1 Simulating normal (Gaussian) rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions

Transcription

1 Copyright c 2007 by Karl Sigman 1 Simulating normal Gaussian rvs with applications to simulating Brownian motion and geometric Brownian motion in one and two dimensions Fundamental to many applications in financial engineering is the normal Gaussian distribution It is the building block for simulating such basic stochastic processes as Brownian motion and geometric Brownian motion In this section, we will go over algorithms for generating univariate normal rvs and learn how to use such algorithms for constructing sample paths of Brownian motion and geometric Brownian motion, in both one and two dimensions, at a desired sequence of times t 1 < t 2 < < t k 11 Generating a univariate normal random variable We focus here on the generation of a rv Z N0, 1 for we can always then transform it into an X Nµ, σ 2 via X = σz + µ For example, if {Bt : t 0} denotes a standard Brownian motion BM, then for any fixed t > 0, Bt N0, t can be constructed via Bt = tz If it is desired to simulate the pair Bt 1, Bt 2 where 0 < t 1 < t 2, then we can use the recursion Bt 2 = Bt 1 +Bt 2 Bt 1 and the basic stationary and independent increments properties of BM: Generate two iid N0, 1 rvs, Z 1, Z 2 and set Bt 1 = t 1 Z 1, Bt 2 = t 1 Z 1 + t 2 t 1 Z 2 The point is that Bt 1 N0, t 1, and independently Bt 2 Bt 1 N0, t 2 t 1 This method extends in the obvious fashion to the generation of the k dimensional multivariate normal vector Bt 1, Bt 2,, Bt k, in which 0 < t 1 < t 2 < < t k, k 2 N0, 1 density and cdf The density of Z is given by fx = 1 2π e x2 2, x R, and the cdf by Θx = P Z x = 1 2π x e y2 2 dy, x R Using the inverse transform method is not possible here since we do not have an explicit closed formula for Θ 1 y One could of course approximate this inverse function by some closed form function and then use that, and there are some algorithms in the literature for doing that But it is possible to exactly generate a Z N0, 1 using alternative clever algorithms, so that is what we will do Acceptance rejection algorithm for Z Recall from our lecture on the acceptance rejection method that we already have an algorithm for generating a rv Z N0, 1: Algorithm for generating Z N0, 1 via the acceptance rejection method 1 Generate two independent exponentials at rate 1; Y 1 = lnu 1 and Y 2 = lnu 2 1

2 2 If Y 2 Y /2, set Z = Y 1 ; otherwise go back to 1 3 Generate U Set Z = Z if U 05, set Z = Z if U > 05 On average, this algorithm requires 2 2e/π uniforms to produce one copy of Z Since we of course will use the algorithm many many times over and over again, we can do even better by using the free independent exponential produced as overshoot in a success at step 2, Y def = Y 2 Y /2, as the Y 1 in step 1 in the next round; this reduces the expected number to 264 We next introduce another algorithm that generates pairs Z 1, Z 2 of independent normals each time Called the polar method, it requires some more complex calculations sin, cos, etc but has an advantage of only requiring one uniform for each copy of Z, not a random number In a variety of real applications, it is very desirable to try to synchronize the use of random numbers across different runs in a simulation so as to reduce the variance of an estimate, and thus a non-random algorithm might well be useful at times So we include it here in our study Algorithm for generating Z N0, 1 via the polar method 1 Generate an exponential at rate 1/2, D = 2 lnu 1, and a uniform over 0, 2π, Θ = 2πU 2 2 Set Z 1 = D cos Θ, Z 2 = D sin Θ The polar method is easily derived proved via starting with an iid N0, 1 pair Z 1, Z 2, mapping the vector Z 1, Z 2 into polar coordinates, D = Z1 2 + Z2 2 the squared distance from the origin, Θ = arctan Z 2 /Z 1 the angle, counterclockwise, from the horizontal axis to the point Z 1, Z 2, then deriving using elementary multivariate calculus, with the Jacobian, etc the joint density function gd, θ of D, Θ, to reveal that D is an exponential at rate 1/2 and independent of D, the angle Θ is uniform over 0, 2π; gd, θ = 1 2 e d/2 1 2π, d > 0, θ 0, 2π Using this fact inversely then yields the algorithm: we can produce Z 1, Z 2 by mapping an independent pair D, Θ back into rectangular coordinates; that is exactly what step 2 in the algorithm does 12 Simulating 1-dimensional Brownian motion BM and geometric Brownian motion GBM 121 Standard BM A stochastic process B = {Bt : t 0} possessing wp1 continuous sample paths is called standard Brownian motion BM if 1 B0 = 0 2 B has both stationary and independent increments 3 Bt Bs has a normal distribution with mean 0 and variance t s, 0 s < t 2 and 3 together can be summarized by: If t 0 = 0 < t 1 < t 2 < < t k, then the increment rvs Bt i Bt i 1, i {1, k}, are independent and distributed as N0, t i t i 1 In particular, Bt i Bt i 1 is independent of Bt i 1 = Bt i 1 B0 If we only wish to simulate Bt at one fixed value t > 0, then we need only generate a unit normal Z N0, 1 and set Bt = tz But typically, we will want to simulate say k values at times t 0 = 0 < t 1 < t 2 < < t k and we can easily do so as follows: 2

3 Generate k iid unit normals Z 1, Z 2,, Z k, then construct the independent increments via Bt i Bt i 1 = t i t i 1 Z i, i = 1,, k Thus to simulate the values Bt 1,, Bt k, we sequentially generate unit normals, Z 1, Z 2,, Z k, and use the recursion Bt i+1 = Bt i + Bt i+1 Bt i = Bt i + t i+1 t i Z i+1, i {0, k 1} Simulating Standard BM at times 0 = t 0 < t 1 < t 2 < < t k : Sequentially generate unit normals Z 1, Z 2,, Z k, and recursively define Bt 1 = t 1 Z 1 Bt 2 = Bt 1 + t 2 t 1 Z 2 = t 1 Z 1 + t 2 t 1 Z 2 Bt k = k ti t i 1 Z i i=1 In the end, we see that to simulate BM at a collection of specific times we need only generate unit normals 122 BM with drift Xt = σbt + µt will denote the BM with drift µ and variance term σ > 0 It has continuous sample paths and is defined by 1 X0 = 0 2 X has both stationary and independent increments 3 Xt Xs has a normal distribution with mean µt s and variance σ 2 t s, 0 s < t Xt Xs thus can be constructed simulated by generating a standard normal rv Z and setting Xt Xs = σ t sz + µt s Again, by the stationary and independent increments, we can simulate such a BM at times 0 = t 0 < t 1 < t 2 < < t k, by generating k iid unit normals Z 1, Z 2,, Z k and using the recursion Xt i+1 = Xt i + Xt i+1 Xt i = Xt i + σ t i+1 t i Z i+1 + µt i+1 t i Simulating BM with drift µ and variance term σ at times 0 = t 0 < t 1 < t 2 < < t k : Sequentially generate unit normals Z 1, Z 2,, Z k, and recursively define Xt 1 = σ t 1 Z 1 + µt 1 Xt 2 = Xt 1 + σ t 2 t 1 Z 2 + µt 2 t 1 = σ t 1 Z 1 + µt 1 + σ t 2 t 1 Z 2 + µt 2 t 1 Xt k = k σ t i t i 1 Z i + µt i t i 1 i=1 3

4 123 Geometric BM Geometric Brownian motion GBM is given by St = S0e Xt, t 0, where Xt = σbt + µt, t 0, is a BM e Xt has a lognormal distribution for each fixed t > 0 In general if Y = e X is lognormal with X Nµ, σ 2, then we can easily simulate Y via setting Y = e σz+µ, with Z N0, 1 Moreover, for any 0 s < t it holds that St = S0 Ss S0 St Ss = S0eXs e Xt Xs, and since the increment Xs is independent of the increment Xt Xs, we conclude that the consecutive ratios Ss St S0 and Ss are independent lognormals We can thus simulate the pair Ss, St by generating two iid N0, 1 rvs, Z 1, Z 2 and setting Ss = S0e σ sz 1 +µs, St = Sse σ t sz 2 +µt s = S0e σ sz 1 +µs e σ t sz 2 +µt s More generally, for 0 = t 0 < t 1 < t 2 < < t k, define Y i = St i /St i 1, i {1, 2,, k} Then we can write St 1 = S0Y 1 St 2 = St 1 Y 2 = S 0 Y 1 Y 2 St k = St k 1 Y k = S 0 Y 1 Y 2 Y k The Y i are independent lognormal rvs and can be constructed by generating k iid N0, 1 rvs, Z 1, Z 2, Z k and setting Y i = e σ t i t i 1 Z i +µt i t i 1, i {1, 2,, k} 1 Simulating paths of GBM is thus an easy consequence of our algorithm for simulating paths of BM since for 0 = t 0 < t 1 < t 2 < < t k, the following recursion holds St i+1 = St i e Xt i+1 Xt i, i {0, 1,, k 1} Simulating Geometric BM with drift µ and variance term σ at times 0 = t 0 < t 1 < t 2 < < t k : Sequentially generate unit normals Z 1, Z 2,, Z k, and set the Y i as in 1 Then recursively define St 1 = S0Y 1 St 2 = St 1 Y 2 = S 0 Y 1 Y 2 St k = St k 1 Y k = S 0 Y 1 Y 2 Y k 4

5 13 Simulating correlated BM and GBM in two dimensions in which the two BM s have a specific desired correlation ρ In many financial applications, there are several correlated assets that make up a portfolio or from which certain options/derivatives are created spread options, basket options, etc Here we focus on the case of two correlated assets Let W 1 t and W 1 t denote standard Brownian motions Consider two Brownian motions X 1 t = σ 1 W 1 t+µ 1 t, and X 2 t = σ 2 W 1 t+µ 2 t Xt = X 1 t, X 2 t T is a two-dimensional Brownian motion, where we shall assume the coordinates have correlation coefficient ρ: For a given 1 < ρ < 1, CovX 1 t, X 2 t σ 1 t σ2 t = ρ, t > 0 2 To construct this BM, we start with two independent standard BM s, B 1 t and B 2 t, define Bt = B 1 t, B 2 t T, define the 2 2 matrix and construct where µ = µ 1, µ 2 T Letting A = σ1 0 σ 2 ρ σ 2 1 ρ 2 Xt = ABt + µt, t 0, Σ = AA T = σ 2 1 σ 1 σ 2 ρ σ 1 σ 2 ρ σ 2 2 we say that X is a two-dimensional BM with drift vector µ and covariance matrix Σ; we denote this by X BMµ, Σ From 2, this implies that for each t > 0, CovX 1 t, X 2 t = ρσ 1 σ 2 t As with BM in 1-dimension, X has both stationary and independent increments If we wish to simulate Xt at a fixed time t > 0, then we can generate Z 1, Z 2 iid N0, 1, define the vector Z = Z 1, Z 2 T, then set Xt = taz + µt For generating the pair Xt 1 and Xt 2 at the times 0 < t 1 < t 2, we generate a pair of N0, 1 vectors, Z 1 = Z 1,1, Z 1,2 T, Z 2 = Z 2,1, Z 2,2 T and define Xt 1 = t 1 AZ 1 + µt 1 Xt 2 = Xt 1 + t 2 t 1 AZ 2 + µt 2 t 1 = t 1 AZ 1 + µt 1 + t 2 t 1 AZ 2 + µt 2 t 1 This then yields,, 5

6 An algorithm for simulating X BMµ, Σ at times 0 = t 0 < t 1 < t 2 < < t k : Sequentially generate k iid pairs of independent N0, 1 rvs; Z 1 = Z 1,1, Z 1,2 T, Z 2 = Z 2,1, Z 2,2 T,, Z k = Z k,1, Z k,2 T, then recursively define Xt 1 = t 1 AZ 1 + µt 1 Xt 2 = Xt 1 + t 2 t 1 AZ 2 + µt 2 t 1 = t 1 AZ 1 + µt 1 + t 2 t 1 AZ 2 + µt 2 t 1 Xt k = Xt k 1 + k t k t k 1 AZ k + µt k t k 1 = t i t i 1 AZ i + µt i t i 1 i=1 From here we can then define correlated geometric BM s GBM S 1 t = S 1 0e X 1t, S 2 t = S 2 0e X 2t To simulate these, we use the algorithm above for the BM, coordinate by coordinate, utilizing the rows of the matrix A = A i,j given above, together with the algorithm at the end of Section 123 for simulating a one dimensional GBM An algorithm for simulating two correlated GBMs with underlying two-dimensional X BMµ, Σ at times 0 = t 0 < t 1 < t 2 < < t k : Sequentially generate k iid pairs of independent N0, 1 rvs; Z 1 = Z 1,1, Z 1,2 T, Z 2 = Z 2,1, Z 2,2 T,, Z k = Z k,1, Z k,2 T, then recursively define 2 t1 j=1 S 1 t 1 = S 1 0e A 1,jZ 1,j +µ 1 t 1 2 t2 t 1 j=1 S 1 t 2 = S 1 t 1 e A 1,jZ 2,j +µ 1 t 2 t 1 2 tk t k 1 j=1 S 1 t k = S 1 t k 1 e A 1,jZ k,j +µ 1 t k t k 1 2 t1 j=1 S 2 t 1 = S 2 0e A 2,jZ 1,j +µ 2 t 1 2 t2 t 1 j=1 S 2 t 2 = S 2 t 1 e A 2,jZ 2,j +µ 2 t 2 t 1 2 tk t k 1 j=1 S 2 t k = S 2 t k 1 e A 2,jZ k,j +µ 2 t k t k 1 Remark 11 The case of three or more dimensions can be handled analogously to our method here; we will cover this in more detail later 6