10. Monte Carlo Methods

Transcription

1 10. Monte Carlo Methods 1. Introduction. Monte Carlo simulation is an important tool in computational finance. It may be used to evaluate portfolio management rules, to price options, to simulate hedging strategies and to estimate Value at Risk. Advantages: generality, relative ease of use and flexibility. It may take into account stochastic volatility and many complicated features of exotic options and it lends itself to treating high dimensional problems, where the lattice and the PDE framework cannot be applied. Drawback: computational cost. An increasing number of replications is needed to find accurate solutions and to refine the confidence interval of the estimates we are interested in. 1

2 Even if we use the more appealing terms simulation or sampling, Monte Carlo methods are conceptually a numerical integration tools. Classical approaches to numerical integration based on quadrature formulae are deterministic (just as quasi-monte Carlo methods). Monte Carlo methods are based on random sampling, and so some connection with statistic is expected. Remark. Numerical integration may be implicitly used to estimates probabilities. In fact if A is an event which may occur or not depending on a random variable X, then P(A) = I A (x)f X (x) dx, where f X (x) is the probability function of the random variable X and I A (x) is the indicator function for event A (taking the value 1 if A occurs when X = x, 0 otherwise). 2

3 We have seen that option pricing requires computing an expected value under a risk-neutral measure, but an expected value is actually an integral. In fact the expected value of a function g( ) of a random variable X with probability function f X (x) is: E[g(X)] = g(x)f X (x) dx. In one-dimensional cases we my find an analytical solution, like in the Black- Scholescase, butthisisdifficultingeneral. IftherandomvariableX isascalar, classical deterministic methods work very well, but when expectation is taken with respect to a random vector and we must integrate over a high-dimensional space, random sampling may be necessary. Random sampling is a natural way to simulate dynamics affected by uncertainty, such as prices modeled by stochastic differential equation. Natural applications are: option pricing, portfolio optimization, risk management and estimation of Value at Risk. 3

4 Monte Carlo simulation is based on random number generation, actually we must speak of pseudo-random numbers, since nothing is random on a computer. If we put random numbers into a simulation procedure, the output will be a sequence of random variables. Given this output, statistical techniques are used to build an estimate of a quantity of interest. It is interesting to evaluate the reliability of this estimate in some way (i.e. by a confidence interval) and to carry out the simulation experiments in such a way that the estimation error is controlled. Some techniques deal with the issue of setting the number of simulation experiments (replications) properly. Intuitively, the more replications we run, the more reliable our estimates will be. Unfortunately, reaching a suitable precision might require a prohibitive number of experiments. Note that simulation may be used to evaluate the consequences of a certain policy, but it cannot generate the policy itself. 4

5 2. Monte Carlo integration. The definite integral of a function is a number, and computing that number is a deterministic problem involving no randomness. Nevertheless we may cast the problem within a stochastic framework by interpreting the integral as an expected value. For example let us consider an integral on the unit interval [0,1]: I = 1 0 g(x) dx. WemaythinkofthisintegralastheexpectedvaleE[g(U)],whereU isauniform random variable on the interval [0,1], i.e. U U[0,1]. We may estimate the expected value (a number) by a sample mean (a random variable). 5

6 What we have to do is generating a sequence {U i }, i = 1,...,m, of independent random samples from the uniform distribution and then evaluate the sample mean: Î m = 1 m m g(u i ). i=1 The strong law of large numbers implies that, with probability 1, lim Î m = I. m + 6

7 For one-dimensional integration, Monte Carlo is hardly competitive with deterministic quadrature, but when computing a multidimensional integral it may be the only viable option. In general, if we have an integral like: I = A φ(x) dx, where A R n, we may estimate I by randomly sampling a sequence of points x i A, i = 1,...,m, and building the estimator Î m = vol(a) m φ(x i ), m where vol(a) denotes the volume of the region A. To understand the formula, we should think that the ratio 1 m φ(x i ) estimates the average value of the function, which must multiplied by the m i=1 volume i=1 of the integration region in order to get the integral. 7

8 Random sampling, which is where Monte Carlo comes from, is nor really possible with a computer, but we can generate a sequence of pseudo-random numbers using generators provided by most programming languages and environments. For example in MATLAB these generators are actually part of the Statistical Toolbox, but the core MATLAB environment also offer some functions (e.g. rand, randn) to sample from common distribution. Matlab Exercise 1. Calculate I = 1 Matlab Exercise 2. Given an estimate of π. 0 e x dx

9 3. Generating Pseudo-Random Variates. In order to use simulation techniques, we first need to generate independent samples from some given distribution functions. The usual way to generate pseudo-random variates, i.e. sample from a given probability distribution, starts from the generation of pseudo-random numbers, which are simply variates from the uniform distribution on the interval [0, 1]. Then suitable transformations are applied in order to obtain the desired distribution. 9

10 Uniform Random Number Generators. The simplest and the most important distribution in simulation is the uniform distribution U[0, 1]. Note that if X is a U[0,1] random variable, then Y = a+(b a)x is a U[a,b] random variable. We focus on how to generate U[0,1] random variables. Examples. - 2 outcomes, p 1 = p 2 = 1 (coin) 2-6 outcomes, p 1 =... = p 6 = 1 6 (dice) - 3 outcomes, p 1 = p 2 = 0.49, p 3 = 0.05 (roulette wheel) 10

11 Linear Congruential Generator (LCG). A common technique to produce U[0,1] variates is to generate a sequence of integers n i, defined recursively by n i = (an i 1 ) mod m (1) for i = 1,2,...,N, where n 0 ( 0) is called the seed, a > 0 and m > 0 are integers such that a and m have no common factors and mod denotes the reminder of integer division (e.g. 15 mod 6=3). Equation (1) generates a sequence of numbers in the set {1,2,...,m 1}. Note that n i are periodic with period m 1, this is because there are not m different n i and two in {n 0,...,n m 1 } must be equal: n i = n i+p with p m 1. If the period is m 1 then (1) is said to have full period. The condition of full period is that m is a prime, a m 1 1 is divisible by m, and a j 1 is not divisible by m for j = 1,...,m 2. 11

12 Example. n 0 = 35, a = 13, m = 100. Then the sequence is {35,55,15,95,35,...}. So p = 4. 12

13 Pseudo Uniform Random Numbers. If we define x i = n i m then x i is a sequence of numbers in the interval (0,1). If (1) has full period then these x i s are called pseudo-u[0,1] random numbers. In view of the periodic property, the number m should be as large as possible, becauseasmallsetofnumbersmakestheoutcomeeasiertopredict acontrast to randomness. The main drawback of linear congruential generators is that consecutive points obtained lie on parallel hyperplanes, which implies that the unit cube cannot be uniformly filled with these points. For example, if a = 6 and m = 11, then the ten distinct points generated lie on just two parallel lines in the unit square. x i+1 13 x i

14 Choice of Parameters. Agoodchoiceofaandmisgivenbya = 16807andm = = The seed n 0 can be any positive integer and can be chosen manually. This allows us to repeatedly generate the same set of numbers, which may be useful when we want to compare different simulation techniques. In general, we let the computer choose n 0 for us, a common choice is the computer s internal clock or the current clock value. For state of the art random number generators (linear and nonlinear), see the plab website at It includes extensive information on random number generation, as well as links to free software in a variety of computing languages. 14

15 Normal Random Number Generators. Once we have generated a set of pseudo-u[0,1] random numbers, we can generate pseudo-n(0, 1) random numbers. Again there are several methods to generate a sequence of independent N(0,1) random numbers. Note that if X is a N(0,1) random variable, then Y = µ+σx is a N(µ,σ 2 ) random variable. 15

16 Convolution Method. This method is based on the Central Limit Theorem. We know that if X i are independent identically distributed random variables, with finite mean µ and finite variance σ 2, then n X i nµ Z n = i=1 σ (2) n converges in distribution to a N(0, 1) random variable, i.e. P(Z n x) Φ(x), n. If X is U[0,1] distributed then its mean is 1 2 and its variance is If we choose n = 12 then (2) is simplified to Z n = 12 i=1 X i 6. (3) 16

17 Note that Z n generated in this way is only an approximate normal random number. This is due to the fact that the Central Limit Theorem only gives convergence in distribution, not almost surely, and n should tend to infinity. In practice there are hardly any differences between pseudo-n(0, 1) random numbers generated by (3) and those generated by other techniques. The disadvantage is that we need to generate 12 uniform random numbers to generate 1 normal random number, and this does not seem efficient. 17

18 Box Muller Method. This is a direct method as follows: Generate two independent U[0,1] random numbers X 1 and X 2, define Z = h(x) where X = (X 1,X 2 ) and Z = (Z 1,Z 2 ) are R 2 vectors and h : [0,1] 2 R 2 is a vector function defined by ( (z 1,z 2 ) = h(x 1,x 2 ) = 2 lnx1 cos(2πx 2 ), ) 2 lnx 1 sin(2πx 2 ). Then Z 1 and Z 2 are two independent N(0,1) random numbers. 18

19 This result can be easily proved with the transformation method. Specifically, we can find the inverse function h 1 by (x 1,x 2 ) = h 1 (z 1,z 2 ) = ( e 1 2 (z2 1 +z2 2 ), 1 ) 2π arctanz 2. z 1 The absolute value of the determinant of the Jacobian matrix is (x 1,x 2 ) (z 1,z 2 ) = 1 1 e 1 2 z2 1 e 1 2 z2 2. (4) 2π 2π From the transformation theorem of random variables and (4) we know that if X 1 and X 2 are independent U[0,1] random variables, then Z 1 and Z 2 are independent N(0, 1) random variables. 19

20 Correlated Normal Distributions. Assume X N(µ,Σ), where Σ R n n is symmetric positive definite. To generate a normal vector X do the following: (a) Calculate the Cholesky decomposition Σ = LL T. (b) GeneratenindependentN(0,1)randomnumbersZ i andletz = (Z 1,...,Z n ). (c) Set X = µ+lz. Example. To generate 2 correlated N(µ,σ 2 ) RVs with correlation coefficient ρ, let (Y 1,Y 2 ) = (Z 1,ρZ ρ 2 Z 2 ) and then set (X 1,X 2 ) = (µ+σy 1,µ+σY 2 ). 20

21 Inverse Transformation Method. To generate a random variable X with known cdf F(x) = P(X x), the inverse transform method sets X = F 1 (U), where U is a U[0,1] random variable (i.e. X satisfies F(X) = U U[0,1])). The inverse of F is well defined if F is strictly increasing and continuous, otherwise we set F 1 (u) = inf{x : F(x) u}. Exercise. Use the inversion method to generate X from a Rayleigh distribution, i.e. ( ) x 2 F(x) = 1 exp. 2σ 2 21

22 It is easy to see that the random variable X generated by this method is actually characterized by the distribution function F: X = F 1 (U) P(X x) = P(F 1 (U) x) = P(U F(x)) = F(x) 0 = F(x) 1 du X F where we have used the monoticity of F and the fact that U is uniformly distributed. 22

23 Examples. 1. Exponential with parameter a > 0: F(x) = 1 e ax, x 0. Let U U[0,1] and F(X) = U 1 e ax = U X = 1 a ln(1 U) = 1 a lnu. Hence X Exp(a). Remark. Generating exponential random variable is useful when you have to simulate a Poisson process, which is a possible model for shocks in asset prices or credit rating. 23

24 2. Arcsin law: Let B t be a Brownian motion on the time interval [0,1]. Let T = argmax 0 t 1 B t. Then, for any t [0,1] we have that P(T t) = 2 π arcsin t =: F(t). Similarly, let L = sup{t 1 : B t = 0}. Then, for any s [0,1] we have that P(L s) = 2 π arcsin s = F(s). How to generate from this distribution? Let U U[0,1] and Hence X F. F(X) = U 2 π arcsin X = U ( X = sin πu ) 2 = cos(πu). 24

25 Acceptance-Rejection Method. Suppose we must generate random variates according to a probability density function (pdf) f(x) and the difficulty in inverting the corresponding distribution function makes the inverse transform method unattractive. Assume X has pdf f(x) on a set S, g is another pdf on S from which we know how to generate samples, and f is dominated byg on S, i.e. there exists a constant c such that f(x) cg(x), x S. If the distribution g(x) is easy to simulate, it can be shown that the following acceptance-rejection method generates a random variable X distributed according to the density f: generates a sample X from g and a sample U from U[0,1], and accepts X as a sample from f if U f(x) cg(x) and rejects X otherwise, and repeats the process. 25

26 Exercise. Suppose the pdf f is defined over a finite interval [a,b] and is bounded by M. Use the acceptance-rejection method to generate X from f. 26

27 Let us prove that the random variate generated by the acceptance-rejection method has pdf f. Let Y be a sample returned by the algorithm. Then Y has the same distribution of X conditional on U f(x) cg(x). So for any measurable subset A S, we have f(x) cg(x) ) P(Y A) = P(X A U f(x) cg(x) ) = P(X A & U P(U f(x) cg(x) ). Now, as U U[0,1], P(U f(x) cg(x) ) = which yields S P(U f(x) )g(x) dx = cg(x) P(Y A) = cp(x A & U f(x) cg(x) ) = c As A was chosen arbitrarily, we have that Y f. A S f(x) cg(x) g(x) dx = 1 c f(x) cg(x) g(x) dx = A Sf(x) dx = 1 c f(x) dx. 27

28 4. Monte Carlo Methods for Option Pricing. A wide class of derivative pricing problems come down to the evaluation of the following expectation E[f(Z(T;t,z))], where Z denotes the stochastic process that describes the price evolution of one or more underlying financial variables such as asset prices and interest rates, under the respective risk neutral probability distributions. The process Z has the initial value z at time t. The function f specifies the value of the derivative at the expiration time T. Monte Carlo simulation is a powerful and versatile technique for estimating the expected value of a random variable. Examples. E[e r(t t) (S T X) + ], where S T = S t e (r 1 2 σ2 )(T t)+σ T tz, Z N(0,1). E[e r(t t) (max t τ T S τ X) + ], where S τ as before. 28

29 Simulation Procedure. (a) Generate sample paths of the underlying state variables(asset prices, interest rates, etc.) according to risk neutral probability distributions (for example, if we assume that the asset price follows a geometric Brownian motion, this means that the drift µ for the asset price must be replaced by the risk-free rate r). Depending on the nature of the option at hand, we may need to generate the full sample paths, or simply the terminal asset price. (b) For each simulated sample path, evaluate discounted cash flows of the derivative. (c) Take the sample average of the discounted cash flows over all sample paths. 29

30 Matlab Exercise. Write a program to simulate paths of stock prices S satisfying the discretized SDE: S = rs t+σs tz where Z N(0,1). The inputs are the initial asset price S, the time horizon T, the number of partitions n (time-step t = T ), the interest rate r, the n volatility σ, and the number of simulations M. The output is a graph of paths of the stock price (or the data needed to generate the graph). 30

31 5. Pricing European Options. If the payoff is a function of the underlying asset at one specific time, then we can simplify the above Monte Carlo procedure. For example, a European call option has a payoff max(s(t) X,0) at expiry. If we assume that S follows a lognormal process, then S(T) = S 0 e (r 1 2 σ2 )T+σ T Z (5) where Z is a standard N(0,1) random variable. To value the call option, we only need to simulate Z to get samples of S(T). There is no need to simulate the whole path of S. The computational work load will be considerably reduced. Of course, if we want to price path-dependent European options, we will have to simulate the whole path of the asset process S. 31

32 Main Advantage and Drawback. With the Monte Carlo approach it is easy to price complicated terminal payoff function such as path-dependent options. Practitioners often use the brute force Monte Carlo simulation to study newly invented options. However, the method requires a large number of simulation trials to achieve a high level of accuracy, which makes it less competitive compared to the binomial and finite difference methods, when analytic properties of an option pricing model are better known and formulated. Remark. The CLT tells us that the Monte Carlo estimate is correct to order O( 1 M ), where M is the sample size. So to increase the accuracy by a factor of 10, we need to compute 100 times more paths. 32

33 6. Computation Efficiency. Suppose W total is the total amount of computational work units available to generate an estimate of the value of the option V. Assume there are two methods for generating MC estimates, requiring W 1 and W 2 units of computational work for each run. Assume W total is divisible by both W 1 and W 2. Denote by V i 1 and V i 2 the samples for the estimator of V using method 1 and 2, and σ 1 and σ 2 their standard deviation. The sample means for estimating V using W total amount of work are, respectively, 1 N 1 N 1 i=1 V i 1 and 1 N 2 N 2 where N i = W total W i, i = 1,2. The law of large numbers tells us that the above two estimators are approximately normal with mean V and variances σ 2 1 N 1 and σ 2 2 N 2. Hence, method 1 is preferred over method 2 provided that σ 2 1W 1 σ 2 2W i=1 V i 2,

34 7. Setting the Number of Replication. Carrying out a Monte Carlo simulation entails the generation of samples of the quantity of interest and then an estimation of the relevant parameters. One would expect that the larger the number of samples, or replications, the better the quality of the estimates will be. Recall that given a sequence of independent sample X i, i = 1,...,n, drawn from the same underlying distribution, we may build the sample mean: X(n) = 1 n X i, n which is un unbiased estimator of the parameter µ = E[X i ], and the sample variance: S 2 (n) = 1 n 1 i=1 n [X i X(n)] 2. i=1 34

35 We may try to quantify the quality of our estimator by considering the expected value of squared error of estimate: E[( X(n) µ) 2 ] = Var( X(n)) = σ2 n, where σ 2 may be estimate by the sample variance. Clearly increasing the number n of replications improves the estimate, but how can we reasonably set the value of n? There are some techniques that specify how reasonably set the value of the number of replications. For details of these methods see Brandimarte (2006), Hull (2014). 35

36 8. Variance reduction techniques. One way to improve the accuracy of an estimate is to increase the number of replications n, since Var( X(n)) = Var(X i). n However this brute-force approach may require an excessive computational effort. An alternative is to work on the numerator of this fraction and to reduce the variance of the sample X i, i = 1,...,n directly. This may be accomplished in different ways, more or less complicated, and more or less rewarding as well. Some approaches do not require deep knowledge about the systems we are simulating. Better results might be obtained by exploiting some more knowledge. 36

37 Antithetic Variate Method. Suppose {ε i } are independent samples from N(0,1) for the simulation of asset prices, so that S i T = S t e (r σ2 2 )(T t)+σ T tε i, i = 1,...,M, where M is the total number of simulation runs. An unbiased estimator of a European call option price is given by ĉ = 1 M c i = 1 M e r(t t) max(st i X,0). M M i=1 i=1 We observe that { ε i } are also independent samples from N(0,1), and therefore the simulated price S i T = S t e (r σ2 2 )(T t) σ T tε i, i = 1,...,M, is a valid sample of the terminal asset price. 37

38 A new unbiased estimator is given by c = 1 M c i = 1 M e r(t t) max( S T i X,0). M M i=1 i=1 Normally, we expect c i and c i to be negatively correlated, i.e. if one estimate overshoots the true value, the other estimate undershoots the true value. The antithetic variate estimate is defined to be ĉ+ c c =. 2 The antithetic variate method improves the computation efficiency provided that cov(c i, c i ) 0. [Reason: Var(X +Y) = Var(X)+Var(Y)+2cov(X,Y). Var( c) = 1 2 Var(ĉ)+ 1 2 cov(ĉ, c)] 38

39 Control Variate Method. Suppose A and B are two similar options and the analytic price formula for option B is available. Let V A and V A be the true and estimated values of option A, V B and V B are similar notations for option B. The control variate estimate of option A is defined to be Ṽ A = V A +(V B V B ), where the error V B V B is used as an adjustment in the estimation of V A. The control variate method reduces the variance of the estimator of V A when the options A and B are strongly (positively) correlated. 39

40 The general control variate estimate of option A is defined to be where β R is a parameter. Ṽ β A = V A +β(v B V B ), The minimum variance of Ṽ β A is achieved when β = cov( V A, V B ). Var( V B ) Unfortunately, cov( V A, V B ) is in general not available. One may estimate β using the regression technique from the simulated option values V i A and V i B, i = 1,...,M. Note: E(X +β(e(y) Y)) = E(X)+β(E(Y) E(Y)) = E(X) and Var(X+β(E(Y) Y)) = Var(X βy) = Var(X)+β 2 Var(Y) 2βcov(X,Y). 40

41 Other Variance Reduction Procedures. Other methods include - importance sampling that modifies distribution functions to make sampling more efficient, - stratified sampling that divides the distribution regions and takes samples from these regions according to their probabilities, - moment matching that adjusts the samples such that their moments are the same as those of distribution functions, - low-discrepancy sequence thatleads to estimating errorproportional to 1 M 1 rather than, where M is the sample size. M For details of these methods see Hull (2014) or Brandimarte (2006) or Glasserman (2004). 41

42 Matlab Exercise. Write a program to price European call and put options using the Monte Carlo simulation with the antithetic variate method. The standard normal random variables can be generated by first generating U[0, 1] random variables and then using the Box Muller method. The terminal asset prices can be generated by (5). TheinputsarethecurrentassetpriceS 0,theexercisepriceX,thevolatility σ, the interest rate r, the exercise time T, and the number of simulations M. The outputs are European call and put prices at time t = 0 (same strike price). 42

43 9. Quasi-Monte Carlo Simulation. The use of variance reduction techniques is based on the idea that random sampling is really random. However, the random numbers produced by a LCG or more sophisticated algorithms are not random at all. Hence one could take a philosophical point of view and wonder about the very validity of variance reduction methods, or even the Monte Carlo approach itself. Taking a pragmatic point of view, and considering the fact that Monte Carlo methods have proven their value over the years, we should conclude that this shows that there are some deterministic number sequences that work well in generating samples. These sequences are called low-discrepancy sequences. An alternative name for a low-discrepancy sequence is quasi-random sequence, which is way the term quasi-monte Carlo is used. Actually the quasi-random term is a bit misleading, as there is no randomness at all. 43

44 Some theoretical results suggest that low-discrepancy sequences may perform better than pseudorandom sequences obtained through a LCG or its variations. In fact the estimation error with Monte Carlo simulation is something like O( 1 M ), where M is the number of samples. With certain low-discrepancy sequences, itcan be shownthattheerroris somethinglike O(lnM) m /M where m is the dimension of space in which we are integrating. Different sequences have been proposed in literature. We refer, for example, to Brandimarte (2006) for more details. 44