Monte Carlo Path Simulation and the Multilevel Monte Carlo Method. Krister Janzon

Transcription

1 Monte Carlo Path Simulation and the Multilevel Monte Carlo Method Krister Janzon Master s Thesis, 30 ECTS Master of Science Programme in Engineering Physics, 300 ECTS Spring 2018

2

3 Monte Carlo Path Simulation and the Multilevel Monte Carlo Method Krister Janzon September 2018 A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Engineering Physics Umeå University

4 monte carlo path simulation and the multilevel monte carlo method 2018 Krister Janzon Master s Thesis in Engineering Physics, 30.0 ECTS, Department of Physics, Umeå University. Conducted at the Department of Mathematics and Mathematical Statistics, Umeå University. Supervisor: David Cohen, Department of Mathematics and Mathematical Statistics, Umeå University. Examiner: Markus Ådahl, Department of Mathematics and Mathematical Statistics, Umeå University. ii

5 Abstract A standard problem in the field of computational finance is that of pricing derivative securities. This is often accomplished by estimating an expected value of a functional of a stochastic process, defined by a stochastic differential equation (SDE). In such a setting the random sampling algorithm Monte Carlo (MC) is useful, where paths of the process are sampled. However, MC in its standard form (SMC) is inherently slow. Additionally, if the analytical solution to the underlying SDE is not available, a numerical approximation of the process is necessary, adding another layer of computational complexity to the SMC algorithm. Thus, the computational cost of achieving a certain level of accuracy of the estimation using SMC may be relatively high. In this thesis we introduce and review the theory of the SMC method, with and without the need of numerical approximation for path simulation. Two numerical methods for path approximation are introduced: the Euler Maruyama method and Milstein s method. Moreover, we also introduce and review the theory of a relatively new (2008) MC method the multilevel Monte Carlo (MLMC) method which is only applicable when paths are approximated. This method boldly claims that it can under certain conditions eradicate the additional complexity stemming from the approximation of paths. With this in mind, we wish to see whether this claim holds when pricing a European call option, where the underlying stock process is modelled by geometric Brownian motion. We also want to compare the performance of MLMC in this scenario to that of SMC, with and without path approximation. Two numerical experiments are performed. The first to determine the optimal implementation of MLMC, a static or adaptive approach. The second to illustrate the difference in performance of adaptive MLMC and SMC depending on the used numerical method and whether the analytical solution is available. The results show that SMC is inferior to adaptive MLMC if numerical approximation of paths is needed, and that adaptive MLMC seems to meet the complexity of SMC with an analytical solution. However, while the complexity of adaptive MLMC is impressive, it cannot quite compensate for the additional cost of approximating paths, ending up roughly ten times slower than SMC with an analytical solution. Keywords: Multilevel Monte Carlo, computational complexity, option pricing, path approximation, Euler Maruyama, Milstein iii

6 Acknowledgements I would like to express my very great appreciation to David Cohen for providing me with the opportunity to pursue this thesis, as well as for his patient guidance, encouragement, and constructive criticism. Furthermore, I am deeply grateful for all of the support and encouragement from family and friends throughout the duration of this thesis and degree. It is with great joy that they are both finally complete. iv

7 Contents 1 Introduction Background Purpose Goals Outline Preliminaries Two Fundamental Theorems for Monte Carlo Methods The Strong Law of Large Numbers The Central Limit Theorem Stochastic Differential Equations Numerical Methods for SDEs Simulation of Geometric Brownian Motion Big-O Notation Option Pricing The Standard Monte Carlo Method Introduction Fundamental Theory The Crude Estimator Analyzing Efficiency and Cost of the Crude Estimator Variance Reduction Standard Monte Carlo with SDE Path Simulation The Multilevel Monte Carlo Method Introduction Fundamental Theory Forming the MLMC Estimator Analyzing Efficiency and Cost of the MLMC Estimator MLMC Complexity Theorem Adaptive MLMC Numerical Experiments Introduction Numerical Experiment 1: Optimal MLMC Setup Results Discussion Numerical Experiment 2: Standard MC vs Adaptive MLMC Setup Results Discussion Conclusion and Outlook 52 References 54 v

8 Chapter 1 Introduction 1.1 Background A standard problem in the field of computational finance, as well as many others, is that of estimating an expected value. In finance, this is often the case in for example the pricing of derivative securities financial contracts whose values are derived from one or several underlying assets [17]. The usual setting is that the price is found as an expected value of a functional of a random variable or stochastic process, defined by a stochastic differential equation (SDE). Over the years, the random sampling algorithm Monte Carlo (MC) has become an invaluable tool to estimate such quantities, due to its flexibility and relative ease of implementation [12, 17]. In spite of its popularity, MC estimation is inherently slow in its standard form standard MC (SMC) with the variance of the estimation converging as one over the number of samples. Consequently, this implies that the estimation error converges as one over the square root of the number of samples [16]. Additionally, whenever the analytical solution to the underlying SDE is not readily available, numerical approximation of the stochastic process is necessary for SMC estimation. This approximation adds another layer of computational complexity to the SMC algorithm and introduces a bias due to discretization error. Thus, the computational cost of achieving a certain level of accuracy of the estimation using SMC may be relatively high if path approximation is necessary, and it easily escalates when the sought accuracy improves. In order to asses the slow convergence of SMC a number of so-called variance reduction techniques have been developed to be incorporated with SMC. However, most of these techniques do not in effect change the actual computational complexity, they merely reduce the computational cost. On the contrary, a method which does reduce the complexity is the so-called quasi-mc method, which instead of using random sampling selects the samples in very clever ways in order to reduce the estimation error [12]. Unfortunately, quasi-mc requires a lot of additional knowledge compared to SMC in order to be implemented. Finally, to reduce the discretization error, high-order numerical methods have been used to approximate the SDE of interest, such that the approximation is improved and the overall error is reduced. 1

9 1. Introduction Purpose In 2008, Michael B. Giles presented in his article Multilevel Monte Carlo Path Simulation, appearing in Operations Research, a technique called multilevel Monte Carlo (MLMC). This technique is only applicable when the SDE of interest is discretized, and the idea is to perform path simulations at different levels of discretization. At a coarse discretization simulations are computationally cheap, and provide a low estimation accuracy. Conversely, fine discretization provides high accuracy and comes at a high computational cost. By performing most simulations at a coarse discretization and sequentially improving the discretization while decreasing the number of simulations, the computation is optimized, resulting in an overall lower computational complexity. Under certain conditions MLMC can provide a substantial improvement of the computational complexity, compared to SMC [5]. In fact, with a sufficiently good numerical method in conjunction with MLMC, the computational complexity will be comparable to that of SMC when paths need not be approximated [7]. Thus, by using MLMC it is possible to eradicate the extra computational complexity introduced by numerically approximating paths. The introduction of MLMC has sparked intense research of the method and its application in computational finance and several other fields. Much of the research concerns how to effectively apply MLMC on a specific problem, and how to combine known variance reduction techniques with MLMC, see for example [6, 8]. For instance, it has been shown that by combining quasi-mc and MLMC it is possible to achieve an even lower complexity than either of the two methods independently [9]. All of the above motivates the study of MLMC, and shows that this new method of estimation has a lot of potential, and can be used to further improve previously established standards of estimation. 1.2 Purpose This thesis aims to introduce SMC and MLMC in detail, discussing their respective underlying theory in a clear and accessible way. This includes introducing the methods on a conceptual level, as well as deriving their computational complexities under different circumstances. Furthermore, we also aim to illustrate differences in the performance between SMC and MLMC when implemented on a relevant problem from computational finance, as well as compare experimental results to theoretical results. Notably, we are interested in seeing how the choice of the numerical method influences the complexity of SMC and MLMC, as well as how these complexities compare to the complexity of SMC without path approximation. 1.3 Goals A main target of this thesis will be to provide concrete evidence that MLMC yields superior results compared to SMC, with respect to both computational cost as well as achieved accuracy when implemented on a relevant problem from computational finance. Additionally, we would like to answer the question of whether MLMC actually manages in practice to effectively remove the additional computational complexity arising from path approximation. 2

10 1. Introduction Outline 1.4 Outline This section concludes Chapter 1. The outline of the remaining chapters in this thesis are as follows: Chapter 2: Preliminary knowledge necessary in order to comprehend and adequately explain the central parts of this thesis. Notably, SDE theory for modelling of asset price dynamics and numerical methods for SDEs. Chapter 3: Introduces the theory of SMC with and without the need of path approximation as well as a quick discussion of variance reduction. Chapter 4: Introduces the theory of MLMC, including a general complexity theorem as well as an adaptive implementation approach. Chapter 5: Two numerical experiments are introduced, performed and discussed. The first to determine the optimal implementation of MLMC, a static or adaptive approach. The second to illustrate the difference in performance of adaptive MLMC and SMC depending on the used numerical method and if the analytical solution is available. Chapter 6: Concludes the thesis. Summarizes the most interesting results and provides a discussion of them. Further, suggestions of what could be done next in order to expand the thesis is also given. 3

11 Chapter 2 Preliminaries 2.1 Two Fundamental Theorems for Monte Carlo Methods Monte Carlo methods are built upon two fundamental limit theorems of probability theory, the strong law of large numbers and the central limit theorem. Let us introduce them both The Strong Law of Large Numbers The strong law of large numbers is the foundation of Monte Carlo methods, it ensures that the used estimator converges correctly in the limit to the theoretical expectation which is sought. Several versions of the strong law exist, with different assumptions and conditions. The simplest form of the strong law is attributed to the Russian mathematician Andrey Nikolaevich Kolmogorov ( ), and will suffice for our uses. This version of the theorem is given below, based on [12, 13]. Theorem (Kolmogorov s Strong Law of Large Numbers). Let {ξ i : i N} be a sequence of (integrable, real-valued) independent and identically distributed (i.i.d.) random variables (defined on some probability space (Ω, A, P)) with expected value E[ξ i ] = µ. Let the sequence s running average be defined as X n := 1 n n ξ i, (2.1) i=1 then, as n, X n a.s. µ. Thus, the strong law states that by using a sequence of independent and identically distributed random variables, their arithmetic mean will almost surely (a.s.) converge to their common theoretical expectation. If the reader is unfamiliar with almost sure convergence, a definition based on [12] is given below. 4

12 2. Preliminaries Two Fundamental Theorems for Monte Carlo Methods Definition (Almost Sure Convergence). Let {X i : i N} be a sequence of random variables (like those in Eq. 2.1 for example). For a given µ R, we say that the sequence {X n } converges almost surely to µ, if and only if there exist for all δ, ε > 0 an n 0 N such that P[ X n µ > δ, n > n 0 ] < ε The Central Limit Theorem The central limit theorem has an important role in Monte Carlo methods since it justifies the construction of error bounds of the estimator using the normal distribution. Based on [12] we state this theorem below. Theorem (The Central Limit Theorem). Let {ξ i : i N} be a sequence of i.i.d. random variables with expectation E[ξ i ] = µ, and finite variance, V[ξ i ] = σ 2. Let the sequence s running sum be defined as S n := n ξ i, and further let the centralized and normalized running sum be defined as, i=1 X n := S n nµ σ n, (2.2) then, as n, X n i.d. N (0, 1). Thus, as the running sum of a sequence of i.i.d. random variables (with finite expectations and variances) tends to infinity, the corresponding centralized and normalized running sum will converge in distribution (i.d.) to the standard normal distribution. This means that the actual outcomes of such a centralized and normalized sum after many samples will be distributed as the standard normal distribution. If the reader is unfamiliar with the notion of convergence in distribution, a definition based on [10] is given below. Definition (Convergence in Distribution). Let {X i : i N} be a sequence of random variables (like those in Eq. 2.2 for example) and F Xi (x) their respective cumulative distribution function. If for some random variable X with cumulative distribution function F X (x), lim F X n (x) = F X (x), n x where F X (x) is continuous, it is said that X n converges in distribution to X. 5

13 2. Preliminaries Stochastic Differential Equations 2.2 Stochastic Differential Equations One of the most commonly studied problems in finance is the modelling of asset price dynamics. Usually, the dynamics are modelled as a stochastic differential equation (SDE) of the form ds(t) = µs(t)dt + σs(t)dw (t), S(0) = S 0, (2.3) where S(t) is a stochastic process of the asset s price, S0 is some given initial value, µ is the asset s expected percentage return (mean), and σ is the asset s percentage volatility (standard deviation) [17]. Further, W (t) is a specific type of stochastic process called a Wiener process; named after the American mathematician Norbert Wiener ( ). If the reader is unfamiliar with these concepts both are defined and explained below, based on [1, 13]. Definition (Stochastic process). A stochastic process is a family of random variables {X(t, ω) : t τ, ω Ω} defined on some probability space (Ω, A, P). The stochastic process is called discrete if τ is a discrete set and continuous if τ is a continuous set. We will mainly be interested in continuous stochastic processes as defined above, where τ = [0, T ], T > 0, defines a continuous time interval. Such a process is a function, X(t, ω) : τ Ω R, mapping either to a discrete or continuous set of R. Further, for any fixed value t τ, X(t, ) reduces to a random variable, and for any fixed value ω Ω, X(, ω) is said to be a path 1, since it maps the specific evolution of X(, ω) over the interval τ to R. Finally, by convention, the variable ω is generally suppressed, such that X(t) = X(t, ω) denotes a random variable at each time t, and X( ) denotes a path. Continuing, the Wiener process, or Brownian motion as it is also quite commonly called 2, is an important concept in the study of SDEs since it often acts as the stochastic element. Definition (Wiener process). Without loss of generality, assume T > 0 and τ = [0, T ]. real-valued stochastic process, {W (t) : t τ}, with continuous paths, satisfies the following properties: (i) W (0) = 0 almost surely (ii) W (t) W (s) N (0, t s) for 0 s < t T (iii) W (t 2 ) W (t 1 ) and W (t 4 ) W (t 3 ) are independent for 0 t 1 < t 2 t 3 < t 4 T it is said to be a (one-dimensional) Wiener process. A difference such as in (ii) is usually denoted as a Wiener increment, and from (iii) it is evident that Wiener increments are independent of each other. Also, it is said that the increments are stationary, since from (ii) the distribution of an increment, W (t) W (s), only depends on the difference t s, thus them being time-invariant. Further, from (i) and (ii) one may deduce that W (t) N (0, t) for all t τ, and finally, it can be shown that W (t) is almost surely nowhere differentiable [13]. 1 Also known as a trajectory, or realization of the stochastic process. 2 In reference to the similar nature of such phenomena. If a 6

14 2. Preliminaries Stochastic Differential Equations Returning to Eq Formally, the SDE defined by Eq. 2.3 is called geometric Brownian motion (GBM), and its analytical solution, S(t), on some time interval t = [0, T ], T > 0, reads S(t) = S 0 exp ) ] [(µ σ2 t + σw (t). (2.4) 2 This solution can be derived using for example stochastic calculus, see [17, 1]. We can further remark that since the solution S(t) depends upon the Wiener process, W (t), it is itself also a stochastic process, and for a fixed value of t it has the following properties E[S(t)] = S 0 e µt, V[S(t)] = S 2 0e 2µt (e σ2t 1). In general, GBM, Eq. 2.3, is used to model asset price dynamics since neither the expected return nor the volatility depends on the current asset price. Further, the process cannot take on negative values, and because it depends on a Wiener process, it possesses the same irregular paths. All these attributes are typically observed in real asset price dynamics [17] Numerical Methods for SDEs As we have already seen, the analytical solution to the geometric Brownian motion SDE, Eq. 2.3, is given by Eq Generally, this is an exception. Most SDEs are notoriously hard to solve analytically, if it is even possible. Thus, in order to study the solution of an SDE over time, one usually needs to approximate it using some numerical method. In this section we will introduce two numerical methods for discretization of SDEs the Euler Maruyama method and Milstein s method and properties of these. Given a general Wiener-driven SDE of the form dx(t) = a(t, X(t))dt + b(t, X(t))dW (t), X(0) = X 0, (2.5) where t τ = [0, T ], T > 0, and the functions a, b : τ R R. Introduce the equidistant 3 time partition of τ, using K steps, 0 = t 0 < t 1 < < t K 1 < t K = T, such that the difference between adjacent steps, denoted t = T/K, is constant. The Euler Maruyama discretization of Eq. 2.5 using the step size t is for k = 0, 1,..., K 1, given by X t k+1 = Xk t X t 0 = X 0, + a ( k t, X t k ) ( ) t + b k t, X t W t k k, (2.6) where, W t k := W ((k + 1) t) W (k t) N (0, t). Similarly, the Milstein discretization of Eq This is a simplification, in general there is no requirement for equidistant time partitioning. 7

15 2. Preliminaries Stochastic Differential Equations using the step size t is for k = 0, 1,..., K 1, given by X t k+1 = Xk t X t 0 = X 0, + a ( k t, X t k b( k t, X t k ) t + b ( k t, X t ) b ( k t, X t k k ) W t k )[( W t k ) 2 t ], (2.7) where the derivative of the function b(t, X(t)) is with respect to the state variable, X(t). Both methods above are derived in the same way, using a stochastic version of Taylor expansion on Eq From Eq. 2.6 one can see that the Euler Maruyama method only includes first-order terms and ignores higher orders, whereas Milstein s method, Eq. 2.7, also includes a second-order term. Thus, one can see Milstein s method as a direct extension of the Euler Maruyama method, naturally including more information. The discretization schemes above, Eq. 2.6 and Eq. 2.7, respectively provide for each point in time t k, k = 1, 2,..., K, the approximation X t k of X(t k ), i.e. X t k X(t k ). Intuitively, we expect that the approximation improves as the step size decreases, and that thus the discretization error Xk t X(t k) 0 as t 0. We will now introduce two important concepts, strong and weak convergence, which both are used to quantify the convergence of numerical methods, and the speed at which they do so. Definition (Strong Convergence of Numerical Method). Let t be the equidistant step size of a time partition 0 = t 0 < t 1 < < t K = T of the interval [0, T ], T > 0, and X t the corresponding SDE discretization of Eq Then, the strong error is defined as and the numerical method is said to converge strongly if errstrong t := sup E [ X k t X(k t) ], (2.8) 0 k K lim t 0 err t strong = 0. A method which converges strongly is said to have strong order of convergence, or strong convergence rate, q, if for some constants A and t 0 > 0 err t strong A t q, t [0, t 0 ]. From Eq. 2.8 it is somewhat evident that the strong error quantifies the average, overall, pathwise discretization error. Meaning that for it to go to zero, the numerically approximated paths would on average have to converge to the analytical solution at each point. Definition (Weak Convergence of Numerical Method). Let t be the equidistant step size of a time partition 0 = t 0 < t 1 < < t K = T of [0, T ], and X t the corresponding SDE discretization of Eq Further, let f : R R be a 2(p + 1) times continuously differentiable function with at most 8

16 2. Preliminaries Stochastic Differential Equations polynomial growth. Then, the weak error is defined as errweak t := sup [ E f(x t k ) ] E [f(x(k t))], (2.9) 0 k K and the numerical method is said to converge weakly if lim t 0 err t weak = 0. A method which converges weakly is said to have weak order of convergence, or weak convergence rate, p, if for some constants A and t 0 > 0 err t weak A t p, t [0, t 0 ]. (2.10) There is no need to worry about the function f in the definition above, for our purposes it will suffice to let f(x) = x. Using this function, the weak error, Eq. 2.9, quantifies the error between the average numerically approximated path and the average analytical path. This implies that if the weak error would go to zero, it does not ensure that a numerically approximated path approaches the path of the analytical solution, as the strong error did. Hence, the notion of weak. For applications where only the average value of paths is interesting, the weak error will suffice to control the error. However, if one is interested in controlling the error of specific paths, it is crucial to consider the strong error instead. Without stating any proofs, in general, given sufficiently nice conditions, the Euler Maruyama method, Eq. 2.6, has strong order of convergence, q EM = 1/2, and weak order, p EM = 1. On the contrary, Milstein s method, Eq. 2.7, has strong order of convergence, q M = 1, but still weak order, p M = 1 [13]. We attribute the higher strong order to the second-order term in Milstein s scheme Simulation of Geometric Brownian Motion The ability to simulate paths of geometric Brownian motion, Eq. 2.3, will play a large role in this thesis, and in this section we apply the two numerical methods introduced above on this SDE. Also, since the analytical solution to GBM is known, one may question why it is necessary to approximate the SDE using numerical methods. In this case, it is not. However, since the analytical solution is indeed known, it offers great insight into the performance of the numerical methods, since they can be compared to the analytical solution. Firstly, we quickly explain how to use the analytical solution to simulate a path on the time interval [0, T ], T > 0. Since there is no need to partition the time, one simply chooses a sufficient number of time points to evaluate the path at, using Eq Since the Wiener process present in the analytical solution is normally distributed as, W (t) N (0, t), it is evaluated as a standard normal realization scaled by t, to obtain the correct variance. The advantage of the analytical solution is that it is not necessary to simulate the path at prior time points, if one is only interested in for example the end time, T. On the contrary, using either numerical method, one must generate the path up to the point of interest. 9

17 2. Preliminaries Big-O Notation If we instead want to simulate approximated paths on the time interval [0, T ], we first partition this interval into equidistant time steps as, 0 = t 0 < t 1 < < t K = T, where the step size is given by t = T/K. Applying the Euler Maruyama scheme, Eq. 2.6, on GBM, Eq. 2.3, we can directly see that by comparison with Eq. 2.5, we must have a(t, X(t)) = µs(t), and b(t, X(t)) = σs(t). Thus, the Euler Maruyama discretization of GBM reads Sk+1 t = Sk t S0 t = S 0. + µs t k t + σs t k Wk t, (2.11) Similarly, by applying Milstein s scheme, Eq. 2.7, on GBM, Eq. 2.3, and using b ( ) k t, Xk t = σ, we obtain the Milstein discretization of GBM as S t k+1 = Sk t S t 0 = S 0. + µs t k σ2 S t k t + σs t k Wk t [( ) W t 2 ] t k (2.12) Finally, we only have one problem left, the Wiener increments, W t, present in both discretizations. Happily, as we noted earlier, these are defined as W t k k := W ((k + 1) t) W (k t) N (0, t). Thus, they can be created by using standard normal realizations and scaling them by t to obtain the appropriate variance of t. This is basically the same as done for the analytical solution earlier. In conclusion, in order to generate a path of GBM using either Eq or Eq. 2.12, one uses the initial value to get the path started, and recursively update the value in each step, where one also needs to generate a Wiener increment as suggest above before the next step can be calculated. 2.3 Big-O Notation On many occasions in this thesis we will be interested in what happens to a function in the limit towards infinity or zero. To describe this limiting behaviour the so-called big-o notation is used. Definition (Big-O Notation). Let the functions f, g : R R. We write f(x) = O(g(x)) as x, if and only if there exists real constants k > 0 and x 0, such that for all x > x 0 f(x) k g(x). Similarly, we write f(x) = O(g(x)) as x 0, if and only if there exist positive real constants k and d, such that for all x satisfying 0 < x < d, f(x) k g(x). Informally, we simply write f(x) = O(g(x)) for either case since it is usually clear from the context which is meant. The interpretation of this notation is that the function f(x) is bounded from the above by some multiple of g(x) in the limit, and thus f(x) will never exceed the growth rate of g(x). 10

18 2. Preliminaries Option Pricing This notation is simple to use, and it has the following two properties which are often used: (i) If f(x) = O(g(x)), then for some constant c, cf(x) = O(cg(x)) = O(g(x)). (ii) If f 1 (x) = O(g 1 (x)) and f 2 (x) = O(g 2 (x)), then f 1 (x)f 2 (x) = O(g 1 (x)g 2 (x)). 2.4 Option Pricing In the numerical experiments of this thesis we will be pricing a European call option. A European call option is a financial contract contingent on the evolution of an underlying stock process, S(t). The holder of such a contract has the right, but not the obligation, to buy the underlying stock of the contract at a predetermined time, the so-called maturity, T, for a predetermined price, the so-called strike price, K. The nomenclature call refers to the option to buy the underlying stock. The opposite, a so-called put, gives the option to sell the underlying stock under similar circumstances. The nomenclature European refers to the ability to only exercise the option at the maturity, T. In addition to European-style options a wide range of option-styles exist with more elaborate exercising-schemes. An example is American options, which may be exercised by the holder whenever seems suitable. For further details on different option-styles, see for example [17]. Rationally, a European call option will only be exercised if the stock s price at maturity is higher than the strike price, thus yielding a payoff of S(T ) K. If the option is not exercised the payoff is evidently zero. To price this kind of contract, the expected payoff at maturity must be reflected in its price, otherwise the price would not be fair. Luckily, for a European call option it is possible to derive an analytical formula of its price, the so-called Black Scholes formula [3]. The Black Scholes formula reads O c (S(t 0 ), t 0 ) = N(d 1 )S(t 0 ) N(d 2 )Ke r(t t0), (2.13) where O c (S(t 0 ), t 0 ) is the European call option s price at the sought time t = t 0, and at the current price S(t 0 ) of the underlying stock. Further, the function N( ) is the standard normal cumulative distribution function, and the remaining variables of Eq are 1 d 1 = σ T t 0 [ log d 2 = d 1 σ T t 0, ( S(t0 ) K ) + ) ] (r + σ2 (T t 0 ), 2 where σ is the underlying stock s volatility, T is the option s maturity, K is the option s strike price, and r is the so-called risk-free interest rate. It is important to note that r does not denote the underlying stock s expected return, but instead the risk-free interest rate, which is basically the expected return rate of money invested in a risk-free fashion. This parameter is used instead of the stock s expected percentage return due to an important step in the derivation of Eq. 2.13, see [3] for details. It is also possible to price a European call option by random sampling simulation. In order to do so, one 11

19 2. Preliminaries Option Pricing needs to generate paths of the underlying stock process, S(t), from the current time up until maturity. In general, there are many possibilities of how to model the underlying stock process, but the most recognized is to assume that the process is a GBM, Eq Paths may then be simulated using one of the ways described in Section However, if we are to price a European call option, the underlying stock process must use the risk-free interest rate, r, as its expected percentage return, instead of the stock s actual expected rate of return, µ, for the pricing to be correct. This follows from the same argument which states that the risk-free interest rate should be used in Eq To obtain the fair price of the option, the expected payoff is determined using for example either method in Chapter 3 or Chapter 4 by evaluating the payoff function, max{s(t ) K, 0}, for each path. The obtained price then needs to be adjusted to reflect the fair price at the sought time, t = t 0, and thus the price is discounted to this time by multiplying it with a factor of e r(t t0). 12

20 Chapter 3 The Standard Monte Carlo Method 3.1 Introduction Often regarded as one of the earliest recorded resemblances of a Monte Carlo algorithm is the experiment proposed in 1733, and published 1777 by French scientist George-Louis Leclerc, Comte de Buffon ( ) [2]. In the article [2] by Behrends and Buescu, Buffon s experiment is summarized, as well as his intentions with the experiment and who actually made concrete use of it. The experiment s original description regards randomly throwing a stick of length 2r onto a floor of wooden planks of width a, and the probability, P, of the stick intersecting more than one plank upon landing, given that 2r < a. A common misconception is that this experiment was designed in order to estimate π, but this is not really the case. In fact, Buffon had the intention of creating a fair game to gamble with, and accurately derived an expression for the probability P in order to design such a game. However, and much why this experiment is famous; in 1812 French mathematician Pierre-Simon de Laplace ( ) published another derivation of the probability P, such that P = 4r/(πa). Laplace further remarked that by repeating the experiment many times an estimate of P could be obtained as the quotient between the number of intersections and total throws, and thus an approximation of π could be derived [2]. This experiment has later become known as Buffon s needle. In essence, it highlights the underlying idea of Monte Carlo methods; that through repeated random sampling a quantity of interest is estimated based on some probabilistic definition. Regarding Buffon s needle we can draw two conclusions. First, the expected outcome of the experiment, i.e. the probability P, is already known beforehand. Thus, the experiment would merely confirm the already derived theoretical answer. Secondly, one would physically have to throw the sticks or needles oneself quite tiresome and not at all very efficient. Happily, the modern approach of Monte Carlo methods is the opposite of both these conclusions. Today, the aim is mostly to find a probabilistic definition of a problem, and use Monte Carlo to estimate and study the unknown solution, which may be too difficult or impossible to solve in an analytical matter [12]. Further, the random sampling is thankfully done by computers. 13

21 3. The Standard Monte Carlo Method Fundamental Theory Modern Monte Carlo methods have their origins at the Los Alamos national laboratory in the late 1940s, during the development of the hydrogen bomb and problems regarding neutron diffusion [15]. Mathematician Stanislaw Ulam ( ) first conceptualized the method in 1946 and later the same year described it to John von Neumann ( ), who constructed a working algorithm of it for the first electronic computer, the ENIAC [4]. The method was needless to say a great success. Soon after, in 1949, the first publication of the method appears in the article The Monte Carlo Method by Nicholas Metropolis ( ) and Ulam, in the Journal of the American Statistical Association [13]. It is now suitable to remark that the name Monte Carlo was suggested by Metropolis to Ulam in reference to the somewhat gambling nature of random sampling, and the fact that Ulam s uncle often would go to the then centre of gambling the Monte Carlo casino [15]. After the initial development of the Monte Carlo methodology the method quickly grew and expanded to new fields and problems. For example, the first use of Monte Carlo methods in finance is attributed to Phelim Boyle (1941-), who in 1977 published the article Options: A Monte Carlo approach, appearing in the Journal of Financial Economics, where a framework for option valuation using a Monte Carlo method is developed [17]. Furthermore, with the rise of computers the usage of Monte Carlo methods has only increased, and today they act as an invaluable tool due to their inherent simplicity and applicability, in whichever areas where sampling, estimation, or optimization is of the essence [14]. In this chapter the so-called standard Monte Carlo method will be considered, where the reader is only assumed to have a basic background of probability theory. Initially, the fundamental theory will be introduced, along with a brief discussion of the method s efficiency and the notion of variance reduction techniques. After this, the theory of the standard Monte Carlo method is expanded such that an individual sample consist of an approximated path of the solution of a stochastic differential equation (SDE). We revisit the theory in this setting and also note how the efficiency has declined due to the introduction of a bias discretization error. The theory of this chapter is largely based on the corresponding sections in the books by Jäckel [12] and Korn et al. [13], respectively, unless stated otherwise. 3.2 Fundamental Theory As explained in the previous section, there exist a vast amount of different Monte Carlo methods, each to be used in a different setting and with varying estimation efficiency. In this section we develop the standard Monte Carlo method using the so-called crude estimator 1, and explain the supporting theory for its use when estimating the expected value of a function of a random variable, E[g(X)] The Crude Estimator Assume that we have a nice function, g : R R, which depends on some random variable, X, with cumulative distribution function F X (x). This in effect makes g(x) a random variable. Assume further 1 The nomenclature refers to its direct application and relative inefficiency compared to more sophisticated estimators. 14

22 3. The Standard Monte Carlo Method Fundamental Theory that g(x) has a finite variance and that we are interested in the unknown expected value of g(x), E[g(X)] := θ. This is a rather common setting especially in finance where X may be an asset s price at a certain point in time and the function g some general payoff function. First we will introduce the theoretical framework, and then the simulation approach to estimate θ, concluding with a discussion of the efficiency of the crude estimator. Since X F X (x), if we obtain a set of N 1 i.i.d. random variables such that X i F X (x), i = 1, 2,..., N, the crude estimator of θ is defined as ˆΘ N := 1 N N g(x i ). (3.1) i=1 Calculating the expected value of the estimator, we obtain [ ] E[ ˆΘ 1 N N ] = E g(x i ) = 1 N N i=1 N E[g(X i )] = θ, (3.2) i=1 and we note that the bias of the crude estimator is zero, Bias[ ˆΘ N ] := E[ ˆΘ N θ] = E[ ˆΘ N ] θ = 0, (3.3) meaning that the estimator is unbiased with respect to θ, i.e. that it has no systematic error. Thus, when using the estimator in simulation, it will on average estimate θ correctly. Eq. 3.2 also tells us that ˆΘ N a.s. θ as N by the strong law of large numbers. Further, since realizations are i.i.d., the variance of ˆΘ N can be shown to be [ ] V[ ˆΘ 1 N N ] = V g(x i ) N i=1 = 1 N 2 N i=1 V[g(X i )] = V[g(X)] N. (3.4) Finally, using Eq. 3.4 and the fact that since N 1, by the central limit theorem, the crude estimator, Eq. 3.1, approximately converges in distribution as ˆΘ N ( i.d. N θ, V[g(X)] ). N This justifies the construction of a 100(1 α) % confidence interval for θ as [ ] V[g(X)] ˆΘ N z 1 α/2, ˆΘ V[g(X)] N + z 1 α/2, (3.5) N N where z 1 α/2 is the (1 α/2) quantile of the standard normal distribution. 15

23 3. The Standard Monte Carlo Method Fundamental Theory Simulation Approach Utilizing the above framework, a straightforward way to estimate the sought expected value, θ, is to perform a Monte Carlo simulation. First generate N 1 independent realizations of X F X (x), x 1, x 2,..., x N, which, without going deeper on exactly how this is accomplished, should be a fairly easy task assuming F X (x) is known using some built-in function of the used language. realization of X as x i, one estimates θ with the point estimator Then, if we denote the i:th ˆθ N = 1 N N g(x i ). i=1 Further, by estimating V[g(X)] using the sample variance (as a point estimate of V[g(X)]) obtained from the N evaluations of g(x i ), as ( ) ( 2 ˆσ g,n 2 = 1 N [g(x i )] N 2 1 N g(x i )), N i=1 we can [12] obtain an approximate 100(1 α) % confidence interval of θ by considering ˆθN z 1 α/2 ˆσ 2 g,n N i=1, ˆθ ˆσ g,n 2 N + z 1 α/2. N In principal, this is how a Monte Carlo simulation is performed. Conceptually quite easy. Now that we have laid the foundation of how a Monte Carlo simulation is performed and motivated, we are in a good position to discuss the performance of the crude estimator Analyzing Efficiency and Cost of the Crude Estimator It is important to recall that the crude estimator, Eq. 3.1, is a random quantity, and that this implies that the difference between the estimator and the true value, the so-called Monte Carlo error, ˆΘ N θ, also is a random quantity. A common way of quantifying the Monte Carlo error for an arbitrary estimator, Θ N, of the unknown parameter θ, is by the mean square error (MSE), MSE [ ΘN ] := E [ ( ΘN θ ) 2 ] = V [ ΘN ] + ( E [ ΘN θ ]) 2, (3.6) which, as seen, is equal to the sum of the variance and squared bias of the estimator, respectively. Thus, the MSE is a quantity which incorporates both the systematic error, i.e. bias, and the size of random fluctuations in the form of variance. For this reason, the MSE commonly acts as a measure of an estimator s efficiency, i.e. how good it is at estimating θ. Generally, when choosing between two estimators the one with the smaller MSE is considered the more efficient. 16

24 3. The Standard Monte Carlo Method Fundamental Theory Since the crude estimator is unbiased by Eq. 3.3, we can, for the moment, disregard the bias term in Eq. 3.6, and obtain with the help of Eq. 3.4 its MSE as [ ] [ ] MSE ˆΘN = V ˆΘN = V[g(X)] N. Because the MSE effectively has the squared dimensions of the crude estimator, it is not appropriate to use as a measure of accuracy per se. Instead, the square root, the so-called root mean square error (RMSE) is often used. In the following, we adopt this definition of accuracy and take [ ] [ ] [ ] V[g(X)] ɛ = RMSE ˆΘN := MSE ˆΘN = V ˆΘN =, (3.7) N as a measure of accuracy of the crude estimator. From Eq. 3.7 one sees that it is exactly the RMSE which determines the width of the confidence interval for the crude estimator, Eq. 3.5, for a given quantile. Eq. 3.7 effectively is the same as the half-width when z 1 α/2 = 1, corresponding roughly to a 68 % confidence interval. This means that if repeating the estimation independently a large number of times, the estimate would fall within ±ɛ of the exact answer, θ, with a frequency of at least In Eq. 3.7, since V[g(X)] is a constant, the only possibility in order to improve the accuracy, ɛ, is to incorporate more i.i.d. random variables, i.e. to increase N. It is also evident that ɛ 0 as N, meaning that the inherent uncertainty in the crude estimator decreases as N increases. Accordingly, the accuracy is proportional to 1/ N, as [ ] ( ) V[g(X)] 1 ɛ = RMSE ˆΘN = = O, N N or more importantly, N is proportional to ɛ 2 as N = O(ɛ 2 ). (3.8) Eq. 3.8 has an important implication. Assume for example that one wishes to improve the accuracy by two decimals, i.e. reduce ɛ to a hundredth of its size. For this to be possible it is necessary to increase N by factor of 10 4 (!), which may, or may not, be feasible, depending on the original size of N, and the computer s RAM. An important remark is in order here. Despite the seemingly slow convergence evident in Eq. 3.8, which compared to other estimation methods, for example finite-difference or quadrature methods, actually is quite bad, it shines when the dimensionality, d, increases. Here, a concrete example of a higher dimensionality would be some function, f, dependent on five individual random variables. In such circumstances, and in general whenever d > 4, the standard Monte Carlo method surpasses its competitors, due to the fact that its convergence is independent of the dimensionality, whereas the other methods respective convergence is not [12, 17]. 17

25 3. The Standard Monte Carlo Method Variance Reduction Finally, what just has been discussed above of course carries over in simulation, with the only exception that V[g(X)] is approximated by the sample variance, ˆσ g,n 2, as noted earlier. Computational Complexity For the above setting, in order to achieve an accuracy proportional to, or of order, ɛ, it is evident from Eq. 3.8 that the number of realizations, N, needs to be in proportion to ɛ 2. This in turn implies that the computational cost, C, i.e. the needed number of computations, to achieve an accuracy of order ɛ, is also in proportion to N and ɛ 2. Thus, we obtain C = O(N) = O(ɛ 2 ), and we say that the computational complexity for the standard Monte Carlo method in order to achieve a RMSE accuracy of ɛ is O(ɛ 2 ). 3.3 Variance Reduction A more elegant way of improving the accuracy, Eq. 3.7, is instead to find a more efficient estimator to use, contrary to the crude one. If one is able to find a new estimator Θ N, such that MSE[ Θ N ] < V[ ˆΘ N ], or equivalently V[ Θ N ] < V[ ˆΘ N ], if Θ N is unbiased, it is evident from Eq. 3.7 that the accuracy must improve as well, independent of N. In this section we will for the sake of clarity introduce the rather simple variance reduction technique of control variates, which will facilitate the understanding and motivation of the main focus in the next chapter, the multilevel Monte Carlo method. In addition to control variates, there are many different techniques, for example; common random numbers, antithetic variates, importance sampling and stratified sampling. These are all included in most literature, for example [13]. The idea of control variates is that we make use of another random variable, h(x), with a known mean µ, which we think is related to our random variable of interest, g(x). The random variable h(x) is then taken as the control variate. Using the random variables g(x) and h(x) we can create a new random variable, Z(X), as Z(X) = g(x) β(h(x) µ), β R, such that E[Z(X)] = θ. Further, the variance of Z(X) is given by V[Z(X)] = V[g(X)] 2βCov[g(X), h(x)] + β 2 V[h(X)], (3.9) where the parameter β may be chosen arbitrarily so as to minimize V[Z(X)]. The value for which the variance is minimized is denoted by β and is given by β = Cov[g(X), h(x)]. V[h(X)] 18

26 3. The Standard Monte Carlo Method Standard Monte Carlo with SDE Path Simulation Inserting β in Eq. 3.9 the variance is simplified to V[Z(X)] = V[g(X)] (Cov[g(X), h(x)])2 V[h(X)] = V[g(X)](1 ρ 2 gh), where ρ gh is the linear correlation between the random variables g(x) and h(x). Thus, we can see that as long as there is any correlation at all, the variance of Z(X) will always be less than that of g(x). An explanation of this is the way Z(X) is constructed, where a deviating value of g(x) will be compensated by the value (h(x) µ), if they are correlated. In light of the previous example, if we can obtain a set of N 1 i.i.d. X i F X (x), i = 1, 2,..., N, a control variate estimator of θ will be random variables such that ˆΘ CV,N = 1 N N Z(X i ), i=1 where as before, ˆΘ CV,N the estimator ˆΘ CV,N is found to be a.s. θ as N by the strong law of large numbers. Further, the variance of V[ ˆΘ CV,N ] = V[g(X)] N (1 ρ2 g,h). Thus, in order to obtain a more efficient estimator than the crude estimator, we need only choose the random variable h(x) such that ρ gh 0. For this purpose basically any random variable dependent on X will suffice. Note though, Eq. 3.8 will still apply using a control variate, but the accuracy for a given N will be better. Simulation Efficiency Even if a new estimator is more efficient in theory, it may come with an additional cost in implementation. This implies that we need to also take into consideration the computational effort needed to use a more efficient estimator, otherwise the simulation efficiency may be worse of than before. Without trying to derive an expression to quantify the simulation efficiency, we can simply conclude that an estimator which achieves the same accuracy, but in less time than another, is evidently more efficient in simulation. Thus, a straightforward way of quantifying the simulation efficiency is simply to measure the total computation time needed in order to obtain the given accuracy. 3.4 Standard Monte Carlo with SDE Path Simulation What if we now would replace the random variable, X, used in the previous sections with a stochastic process. Then, instead of realizations of the random variable being used in the estimation process, paths of the stochastic process would be used. To introduce this case further, assume once again that we have a nice function, g : R R, but this time dependent on the solution, X(t), of some SDE defined on the 19

27 3. The Standard Monte Carlo Method Standard Monte Carlo with SDE Path Simulation time interval [0, T ], T > 0. We are interested in estimating by simulation the expected value E[g(X(t))], for some specific point in time of the stochastic process. As noted earlier, the situation above is a rather common setting in finance, where in that case the SDE usually would be GBM, Eq. 2.3, and the function g some general payoff function. For simplicity and illustrative purposes, we choose to work with GBM with its solution denoted by S(t) and take t = T as the time of interest, i.e. we are interested in estimating E[g(S(T ))] := θ. However, it should be noted that the soon to be developed framework will of course also hold true for other SDEs, and for other points in time. Furthermore, if we for the GBM are using the analytical solution, Eq. 2.4, it all reduces down to exactly the same framework as we derived earlier, since S(T ) simply will be a random variable. On the other hand, if we approximate the SDE using a numerical method, the theory becomes slightly more complicated. This follows since we are introducing a bias in the form of discretization error by approximating the SDE, and this bias needs to be included in the analysis. Thus, we need to revisit the theory regarding the crude estimator and subsequent sections about the efficiency of the crude estimator, under the assumption that the bias now is non-zero. We start by partitioning the time interval [0, T ] into K +1 equidistant points 0 = t 0 < t 1 < < t K = T, with constant step size t = T/K. Further, we use the notation S t k for the numerical approximation of S(t k ) using a step size of t at the k:th step. However, since we are only interested in the point at which k = K, we for notational simplicity define S t := S t K S(t K) = S(T ). Now, the crude estimator, Eq. 3.1, will no longer solely depend on the number of i.i.d. random variables (i.e. in this case individual paths), N, but also the step size which is used in order to approximate the paths of the underlying process, S(t). For this reason, we denote in this context the crude estimator of θ as ˆΘ t N = 1 N N i=1 g(s t i ), (3.10) where the index i refers to individual paths. To evaluate Eq a path is first simulated using for example the Euler Maruyama method or Milstein s method, and then the function g is evaluated at the end point, t = T. This is then repeated N times. A quick look at the crude estimator shows that the expected value becomes E [ [ ] 1 ˆΘ t N = E N N i=1 thus, by the strong law of large numbers g(s t i ) ˆΘ t N ] = 1 N N i=1 E [ g(s t i ) ] = E [ g(s t ) ], (3.11) a.s. E [ g(s t ) ] as N. We note that because of the discretization error in the path approximation, E[g(S t )] θ. This is why we obtain the bias of our estimator using Eq and the definition of θ, as Bias [ ] [ ] ˆΘ t N := E ˆΘ t N θ = E [ g(s t ) ] E [g(s(t ))]. (3.12) 20