V m n 1 2 S r S n = 0. This scheme is called implicit because at time step t m the new iterate is only given implicitly and one needs to solve the following equation to advance Here, the coefficients are a m n Vn 1 m + b m n Vn m + c m n Vn+1 m = Vn m+1. a m n b m n c m n = 1 2 n2 σ 2 t nr t = 1 + n 2 σ 2 t + r t = 1 2 n2 σ 2 t 1 2 nr t. 11

17 The explicit scheme is less costly in terms of computational effort, but requires rather strict conditions on the size of the time step for it to be stable. The fully implicit method is usually unconditionally stable. They both show first order convergence with respect to the time discretization. One can keep the stability but improve on the convergence properties by moving to the so-called θ-scheme. The θ-scheme is a weighted average of the explicit and the implicit Euler scheme with weight factor θ [0, 1], i.e. a m n V m 1 n 1 + b m n V m 1 n + c m n Vn+1 m 1 = A m n Vn 1 m + Bn m Vn m + Cn m Vn+1 m (2.10) with coefficients A m n = 1 2 (1 θ) t ( n 2 σ 2 nr ) Bn m = 1 (1 θ) t ( n 2 σ 2 + r ) Cn m = 1 2 (1 θ) t ( n 2 σ 2 + nr ) a m n = 1 2 θ t ( n 2 σ 2 nr ) b m n = 1 + θ t ( n 2 σ 2 + r ) c m n = 1 2 θ t ( n 2 σ 2 + nr ). Alternatively, the procedure in (2.10) can be written in matrix form as M 1 V m 1 = M 2 V m, (2.11) where the matrices M 1 and M 2 are tridiagonal. Note that in setting θ = 0 or θ = 1 one recovers the explicit or implicit schemes from above. We usually work with θ = 1/2, a choice for which the method is called the Crank-Nicolson scheme. Recall, that in order to find the value of a European put, we start from the terminal condition Vn M = max (K S n, 0) and iterate backwards in time towards V 0 using the θ-scheme (2.11). An American option can be exercised at any time and thus it must always be worth at least as much as its immediate payoff g. The immediate payoff of an American put option is given by g n = max (K S n, 0). We incorporate this into the method 12

18 by dynamic programming and force the option value to be worth at least as much as its immediate payoff at each iterate, i.e. we modify the algorithm towards M 1 ˆV m 1 = M 2 V m (2.12) ) V m 1 = max (g, ˆV m 1. (2.13) To be precise, this only approximates the value of a Bermudan option with possible exercise date on each t i. approximate the value of an American option. In fact, the method has an O ( t) bias when used to For small t this is still a good approximation for the American case where exercising is possible on a continuous time scale. Although, using (2.12) and (2.13) might not the most effective way to price an American option with a finite difference method, it suffices completely for our purposes. For details on how to improve the scheme (e.g. by Rannacher time stepping) we refer the reader to the Oxford lecture notes of Dr. Christoph Reisinger on finite difference methods. The finite difference method is a powerful pricing algorithm for American options written on a single underlying. For models depending on multiple stochastic factors, however, the computational cost of finite difference methods increases drastically to the point where it becomes prohibitive. 2.4 The Longstaff-Schwartz algorithm In this section we present the ideas behind the Longstaff-Schwartz algorithm. We mainly follow the notation used in [Gla04]. The Longstaff-Schwartz algorithm is used to approximate the value of Bermudan options. In the limit, i.e. for a growing number of possible exercise dates with shrinking time difference between them, this is a reasonable approximation for the pricing problem of American options. Let us denote by Ṽi(x) the fair value of the option at time t i assuming it has not been exercised previously. At maturity, this value is determined solely by the payoff function h m (x). We are ultimately interested in Ṽ0(X 0 ) and can get this value according to the dynamic programming formulation Ṽ m (x) = h m (x) Ṽ i 1 (x) = max { hi 1 (x), E [ ]} D i 1,i Ṽ i (x) X i 1 = x, 13

19 where X i is the underlying price at time t i, h i (x) is the exercise value of the option at time t i and D i 1,i denotes the discount factor for the interval [t i, t i+1 ]. If we set V i (x) = D 0,i Ṽ i (x) h i (x) = D 0,i hi (x), with D 0,i = D 0,1 D 1,2... D i 1,i, the formulation can be simplified to V m (x) = h m (x) (2.14) V i 1 (x) = max {h i 1 (x), E [V i (x) X i 1 = x ]}. (2.15) In a first step, the Longstaff-Schwartz algorithm iterates backwards in time to approximate the continuation values C i (x) = E [V i+1 (x) X i = x ] of the option. It assumes that C i (x) can be expanded in terms of an appropriate set of basis functions. This is certainly true for each element of a Hilbert space because every Hilbert space has a countable orthonormal basis. The chosen basis is then truncated to a finite number, R + 1 say, basis functions and the algorithm works with The corresponding coefficients β l namely by minimizing C i (x) Ĉi(x) R+1 = β l φ l (x) l=1 are found by a least squares regression process, [ ( ) ] 2 E E [V i (X i ) X i 1 ] Ĉi 1(X i 1 ). Taking the derivatives with respect to the β l and setting them to zero yields the following matrix-vector equation B φ,φ β = B V,φ, with (B φ,φ ) r,s = E [φ r (X i 1 ) φ s (X i 1 )] and (B V,φ ) r = E [V i (X i ) φ r (X i 1 )]. (2.16) In the numerical implementation the expected values in (2.16) are approximated as Monte Carlo averages over N paths, for example (B φ,φ ) r,s 1 N N n=1 [ ] E φ r (X (n) i 1 ) φ s(x (n) i 1 ). (2.17) 14

20 Once the approximate continuation values are known for each time step, the Longstaff-Schwartz algorithm uses them in a second Monte Carlo step as a decision guidance on when to exercise. This is done as follows. Starting from expiry and moving backwards, at each time step i the continuation value of the previous step is compared to the immediate payoff h i 1 (X i 1 ). If the immediate payoff is higher than the continuation value then the algorithm advances to V i 1 = h i 1 (X i 1 ), otherwise the algorithm takes V i 1 = V i. In principle, one can use the first set of path simulations not only to approximate the continuation values but also to produce an approximate value of the option. However, this value might be biased high and hence the algorithm is usually split into two parts. For the second part one can also choose to work forward in time and stop each path at the first instance when its immediate payoff is higher than the continuation value. This is more efficient than the backward iteration. For the purposes of this thesis we will only cite some approximate values computed in the original paper [LS01] and will not use our own implementation of the algorithm. 15

21 Chapter 3 Multilevel Monte Carlo method Onions have layers. Ogres have layers. Onions have layers. You get it? We both have layers. Shrek In this chapter we introduce the basic multilevel Monte Carlo method. We follow the presentation and notation of the original papers [Gil08] and [Gil]. These two papers started a series of research papers which have shown that the multilevel Monte Carlo approach is a powerful way to reduce the computational cost in comparison to standard Monte Carlo methods when applied to various option pricing problems. Among the published examples are plain vanilla European options, options with Asian payoff, digital options, lookback options as well as barrier options. In [Gil09] the method has been successfully applied to European type basket options, including lookback, Asian and barrier options. The theoretical analysis of the method was originally only done for payoffs which are globally Lipschitz, although practical numerical examples have shown good convergence properties also for digital options for example. In [GHM09] the authors conduct a deeper numerical analysis of the method. 3.1 The setup We want to compute the value of a financial option, whose payoff P = f(s t ) is given as a functional of the path of a stochastic process S t. We assume that the evolution of the stochastic process S t is governed by the SDE ds t = a(s t, t)dt + b(s t, t)dw t, (3.1) 16

22 for the time interval 0 < t < T with given initial data S 0 and given drift and volatility terms a(s t, t) and b(s t, t). Throughout this thesis we will exclusively be working with geometric Brownian motion, i.e. a(s t, t) = rs t and b(s t, t) = σs t for constant risk free rate r and volatility factor σ. However, the multilevel Monte Carlo method is not restricted to this case and hence we state (3.1) in its more general form. The payoff of a classical European call option with strike K and expiry T, say, is given by max (S T K, 0). In this case the payoff is a simple function only of the final state S T of the stochastic path. Other options can be highly path dependent, for example the payoff of a knock-in barrier option naturally depends on the full path S t and not just its final state. Both cases are included in the above setup. 3.2 Basic multilevel decomposition In order to approximate the value of such an option a standard Monte Carlo method generates various discrete path simulations of (3.1) with a certain time step size, evaluates the payoff for each of those paths and computes the expected value as an average. The chosen step size strongly depends on the desired accuracy of the computed solution. In fact, the smaller the step size the better the results. Instead of sticking to one discretisation time step the multilevel Monte Carlo method uses path simulations corresponding to various nested time grids. The idea is to reduce the variance of the finest grid approximations using information from coarser levels in a way that minimizes the overall computational cost. To be more precise, suppose we are given different time discretisations with step sizes h l = 2 l T for l = 0... L. For any given Brownian path W t we denote by P the payoff of the option and by ˆP l its numerical approximation using a discretisation with timestep h l. By linearity we can decompose the expected value on the finest level as follows E[ ˆP L ] = E[ ˆP L 0 ] + E[ ˆP l ˆP l 1 ]. (3.2) In fact, the decomposition (3.2) can (and in our case will) be made a bit more general, in the sense that the numerical approximation ˆP l does not need to be the same for the correction terms E[ ˆP l ˆP l 1 ] and E[ ˆP l+1 ˆP l ]. Instead, we distinguish between two possibly different numerical approximations ˆP l,f and ˆP l,c with step size h l and write E[ ˆP L ] = E[ ˆP L 0 ] + E[ ˆP l,f ˆP l 1,c ]. (3.3) l=1 l=1 17

23 As long as ˆP l,f and ˆP l,c have identical expectation, i.e. E[ ˆP l,f ] = E[ ˆP l,c ], the sum is still telescoping and (3.3) is correct. The additional subscripts f and c were chosen to resemble the labels fine and coarse for an approximation on any given level l > 0. The multilevel Monte Carlo method is based on the decomposition in (3.3) and independently evaluates each expected value on the right hand side of the equation in a way that minimizes the total variance for a given computational cost. This is done as follows. 3.3 Monte Carlo estimators We denote by Ŷ0 an estimator for E[ ˆP 0 ] which is based on N 0 sample paths. Similarly, for l > 0 we denote by Ŷl independent estimators for E[ ˆP l,f ˆP l 1,c ], each one using N l sample paths. The estimators Ŷl usually are computed as means taken over N l sample paths, i.e. for l > 0 the estimator has the form Here, the trick is that ˆP (i) l,f the same Brownian path. Ŷ l = 1 N l (i) and ˆP l 1,c N l i=1 ( ˆP (i) l,f ) (i) ˆP l 1,c. (3.4) are constructed using different time steps but We combine the Ŷl and get the overall estimator L Ŷ = Ŷ l (3.5) l=0 for the option price. Note that Ŷ is in fact a feasible estimator if the Ŷl are unbiased because then we have ] E [Ŷ = E[Ŷ0] + = E[ ˆP 0 ] + [ ] = E ˆPL. L l=1 L l=1 ] E [Ŷl [ E ˆPl,f ˆP ] l 1,c Let us take a closer look at the mean squared error of the estimator [ Ŷ. (Ŷ ) ] [ 2 (Ŷ ]) ] 2 ( ] ) 2 E E [P ] = E E [Ŷ + E [Ŷ E [P ] [ (Ŷ ]) ] 2 ( [ ] ) 2 = E E [Ŷ + E ˆPL E [P ] ] = V [Ŷ + ( [ ] 2 E ˆPL E [P ]). (3.6) 18

24 The first term in (3.6) is just the variance of Ŷ and since all the Ŷl are independent it can be written as [ ] L ] V [Ŷ = V Ŷ l = l=0 L l=0 ] V [Ŷl = L l=0 N 1 l V l, (3.7) where V l is the variance of a single path for estimator Ŷl. Equation (3.7) is one of the starting points of the multilevel algorithm as will be shown in section 3.5. The second term in (3.6) corresponds to how well the payoff can be approximated on the finest grid. 3.4 The multilevel algorithm In this section we present the pseudocode for the actual multilevel algorithm as described in the original paper [Gil08]. The original algorithm takes the desired accuracy ε as an input and computes the required maximal level of refinement L on the fly. This is done based on a heuristic convergence criterion, but has proven to be very effective numerically in the multilevel Monte Carlo research literature. 1. Start with L = Estimate V L using an intial N L samples. 3. Define optimal N l for l = 0,..., L. 4. Evaluate extra samples as needed for new N l. 5. If L 2, test for convergence. 6. If L < 2 or not converged, set L = L + 1 and go back to step 2. The convergence test in step 5 is usually based on the following criterion { 1 } max ŶL 1, ŶL < ε Complexity and convergence theorem In this section we first give a brief and somewhat handwavy explanation of the total cost of the multilevel Monte Carlo algorithm to achieve a certain accuracy of order ε. Then we cite the main theorem from [Gil08] which rigorously explains the convergence and complexity of the multilevel Monte Carlo algorithm for various cases. 19

25 The Ŷl are constructed from N l paths and the computational cost of each path is proportional to h 1 l. Hence, the cost of the combined estimator is proportional to L l=0 N lh 1 l. As mentioned before, we want to minimize the variance in (3.7) for a given cost, i.e. subject to the contraint L l=0 N lh 1 l Const. Treating N l as a continuous variable this minimization can easily be done using a Lagrange multiplier approach. As a result one finds that the variance is minimized if N l is chosen proportional to V l h l. Assume now that the single variance V l is of order O(h l ) (this corresponds to β = 1 in Theorem 3.5.1). By what we just saw, the optimal choice for N l would then also be ] of order O(h l ). If one chooses N l = O(ε 2 L h l ), the variance of Ŷ will be V [Ŷ = O(ε 2 ). If the weak convergence of the estimator is of order O(h L ) = O(2 L ) on the finest level, then by choosing L = O( log(ε) ) we can make the second term in log(2) (3.6) order O(ε 2 ). Thus, for a total cost of L l=0 N l h 1 l = L l=0 O(ε 2 L h l )h 1 l = O(ε 2 L 2 ) = O(ε 2 (log ε) 2 ) [ (Ŷ ) ] 2 the mean squared error E E [P ] would be of order O(ε 2 ). A standard Monte Carlo method has a cost of O(ε 3 ) to achieve the same accuracy. And thus the multilevel Monte Carlo algorithm brings about a drastic improvement in computational cost. The full complexity and convergence properties of the multilevel Monte Carlo method are generalized and summarized in Theorem Theorem (see [Gil08]) Let P denote a functional of the solution of the SDE (3.1) for a given Brownian path W t, and let ˆP l denote the corresponding approximation using a numerical discretisation with timestep h l = 2 l T. If there exist independent estimators Ŷl based on N l Monte Carlo samples, and positive constants α 1/2, β, c 1, c 2, c 3 such that (i) E[ ˆP l P ] c 1 h α l { E[ ˆP0 ], l = 0 (ii) E[Ŷl] = E[ ˆP l ˆP l 1 ], l > 0 ] (iii) V [Ŷl c 2 N l l h β l (iv) the computational complexity C l of Ŷl is bounded by C l c 3 N l h 1 l, 20

26 then there exists a constant c 4 such that for any ε < e 1 there are values L and N l for which the multilevel estimator Ŷ = L l=0 Ŷl has mean-square error with bound [ (Ŷ ) ] 2 MSE E E [P ] < ε 2, with a computational complexity bound c 4 ε 2, β > 1 C c 4 ε 2 (log ε) 2, β = 1 c 4 ε 2 (1 β)/α, 0 < β < 1. 21

27 Chapter 4 Numerical implementation for an American put option under a preset exercise strategy Any general method for pricing American options by simulation requires substantial computational effort Paul Glasserman in [Gla04] The estimators Ŷl used for the multilevel Monte Carlo method need to be constructed carefully in order to avoid the introduction of an undesired bias in the computed solution. In (3.4) we have already given the general form of Ŷl. So far we have not, however, described the numerical approximations ˆP l,c and ˆP l,f in any detail. In general, these will depend on the type of application. Recall that they both should be computed based on a time discretisation with step size h l. For a European call option we could for example just simulate a path with this step size using an Euler- Maruyama scheme and evaluate the payoff at the final timestep for both ˆP l,c and ˆP l,f. It gets more complicated for path dependent options where the two quantities no longer need to be identical. In a sense we have more information about the path available to compute ˆP l,c than ˆP l,f, because we compute ˆP l,c at the same time as we compute ˆP l+1,f, an approximation using a finer timestep. For the remainder of this thesis we will now focus on the valuation of an American put option. In what follows we will describe the construction of the corresponding estimators. 22

28 4.1 Discretizing the SDE The discretization of the SDE (3.1) is crucial. Its convergence properties affect the overall variance and therefore the overall convergence of the multilevel algorithm. In fact, the chosen discretisation scheme for the simulated path strongly influences how the discretised payoff function is constructed and therefore enters into Theorem through the parameters α and β. Suppose we are given a time discretisation of the interval [0, T ] into timesteps 0 = t 0 < t 1 < < t N = T. The most intuitive way to discretize the scalar SDE (3.1) is the Euler-Maruyama scheme Ŝ n+1 = Ŝn + a(ŝn, t n ) (t n+1 t n ) + b(ŝn, t n ) W n, (4.1) where the initial state Ŝ0 = S 0 is known, and where Ŝn indicates the value of the simulated path at time t n. Moreover, W n = W tn+1 W tn is the Brownian increment between timesteps t n and t n+1. Under some weak smoothness conditions on the functions a(s, t) and b(s, t) the Euler-Maruyama scheme is known to be weakly convergent with order one and strongly convergent with order one half. A more refined discretisation can in some cases be achieved using the so-called Milstein scheme. The Milstein scheme approximates the scalar SDE (3.1) via Ŝ n+1 = Ŝn + a (t n+1 t n ) + b W n b b S ( W 2 n (t n+1 t n ) ), (4.2) where Ŝ0 again is the inital state of the path, W n is the Brownian increment as before and the functions a, b and b S are evaluated at (Ŝn, t n ). The Milstein scheme has strong and weak convergence order both equal to one Exact path simulation for geometric Brownian motion Throughout this thesis we assume that the underlying price process is driven by geometric Brownian motion, i.e. that the SDE (3.1) takes the form ds t = r S t dt + σ S t dw t (4.3) with constant r and σ. If we consider the process X t = log(s t ), Ito s lemma tells us that dx t = ) (r σ2 dt + σ dw t, 2 23

29 for which both the above discretisation schemes coincide and yield exact path values at the discrete points, i.e. ) ˆX n+1 = ˆX n + (r σ2 (t n+1 t n ) + σ W n 2 ) n = ˆX n 0 + (r σ2 (t i+1 t i ) + σ W i 2 i=0 i=0 ) = X 0 + (r σ2 t n+1 + σ W n+1 2 = X(t n+1 ). For the purpose of this thesis we will take advantage of the logarithmic transformation and simulate the path X t (and thus S t ) exactly at the discrete points. 4.2 Estimators for a given exercise strategy In what follows we will focus on the example of an American put option under a specified exercise strategy. Suppose we are given an exercise strategy in terms of the exercise region E. According to the mapping implied by (2.6) the exercise region translates into a stopping time τ = inf{t 0 : S t E}. We are interested in computing the value of an American put option under this exercise strategy. This is equivalent to computing the expected value E [P τ ], where the discounted payoff function is P τ = e r τ (K S τ ) + 1 {τ<t } + e r T (K S T ) + 1 {τ T }. (4.4) Here, the notation (K S τ ) + is used to denote max (K S τ, 0) and 1 E is the indicator function of the event E. Note that, in contrast to (2.2) and the equations based thereon, in (4.4) we have incorporated the discount factor directly into P τ. For a given time grid 0 = t 0 < t 1 <... < t N = T and the corresponding discrete path approximations {Ŝi} the naive way to discretise the payoff P τ would simply check at each time step whether or not the exercise region has been reached, i.e. we would work with ˆτ = inf{t i : Ŝi E}. But even if we can simulate the path values at the discrete time steps exactly, this is rather inaccurate, because the path might cross a barrier between two sample points even if the sample points suggest otherwise. So 24

30 convergence would be very slow. We follow an improved approach which is commonly used in the framework of continuously monitored barriers. In case of an American put on a single underlying the exercise region and the hold region of the option are separated by a continuous function B(t), also called the exercise boundary. In fact in this case, the hold region lies above B(t) and the exercise region lies below. We denote by ˆp i the probability that the sample path drops below the exercise boundary (i.e enters the exercise region) in the interval [t i, t i+1 ] conditioned on the values of the sampled path Ŝi and Ŝi+1 at the interval end points, i.e. ( ) S ˆp i = P t B(t) Ŝ for some t [t i, t i+1 ] i, Ŝ i+1 Unfortunately, this probability can only be computed analytically for few special boundary shapes. It can be computed, however, for a linear exercise boundary if the stochastic process has constant drift and diffusion coefficients. If the exercise boundary was in fact linear between t i and t i+1 with end values B(t i ) and B(t i+1 ) and if we modelled the sample path as a Brownian bridge with constant drift r Ŝi and constant diffusion coefficient σ Ŝi in the interval [t i, t i+1 ], then the probability ˆp i would equal (see for example [Huh07] or [BN05]) ( ) 2 (Ŝi B(t i )) + (Ŝi+1 B(t i+1 )) + exp. (4.5) (t i+1 t i ) σ 2 Ŝ2 i For our application, however, we can do even better. The path process S t follows a geometric Brownian motion and as explained in ( section ) we can simulate the process X t = log(s t ) exactly. Moreover, the drift r σ2 and diffusion σ coefficients 2 of X t are indeed constant. So instead of using (4.5), we compute the probability based on the path X t. This is possible because the logarithm is strictly increasing for positive arguments. Hence, if the exercise boundary is positive on the subinterval [t i, t i+1 ] we can write ( ˆp i = P ( = P S t B(t) for some t [t i, t i+1 ] X t log (B(t)) for some t [t i, t i+1 ] ) Ŝ i, Ŝ i+1 ) ˆXi = log(ŝi), ˆXi+1 = log(ŝi+1). 25

31 Consequently, if both B(t i ) and B(t i+1 ) are positive, there holds ( ) ) + ( ) 2 log (Ŝi log(b(t i )) log (Ŝi+1 log(b(t i+1 )) ˆp i = exp (t i+1 t i ) σ 2 2 = exp ( / ) + ( / log (Ŝi B(t i ) log (Ŝi+1 B(t i+1 ) (t i+1 t i ) σ 2 ) + ) + (4.6) In using (4.6) the only potentially remaining source of error is if the exercise boundary is not linear. And although this linearity assumption about the exercise boundary will in most cases not be fullfilled we work with (4.6) as an approximate value for the hitting probability in the interval [t i, t i+1 ]. Recall that we want to compute the expected value [ [ ]] E [P τ ] = E E P τ {Ŝi}. [ ] Thus, we concentrate on the conditional expectation E P τ {Ŝi}. The first indicator function in (4.4) can be split up by subintervals This enables us to approximate P τ = N 1 i=1 1 {τ<t } = N 1 i=1 1 {τ [ti,t i+1 ]}. exp ( r τ) (K S τ ) + 1 {τ [ti,t i+1 ]} + e r T (K S T ) + 1 {τ T } N 1 exp ( r τ i ) (K S τi ) + 1 {τ [ti,t i+1 ]} + e r T (K S T ) + 1 {τ T } i=1 for some approximate value τ i of the actual crossing time. In our implementation we choose the interval midpoint τ i = t i+t i+1. At the crossing point τ 2 i the path touches the barrier and we implement ( ) ti + t i+1 S τi = B(τ i ) = b. 2 26

32 The conditional expectation can then be approximated by [ ] N 1 [ ] E P τ {Ŝi} exp ( r τ i ) (K S τi ) + E 1 {τ [ti,t i+1 ]} {Ŝi} = i=1 [ ] + exp ( r T ) (K S T ) + E 1 {τ T } {Ŝi} ( ) i 1 exp ( r τ i ) (K S τi ) + ˆp i (1 ˆp j ) N 1 i=1 j=0 ( N 1 + exp ( r T ) (K S T ) + j=0 (1 ˆp j ) ) (4.7) The estimator ˆP l,f from Equation (3.3) is contructed according to (4.7) using the probabilities ˆp i according to (4.5). For the coarse path estimator ˆP l,c we work with an additional idea used in [Gil] for path dependent payoffs. Recall, that the estimator corresponds to a step size h l and is computed at the same time as ˆP l+1,f. On the interval [t i, t i+1 ] of length h l we simulate an additional midpoint Ŝn+ 1 based on a Brownian bridge construction. 2 The exact definition depends on the chosen time discretization and will be given in sections and We then look at the two probabilities that the sample path drops below the barrier on either side of the midpoint conditioned on the sampled values, i.e. and exp ( ) 2 ( Ŝ i B(t i )) + (Ŝi+ exp 1 B avg ) + 2 (t i+ 1 t i ) σ 2 Ŝ2 i 2 (4.8) ( ) 2 ( Ŝ i+ 1 B avg ) + (Ŝi+1 B(t i+1 )) + 2. (4.9) (t i+1 t i+ 1 ) σ 2 Ŝ2 i 2 The exact value of B avg again depends on the chosen time discretization and will be given along with the definition of Ŝn+ 1 in sections and In both probability 2 formulae there is a factor of Ŝi in the denominator. This stems from the fact that the underlying Brownian bridge is assumed to have the same volatility coefficient σ Ŝi on the entire interval [t i, t i+1 ]. As before, we can do better than this by moving to the 27

33 transformed path X t = log(s t ). Instead of (4.8) and (4.9) we use ( ) ) + ( ) 2 log (Ŝi log(b(t i )) log (Ŝi+ 1 log(b avg ) 2 ˆp i,1 = exp (t i+ 1 t i ) σ 2 2 ( ) ) + ( ) 2 log (Ŝi+ 1 log(b avg ) log (Ŝi+1 2 ˆp i,2 = exp (t i+1 t i+ 1 ) σ 2 2 ) + ) + log(b(t i+1 )) (4.10). (4.11) The probability that the path crosses the exercise boundary in the entire interval is now easily computed via ˆp i = (1 (1 ˆp i,1 ) (1 ˆp i,2 )). (4.12) We construct the coarse estimator ˆP l,c according to (4.7) using the probabilities ˆp i from (4.12). 4.3 Numerical example As a numerical example we apply the algorithm to an American put option under the exercise strategy given by the exercise boundary B(t) = K 0.8 T t. Our results are presented in the spirit of [Gil08] and [Gil] and were obtained using the parameters T = 1.0, K = 1.0, S 0 = 1.0, r = 0.05 and σ = A brief aside on Brownian bridge construction and Brownian interpolation Suppose X t is a simple Brownian motion with drift µ and diffusion coefficient ρ, i.e. dx t = µ dt + ρ dw t = X t = µ t + ρ W t. Suppose further that we have sampled S t at times t 1 < t 2, say X t1 = a and X t2 = b. Then the conditional distribution of X t at time t = t 1 + λ (t 2 t 1 ) for 0 < λ < 1 is a normal distribution (see e.g. [Gla04]) with mean a + t t 1 t 2 t 1 (b a) and variance ρ 2 (t 2 t) (t t 1 ) t 2 t 1. 28

34 Moreover, if we combine the equations for X(t), X(t n ) and X(t n+1 ) we find that (see e.g. [Gil09]) X t = X tn + λ { X tn+1 X tn } + ρ {Wt W n λ (W n+1 W n )}. (4.13) This means that X t deviates from a linear interpolation if and only if W t deviates from a straight line interpolation Equidistant time grids As a first numerical example we implemented the algorithm using equidistant time grids. To be more precise, on level l = 1... L we partition the interval [0, T ] using 2 l +1 equidistant grid points t n = n T / 2 l for n = 0,..., 2 l. In this case the additional midpoint Ŝn+ 1 constructed for the approximation ˆP l,c actually corresponds to the true 2 midpoint (t n + t n+1 ) / 2 of the interval. As for the probabilities in (4.6), (4.10) and (4.11) we use the process X t = log(s t ) and its discretization. In accordance with (4.13) for λ = 1/2 and ρ = σ we compute ( where D n = log (Ŝn+ 1 2 ) ( ˆXn + ˆX n+1 σ D n ), := ˆX n+ 1 2 := 1 2 ) ( W n+ 1 2 ) W n is a N (0, h l ) variable computed based W n+1 W n+ 1 2 on the Brownian increments from the finer path simulation with step size h l+1 = h l / 2. The definition of B avg from (4.8) and (4.9) in this case is log (B avg ) = log (B(t i)) + log (B(t i+1 )). 2 Figure 4.1 shows the results. In the top left plot the behaviour of the variance of both ˆP l,f and ˆP l,f ˆP l 1,c can be seen. The slope of the latter seems to converge to a value slightly higher than 3, i.e. V 2 l = O(h 3/2 ɛ l ) which in the language of Theorem corresponds [ to a value of β 3/2 ɛ for some small ɛ. In the top right plot we see that E ˆPl,f ˆP ] l 1,c is approximately of order O(h l ) and hence α = 1. The [ ] expected value E ˆPl,f seems to be biased high for the coarser levels. This bias is probably due to the fact that the linear approximation of the logarithm of the exercise boundary is too rough for the coarse levels. The bottom two plots show the behaviour of the multilevel algorithm for various levels of desired accuracy ε. The bottom left plot shows the values N l for l = 0,... L. Recall that the optimal value N l was to be chosen proportionally to V l h l. So for decreasing V l and h l the optimal N l is decreasing for larger l. The bottom right plot 29