Chapter 4: Monte-Carlo Methods

Transcription

1 Chapter 4: Monte-Carlo Methods A Monte-Carlo method is a technique for the numerical realization of a stochastic process by means of normally distributed random variables. In financial mathematics, it is used for the computation of the expectation of option prices in situations where explicit solutions like the Black-Scholes formula are not accessible (e.g., exotic options). For simplicity, we consider a European plain-vanilla put option on a stock S t whose value behaves according to the geometric Brownian motion ( ) d S t = r S t dt + σ S t dw t with a risk-free interest rate r 0, constant volatility σ > 0, and a Wiener process W t. We recall that the option price V(S t,t) at t = 0 is given by the discounted expectation ( ) V(S 0,0) = exp( rt) E(V(S T,T)).

2 The Four Basic Steps of the Monte-Carlo Simulation The idea behind the Monte-Carlo method is based on the approximation of the expectation E(V(S T,T)) in ( ) by the simulation of M pathes {S t 0 < t < T} of the value of the stock: Step 1: Simulation of the pathes For M independent pathes, compute the solution S (k) t,1 k M, of the geometric Brownian motion ( ). This requires the simulation of M independent realizations of the Wiener process W t and the numerical solution of the stochastic differential equation for the respective pathes. Step 2: Computation of the payoff function For 1 k M, compute the payoff function associated with the path S (k) t : V (k) T = (K S (k) T ) +. Step 3: Computation of an estimate of the expectation According to the law of large numbers, the arithmetic mean of equally distributed, independent random variables converges almost surely to the expectation. Hence, an appropriate estimate of the expectation in ( ) is given by V (k) T = (K S (k) T ) +. Step 4: Computation of an estimate of the option price Compute an approximation of the option price by ˆV = exp( rt) Ê(V T ).

3 4.1 Generation of Equally Distributed Random Numbers For the simulation of a Wiener process, we use standardized normally distributed random numbers Z N(0,1) in order to compute the increments W = Z t. Definition 4.1 (Equally Distributed Random Variables) (i) A random variable X is called equally distributed on [a,b] lr, (notation: X U[a,b]), if its density function is given by f(x) = 1/(b a), x [a, b]. (ii) A sequence {X n } n ln of random variables is called F-distributed, if the X n are independent realizations of random numbers with the distribution function F. In the sequel, we will consider the following random number generators: Linear congruential generators The generator RANDU Fibonacci generators

4 4.1.1 Linear Congruential Generators Linear congruential generators are obtained by the following algorithm: Algorithm 4.1: Linear Congruential Generators Given M ln,a,b {0,1,...,M 1},a 0, and a seed X 0 {0,1,...,M 1}, compute For i = 1,2,... X i := (a X i 1 + b) mod M, U i := X i /M. where a mod M := M (a/m a/m ) with a/m : largest integer less or equal a/m. Lemma 4.1 (Properties of Linear Congruential Generators) The random numbers generated by Algorithm 4.1 have the properties (i) X i {0,1,...,M 1}. (ii) The X i are periodic with period less or equal M. (iii) If b = 0 (multiplicative congruential generator), we must exclude the seed X 0 = 0. (iv) If a = 1, we obtain X i = (X 0 + ib) mod M (too easy to predict!).

5 Analysis of Linear Congruential Generators The random numbers X i and U i,i 1, obtained by the linear congruential generator, can be written as m-tuples (X i,x i+1,...,x i+m 1 ) or points (U i,...,u i+m 1 ) [0,1) m. They are situated on (m 1)-dimensional hyperplanes which can be constructed as the following analysis in case m = 2 reveals: X i = (a X i 1 + b) mod M X i = a X i 1 + b k M, k M a X i 1 + b < (k + 1) M. For an arbitrary tuple (z 0,z 1 ) Z 2 we obtain: ( ) z 0 X i 1 + z 1 X i = z 0 X i 1 + z 1 (a X i 1 + b k M) = = X i 1 (z 0 + a z 1 ) + z 1 b z 1 k M = = M (x i 1 z 0 + a z 1 M z 1 k) }{{} =: c + z 1 b. Division by M yields the equation of a straight line in the (U i 1,U i )-plane: ( ) z 0 U i 1 + z 1 U i = c + z 1 b M 1.

6 Analysis of Linear Congruential Generators Question: Does there exist a tuple (z 0,z 1 ) Z 2 such that the intersection of the straight lines with the square [0,1) 2 is non-empty for only a few lines which would violate the requirement of uniform distributions of the points? If we choose (z 0,z 1 ) Z 2 and z 0 +a z 1 = 0 mod M, then c Z. Solving for c in ( ) results in the maximum interval c min c c max such that the associated straight line ( ) has non-zero intersection with [0,1) 2. Figure 4.1: M = 2048, a = 1229, b = 1, X 0 = 1. The figure displays the first 499 points (U i 1,U i ) obtained by the linear congruential generator. These points are lying on 5 parallel straight lines.

7 4.1.2 The Random Number Generator RANDU The generator RANDU is based on a multiplicative congruential method with the data M = 2 31, a = , b = 0. Figure 4.2 suggests that the points (U i 1,U i ) are randomly distributed in the plane. However, Figure 4.3 displays the tuples (U i 2,U i 1,U i ), and we observe that they are lying on only 15 different planes in the unit cube [0,1) 3 which is a serious disadvantage in many applications. Figure 4.2 Figure 4.3

8 4.1.3 Fibonacci Generators Fibonacci generators use the recursively given Fibonacci sequence according to where X 1,X 2 are given seeds. X i := (X i 1 + X i ) mod M, U i := X i /M, i 2, Figure 4.4 displays the first 2000 points (U i 1,U i ) obtained by the Fibonacci generator for M = 2179 and X 1 = X 2 = 1. Significantly less than 2000 points are seen in the unit square due to the fact that the sequence {U i } i ln repeats any 197 entries. Therefore, this Fibonacci generator is less suited for random number generation. Figure 4.4

9 Better and widely used random number generators are the so-called lagged Fibonacci generators: Algorithm 4.2: Lagged Fibonacci Generators Given M ln, µ, ν ln and initial values X 1,...,X max(µ,ν), for i max{µ, ν} compute X i := (X i µ + X i ν ) mod M, U i := X i /M. For M = 2048, µ = 5, ν = 17, Figure 4.5 shows the first 2000 points (U i 1,U i ) obtained by the lagged Fibonacci generator. Figure 4.5

10 4.2 Transformation of Random Variables Normally distributed random variables can be obtained by a transformation of equally distributed random variables using either the inversion of the distribution function or a transformation between random variables Inversion of the Distribution Function Theorem 4.1 (Inversion of the Distribution Function) Assume that U U[0,1] is an equally distributed random variable and assume further that F is a continuous, strongly monotone distribution function. Then, the random variable F 1 (U) is F-distributed. Proof. Since P(U ξ) = ξ, ξ [0,1], we have P(F 1 (U) x) = P(U F(x)) = F(x). Remark: For the normal distribution Φ there are no explicit expressions neither for Φ nor for Φ 1 which means that the nonlinear equation Φ(x) = u must be inverted numerically by, e.g., the application of a Newton-like method. However, for u 1 the problem is ill-conditioned.

11 4.2.2 Transformation of Random Variables Theorem 4.2 (Transformation of Random Variables) Assume that X is a random variable with density function f on A := {x lr n f(x) > 0}. Assume further that the mapping h : A B := h(a) is continuously invertible. Then, the random variable Y = h(x) has the density function where J h 1(y) is the Jacobian of h 1 in y. g(y) := f(h 1 (y)) det J h 1(y), y B, Proof. Applying the transformation rules for multiple integrals, we obtain P(Y = h(x) B) = P(X h 1 (B)) = h 1 (B) f(x) dx = f(h 1 (y)) det J h 1(y) dy. B

12 4.2 Normally Distributed Random Variables The Box-Muller Algorithm We apply Theorem 4.2 in case n = 2, A = S := [0,1] 2, f 1 on S, and h : S lr 2 given by y 1 = h 1 (x 1,x 2 ) = 2 ln(x 1 ) cos(2πx 2 ), y 2 = h 2 (x 1,x 2 ) = 2 ln(x 1 ) sin(2πx 2 ). Since y y 2 2 = 2 ln(x 1 ) and y 2 /y 1 = tan(2πx 2 ), the inverse function can be easily computed x 1 = exp( 1 2 (y2 1 + y 2 2)), x 2 = 1 2π arctan(y 2 y 1 ) with the determinant det J h 1(y) of the Jacobian J h 1(y) given by det J h 1(y) = 1 2π exp( 1 2 (y2 1 + y 2 2)) which is the density of the standard normal distribution in lr 2.

13 Algorithm 4.3: Box-Muller Algorithm Step 1: Generate U 1 U[0,1] and U 2 U[0,1]. Step 2: Set Θ := 2π U 2 and ρ := ln(u 1 ). Step 3: Z 1 := ρ cos(θ) and Z 2 := ρ sin(θ) are normally distributed random variables.. The histogram of Z 1 in Figure 4.6 based on random numbers shows that the algorithm indeed approximately produces normally distributed random variables. Figure 4.6

14 Numerische Mathematik für WiM Marsaglia s Polar Method Marsaglia s variant of the Box-Muller algorithm avoids the evalu For U 1,U 2 U[0,1], we obviously have W i := 2U i 1 [ 1, +1],1 a point in the plane. Only points inside the unit disk K := {(W 1,W 2 ) W W 2 2 < 1} are accepted. A Transformation K S := [0,1] 2 is provided by m x 1 x 2 = Algorithm 4.4: Marsaglia s Polar Method W1 2 + W π arctan(w 2 W 1 ) Step 1: Generate U 1,U 2 U[0,1] and W i := 2U i 1, 1 i 2, a Step 2: Z 1 := W 1 2ln(W W 2 2)//(W W 2 2) and Z 2 := W 2 2 normally distributed random variables. Remark: Marsaglia s polar method is more efficient than the Bo.

15 4.4 Correlated Random Variables We consider a random variable Z = (Z 1,...,Z n ) with density function f. We further denote by µ = E(Z) = (E(Z 1 ),...,E(Z n )) the expectation and by Σ = (Σ ij ) n i,j=1 with Σ ij = (Cov Z) ij := E((Z i µ i )(Z j µ j )) the associated covariance matrix which is symmetric, positive definite. We want to construct an N(µ,Σ)-distributed random variable Y. Theorem 4.3 (Correlated Random Variables) Under the above notations let Σ = LL T be the Cholesky decomposition of Σ with a lower triangular matrix L lr n n. Then, Y = µ + LZ is an N(µ,Σ)-distributed random variable with density function 1 1 f(x) = (2π) n/2 (det Σ) exp( 1 1/2 2 (x µ)t Σ 1 (x µ)).

16 Proof. Setting x = Lz and observing dx = det L dz, we obtain f(z) dz = 1 z (2π) n/2 exp( zt 2 ) dz = 1 x) T L 1 x (2π) n/2 exp( (L 1 ) dz = 2 = 1 (LL T ) 1 x (2π) n/2 exp( xt ) dz = 2 1 Σ 1 x (2π) n/2 det L exp( xt ) dx = 2 = 1 Σ 1 x (2π) n/2 (det Σ) 1/2 exp( xt ) dx. 2 This shows that X = LZ is N(0,Σ)-distributed which readily gives the assertion.

17 Algorithm 4.5: Correlated Random Variables Step 1: Compute the Cholesky decomposition Σ = LL T. Step 2: Compute Z N(0,1) component-wise, i.e., Z i N(0,1), 1 i n, e.g., by Marsaglia s polar algorithm. Step 3: Y = µ + LZ is N(µ, Σ)-distributed. MATLAB program correlated.m randn( state,1) Sigma = [543;454;345] mu = [ 5010] ;N = 10000; L = chol(sigma); X = zeros(3,n); for i = 1 : N X(:.i) = mu + L randn(3,1); end

18 4.5 Sequences of Random Numbers with Low Discrepancy As a measure for the quality of uniform distribution of random numbers we introduce the notion of discrepancy and give examples of sequences of random numbers with low discrepancy. A related issue is that to improve convergence properties in relevant applications such as the numerical integration of high-dimensional integrals Monte Carlo Integration Given a bounded domain Ω lr m and an integrable function f : Ω lr, the Monte Carlo integration provides an approximation of the m-dimensional integral of f over Ω by the weighted sum of the values of the integrand f at randomly chosen points x i,1 i N, in the domain of integration Ω according to I := f(x) dx vol(ω) N f(x i ) =: Q N. Ω lr m N i=1

19 If the random variables x 1,...,x N Ω are independent and uniformly distributed, the law of large numbers implies Q N vol(ω) E(f) = Ω f(x) dx as N. Hence, for the variance of the error ε N := I Q N we obtain Var(ε N ) = E(ε 2 N) E(ε N ) 2 = σ2 (f) N vol(ω)2, σ 2 (f) = f(x) 2 dx ( f(x) dx) 2. Ω Ω The central limit theorem implies the following convergence result for the standard deviation Var(εN ) = O(N 1/2 ) as N, which means slow convergence (improvement of the accuracy by a factor of 10 requires to increase the computational cost by a factor of 100). Another disadvantage is the lack of a strict error bound. Before we discuss methods of variance reduction in section 4.6, we will address the issue to choose the random points x i,1 i N, in such a way that with increasing N the accuracy gets better but clustering is avoided.

20 4.5.2 Discrepancy The idea behind discrepancy is that for a uniformly distributed point set {x 1,...,x N } [0,1] m the fraction of the points lying within some quadrilateral Q [0,1] m should correspond to the volume of the quadrilateral, i.e., card({x i,1 i N x i Q}) N Definition 4.2 (Discrepancy and Star Discrepancy) The discrepancy of a point set {x 1,...,x N } [0,1] m is defined by D N := sup card({x i,1 i N x i Q}) Q [0,1] m N vol(q) vol([0,1] m ). vol(q), where the supremum is taken with respect to all quadrilaterals Q [0,1] m. The star discrepancy D N is defined in the same way, but with the supremum taken with respect to those quadrilaterals Q [0,1] m for which one corner is in the origin, i.e., Q = m i=1 [0,y i ] where (y 1,...,y m ) denotes the diagonally opposite corner.

21 Properties of Discrepancy and Star Discrepancy Theorem 4.4 (Properties of Discrepancy and Star Discrepancy) Let D N and D N be the discrepancy and star discrepancy of a point set {x i } ln,x i lr m,1 i N. Then, there holds Proof. The proof is left as an exercise. Upper Bound for Monte Carlo Integration D N D N 2 m D N. Theorem 4.5 (Theorem of Hlawka and Koksma) Assume that Ω := [0,1] m,m ln, and f : [0,1] m lr is a function of bounded variation BV(f). Then, for the error ε N in the Monte Carlo integration there holds ε N BV(f) D N.

22 Sequences of Low Discrepancy Definition 4.3 (Sequences of Low Discrepancy) A sequence {x i } N i=1 of points or numbers x i lr m,1 i N, is called a sequence of low discrepancy, if there exists a constant C m > 0 such that for all N ln: Remarks: D N C m (ln(n)) m (i) For not too large m ln we have that approximately D N O(N 1 ). Compare this with D N = O(N 1/2 ) for sequences of random numbers in general. (ii) Let U i {0,1/M,..., (M 1)/M} be random numbers obtained by a linear congruential generator. These are not even uniformly distributed, since with Q := [1/(2M + 2),1/(M + 1)] D N card({u i U i Q }) vol(q ) = vol(q 1 ) = N 2M + 2. N.

23 Van der Corput Sequence Remark: The sequence {X (N) i } N i=1 with x (N) i = i/n,1 i N, obviously satisfies D N = 1/N and thus is of low discrepancy. However, it is not suited for Monte Carlo integration, since for each N a new sequence has to be set up. For increasing N, it is more efficient to use already computed numbers. Examples of sequences of low discrepancy: (i) Van der Corput sequence Let i = (d j...d 0 ) 2 := j k=0 d k 2 k be the representation of the integer i ln in fixed point arithmetics with respect to the basis B = 2. Then, the sequence {x i } ln obtained by bit inversion according to is called van der Corput sequence. x i = (.d 0...d j ) 2 := j k=0 d k 2 (k+1)

24 Radical-Inverse Function and the Halton Sequence (ii) Radical-Inverse Function The radical-inverse function Φ B allows to define van der Corput sequences {x i } N i=1 with respect to an arbitrary basis B. Using the same notation as in (i), they are defined by (iii) Halton Sequence Φ B (i) := j k=0 d k B (k+1), x i := Φ B (i). Assume that p 1,...,p m are pairwise prime integers. The Halton sequence is defined as the sequence of vectors x i = {Φ p1 (i),...,φ pm (i)}, i ln.

25 Properties of van der Corput Sequences Theorem 4.6 (Properties of van der Corput Sequences) For the discrepancy of van der Corput sequences there holds lim sup N N D N ln(n) = B 2 4(B+1) ln(b) B 1 4 ln(b), B even,, B odd.

26 4.6 Numerical Integration of Stochastic Differential Equations The Euler-Maruyama Method In case of an ordinary differential equation x (t) = a(t,x(t)), T > 0, where a C(lR +, lr) satisfies a Lipschitz condition in the second argument, the explicit Euler method with uniform step-size h > 0 is given by y i+1 = y i + h a(t i,y i ), t i := i h, i 0. It is well known that it has the order of convergence 1, i.e. sup y i x(t i ) C h. i 0 For the stochastic differential equation dx t = a(t,x t ) dt + b(t,x t ) dw t the counterpart of the explicit Euler method is the Euler-Maruyama method y i+1 = y i + h a(t i,y i ) + b(t i,y i ) W i, W i := W ti+1 W ti.

27 4.6.2 Strong and weak convergence Definition 4.4 (Strong and Weak Convergence) Assume that x h t is an approximation of the solution x t of the stochastic differential equation dx t = a(t,x t ) dt + b(t,x t ) dw t. (i) The approximation x h T is said to converge strongly to x T of order γ > 0, if there exists a constant C > 0, such that for all sufficiently small h > 0 E( x T x h T ) C h γ. (ii) The approximation x h T is said to converge weakly to x T of order κ > 0, if there exists a constant C > 0, such that for all sufficiently small h > 0 E(x T ) E(x h T) C h κ. Theorem 4.7 (Strong and Weak Convergence of the Euler-Maruyama Method) The Euler-Maruyama method is strongly convergent of order γ = 0.5 and weakly convergent of order κ = 1. Proof. We refer to P. Kloeden and E. Platen; Numerical Solution of Stochastic Differential Equations. Springer, Berlin-Heidelberg-New York, 1999

28 4.6.3 The Method of Milstein We recall Itô s Lemma which states: If X t is an Itô process, then for sufficiently smooth f we have that f = f(t,x t ) is an Itô process as well which satisfies df = ( f t + a f x b2 2 f f x2) dt + b x dw t. The associated integral form is t ( ) f(x t ) = f(x t0 ) + (f (x s ) a(x s ) + 1 t 0 2 f (x s ) b(x s ) 2 ) ds + f (x s ) b(x s ) dw s. t 0 In particular, for f(x) = x we obtain ( ) x t = x t0 + a(x s ) ds + b(x s ) dw s. t 0 t 0 Choosing f = a and f = b in ( ) and inserting it into ( ) results in x t = x t0 + t t s t (a(x t0 ) + (a a + 1 t 0 t 0 2 a b 2 ) dz + a b dw z ) ds + t 0 + (b(x t0 ) + (b a + 1 t 0 t 0 2 b b 2 ) dz + b b dw z ) dw s, t 0 where a = a(x z ), b = b(x z ) etc. s s t s t

29 Modifying the double integral w.r.t. dw z dw s by replacing the integrand b (x z )b(x z ) b (x t0 )b(x t0 ), we can write x t = x t0 + a(x t0 ) (t t 0 ) + b(x t0 ) dw s + b (x t0 ) b(x t0 ) t 0 where the remainder is of the (higher) order R = O(h 3/2 ). The double integral can be evaluated as follows: t s t 0 t 0 dw z dw s = t t 0 (W s W t0 ) dw s = = 1 2 (W2 t Wt 2 0 ) t t 0 2 This leads to the method of Milstein t t t s W s dw s W t0 dw s = t 0 t 0 t t 0 t 0 dw z dw s + R, W t0 (W t W t0 ) = 1 2 ((W2 t W 2 t 0 ) t t 0 2 ). y i+1 = y i + h a(t i,y i ) + b(t i,y i ) W b (t i,y i ) b(t i,y i ) (( W) 2 h), where W := Z h, Z N(0,1). Theorem 4.8 (Convergence of the Method of Milstein) The method of Milstein is strongly convergent of order γ = 1.

30 4.7 Reduction of Variances The slow convergence of Monte-Carlo simulations can be explained as follows: We assume that θ n = n k=1 Φ(X k )/n is some stochastic approximation of a stochastic integral θ = E(Φ(x)), where Φ(x) := g(x)/f(x) and X k,1 k n, are independent samples of an F-distributed random variable with F = f. Under the assumption E(Φ(X k )) = θ, Var(Φ(X k )) = σ 2, 1 k n, it follows that n E(θ n ) = 1 E(Φ(X k )) = θ, Var(θ n ) = 1 n Var(Φ(X n k=1 n 2 k )) = σ2 k=1 n. Using the Chebychev inequality for square integrable random variables Y P( Y E(Y) ) δ) Var(Y) δ 2, δ := σ/ ε n, we obtain the error estimate P( θ n θ σ ) ε = P( θ n θ < σ ) 1 ε. εn εn This means that the error gets the smaller the larger the number n of samples is chosen. But in order to reduce the error by a factor of 10, the number of samples must be increased by a factor of 100.

31 4.7.1 Antithetic Variables Theorem 4.9 (Variance reduction by an Antithetic Variable) Assume that the approximation θ n has been generated by a random variable Z N(0,1) and generate an approximation θ n by means of Z N(0,1) such that Var(θ n ) = Var(θ n). Introduce the antithetic variable ˆθ n according to ˆθ n := 1 2 (θ n + θ n). Then, there holds Proof. The fundamental relation implies Taking into account that we thus obtain Var(ˆθ n ) < Var(θ n ). Var(X ± Y) = Var(X) + Var(Y) ± 2 Cov(X, Y) Var(ˆθ n ) = 1 4 Var(θ n + θ n) = 1 4 (Var(θ n) + Var(θ n) + 2 Cov(θ n, θ n)). Cov(X,Y) 1 2 (Var(X) + Var(Y)), Var(ˆθ n ) 1 4 (Var(θ n) + Var(θ n)), if Cov(θ n, θ n) 0, Var(ˆθ n ) < 1 4 (Var(θ n) + Var(θ n)), if Cov(θ n, θ n) > 0, which gives the assertion observing Var(θ n ) = Var(θ n).