Math 416/516: Stochastic Simulation

Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28

Outline 1 Simulation of Itō Diffusions: The Euler Approximation 2 Second Order Approximation: The Milstein Scheme 3 A Summary: Monte Carlo Methods in Sciences and Engineering Haijun Li Math 416/516: Stochastic Simulation Week 13 2 / 28

Numerical Solution of Stochastic Differential Equations SDEs which admit an explicit solution are few exceptions. Therefore numerical techniques for the approximation of the solution to a SDE are often called for. One purpose is to visualize a variety of sample paths of the solution. A collection of such paths is called a scenario, which can be used for some kind of prediction of the stochastic process at future instants of time. A second objective is to achieve reasonable approximations to the distributional quantities (expectations, variances, covariance and higher-order moments) of the solution to a SDE. Only in a few cases one is able to give explicit formulas for these quantities, and even then they frequently involve special functions which have to be approximated numerically. Numerical solutions allow us to simulate as many sample paths as we want; they constitute the basis for Monte-Carlo techniques to obtain the distributional characteristics of Itō diffusions. Haijun Li Math 416/516: Stochastic Simulation Week 13 3 / 28

Euler Approximation Scheme Consider the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. We assume that the coefficient functions µ(x) and σ(x) are Lipschitz continuous, and EX0 2 <, which guarantee the existence and uniqueness of a (strong) solution. 1 To approximate the solution, partition [0, T ] as follows, τ m : 0 = t 0 < t 1 < < t m 1 < t m = T, with i = t i t i 1, 1 i m, and mesh(τ m ) = max 1 i m i. Let i B = B ti B ti 1, 1 i m. 2 Define recursively, 1 i m, X ti = X ti 1 + µ(x ti 1 ) i + σ(x ti 1 ) i B, with initial value X 0. Haijun Li Math 416/516: Stochastic Simulation Week 13 4 / 28

Estimation via Euler Scheme Goal: To estimate v = E[h(X t, 0 t T )], where (X t, 0 t T ) is the solution of the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. Generate n independent sample paths, 1 k n, X (k) t i = X (k) t i 1 + µ(x (k) t i 1 )(t i t i 1 ) + σ(x (k) t i 1 ) t i t i 1 Z (k) with X (k) 0 = X 0, where Z k 1,..., Z k m are i.i.d. standard normal random variables. Use the estimator ˆv n = 1 n approx of (X t,0 t T ) n {}}{ E[h( X (k) t 0, X (k) t 1,..., X (k) t m )] k=1 i, 1 i m Haijun Li Math 416/516: Stochastic Simulation Week 13 5 / 28

Example Consider the following SDE ds t = rs t dt + σs t db t where r is the interest rate and σ is the volatility. Use the Euler scheme: S (k) t 0 = S 0, S (k) t i = S (k) t i 1 + rs (k) t i 1 (t i t i 1 ) + σs (k) t ti i 1 t i 1 Z (k) The estimate of a European call option price: ˆv n = 1 n e rtm (S (k) t n m K ) + k=1 The estimate of an Asian call option price: ( ) + ˆv n = 1 n 1 m e rtm S (k) t n m i K. k=1 i=1 i, 1 i m Haijun Li Math 416/516: Stochastic Simulation Week 13 6 / 28

Discretization Error for v = E[h(X t, 0 t T )] Since we use a time-discretized version of Itō diffusion to approximate the continuous-time Itō diffusion, this would create bias, called the discritization error. That is, in general, ˆv n is a biased estimator. MSE (or L 2 -distance) = E(ˆv n v) 2 = var(ˆv n ) + [E(ˆv n ) v] 2 }{{} bias 2 Such discretization error cannot be eliminated by merely increasing the sample size n. The variance reduction techniques decreases the variance var(ˆv n ), but would not be able to eliminate the bias. Under some mild regularity conditions, when the time discretization becomes finer (m becomes larger), the discretization error converges to zero. Haijun Li Math 416/516: Stochastic Simulation Week 13 7 / 28

Convergence Rate of Numerical Solutions The Euler Approximation Rate is 0.5. Consider the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. Assume that the coefficient functions µ(x) and σ(x) are Lipschitz continuous, and EX0 2 <. The Euler approximation converges strongly with order 0.5. That is, there exists a constant c > 0 such that E X T X (m) T c mesh(τ m ) 0.5. ( 0, as m ) One could use E sup 0 t T X t X (m) t as a more appropriate criteria to describe the pathwise closeness of X and X (m). But this quantity is more difficult to deal with theoretically. Haijun Li Math 416/516: Stochastic Simulation Week 13 8 / 28

160 CHAPTER 3. Numerical Solutions (dashed lines) VS Exact Solution LT-- 00 02 04 06 08 10 00 02 04 06 08 10,! I 8,,,, 00 02 0.4 06 0.8 10 t Figure : dx t = 0.01X t dt + 0.01X t db t, X 0 = 1. Haijun Li Math 416/516: Stochastic Simulation Week 13 9 / 28

Example: Eliminating Discretization Error Consider the Itō diffusion X t that satisfies the SDE dx t = r dt + θ(t)db t, 0 t T, where r is a constant and θ(t) is a deterministic function of t. Rewrite: t X t = X 0 + r t + θ(s)db s 0 Using Itō s Isometry, we have t ( t ) θ(s)db s N 0, θ(s) 2 ds 0 0 When t = T, ( 1/2 T X T = X 0 + r T + θ(s) ds) 2 Z where Z has the standard normal distribution. Haijun Li Math 416/516: Stochastic Simulation Week 13 10 / 28 0

Example: Eliminating Discretization Error (cont d) For a European-type call option, the estimate of the price: ˆv n = 1 n n k=1 e rt (X (k) T K ) + v = E[e rt (X T K ) + ], which does not involve discretization, and has no discretization error. For an Asian-type call option, the estimate of the price: ( ) + ˆv n = 1 n e rt 1 m X (k) t n m i K k=1 i=1 ( ) + ] v = E [e rt 1 T X s ds K, T 0 which still needs discretization. Haijun Li Math 416/516: Stochastic Simulation Week 13 11 / 28

Milstein Approximation Scheme Consider the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. In contrast to the first order approximation, the Milstein approximation exploits a so-called Taylor-Itō expansion that incorporates high order approximation. Heuristics: Apply the Itō lemma to the integrands µ(x s ) and σ(x s ) at each point t i 1 of discretization, and then estimate the higher order terms using separable multiple stochastic integrals. Taylor-Itō expansions involve multiple stochastic integrals. Their rigorous treatment requires a more advanced theory of the stochastic calculus. Haijun Li Math 416/516: Stochastic Simulation Week 13 12 / 28

Milstein Approximation (Grigori Milstein, 1975) 1 To approximate the solution, partition [0, T ] as follows, τ m : 0 = t 0 < t 1 < < t m 1 < t m = T, with i = t i t i 1, 1 i m, and mesh(τ m ) = max 1 i m i. Let i B = B ti B ti 1, 1 i m. 2 With initial value X 0, define recursively, for 1 i m, X ti = X ti 1 + µ(x ti 1 ) i + σ(x ti 1 ) i B + 1 }{{} 2 σ(x t i 1 )σ (X ti 1 )[( i B) 2 i ]. }{{} Euler s first order approx Milstein s second order term Note that where Z N(0, 1). i B = t i t i 1 Z N(0, t i t i 1 ), ( i B) 2 i = (t i t i 1 )(Z 2 1) Haijun Li Math 416/516: Stochastic Simulation Week 13 13 / 28

Estimation via Milstein Scheme Goal: To estimate v = E[h(X t, 0 t T )], where (X t, 0 t T ) is the solution of the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. Generate n independent sample paths, 1 k n, X (k) t i = X (k) t i 1 + µ(x (k) t i 1 )(t i t i 1 ) + σ(x (k) t i 1 ) t i t i 1 Z (k) i + 1 (k) σ(x t 2 i 1 )σ (X (k) t i 1 )(t i t i 1 )[(Z (k) i ) 2 1], with X (k) 0 = X 0, where Z1 k,..., Z m k are i.i.d. standard normal random variables. Use the estimator ˆv n = 1 n approx of (X t,0 t T ) n {}}{ E[h( X (k) t 0, X (k) t 1,..., X (k) t m )] k=1 Haijun Li Math 416/516: Stochastic Simulation Week 13 14 / 28

Convergence Rate of Milstein Scheme The Milstein Approximation Rate is 1.0. Consider the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. Assume that the coefficient functions µ(x) and σ(x) are Lipschitz continuous, and EX0 2 <. The Euler approximation converges strongly with order 1.0. That is, there exists a constant c > 0 such that E X T X (m) T c mesh(τ m ) 1.0. ( 0, as m ) Haijun Li Math 416/516: Stochastic Simulation Week 13 15 / 28

Euler (left 3.4. NUMERICAL column) SOLUTION VS Milstein (right column) 165 I 0.0 0.2 0.4 0.8 0.8 1.0 0.0 0.2 0.4 0.8 0.8 I I 1C I 00 0.2 0.4 0.8 0.n I Figure : dx t = 0.01X t dt + 2X t db t, X 0 = 1. Haijun Li Math 416/516: Stochastic Simulation Week 13 16 / 28

The Lamperti Transform (John Lamperti, 1962, 1972) Consider the SDE: dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. If σ(x) is constant (σ (x) = 0), then the Milstein scheme reduces to the Euler scheme. Example: Langevin Equation dx t = cx t dt + σdb t, t [0, T ]. Can we transform the SDE to a SDE with constant volatility? We standardize it! Use the Lamperti transform Y t = L(X t ), where L(x) = 1 σ(x) dx. The Lamperti transform is widely used in studies of Self-Similar phenomena and Fractals. Haijun Li Math 416/516: Stochastic Simulation Week 13 17 / 28

Standardize SDEs via the Lamperti Transform Consider the SDE dx t = µ(x t )dt + σ(x t )db t, t [0, T ]. Take the derivatives on the Lamperti transform: Use Itō s Lemma, L (x) = 1 σ(x), L (x) = σ (x) σ 2 (x) dy t = dl(x t ) = L (X t )dx t + 1 2 L (X t )dt = [ ] µ(xt ) σ(x t ) σ (x) dt +db t. 2 Since X t = L 1 (Y t ), we have the standardized SDE for Y t : [ µ(l 1 (Y t )) dy t = σ(l 1 (Y t )) σ (L 1 ] (Y t )) dt + db t. 2 Haijun Li Math 416/516: Stochastic Simulation Week 13 18 / 28

Example: Geometric Brownian Motion Consider the linear SDE: with initial value X 0 > 0. Use the Lamperti transform L(x) = x dx t = rx t dt + σx t db t, X 0 1 σs ds = 1 σ log x X 0, Y t = L(X t ) = 1 σ log X t X 0, X t = X 0 e σy t. Since dy t = ( r σ σ ) ( r dt + db t = Y t = 2 σ σ ) t + B t, 2 we obtain the geometric Brownian motion X t = X 0 e (r σ2 /2)t+σB t. Haijun Li Math 416/516: Stochastic Simulation Week 13 19 / 28

Example: Cox-Ingersoll-Ross Process Consider the non-linear SDE: dx t = a(b X t )dt + σ X t db t, with initial value X 0 = 0. Here a, b, σ are positive constants satisfying that 2ab σ 2. Use the Lamperti transform L(x) = x 0 1 σ s ds = 2 x, Yt = L(X t ) = 2 Xt, X t = σ2 Yt 2 σ σ 4. Convert the SDE to a SDE with constant volatility dy t = ( 4ab σ 2 1 2σ 2 ay ) t dt + db t Y t 2 and then use the Euler scheme with no discretization error. Haijun Li Math 416/516: Stochastic Simulation Week 13 20 / 28

Remark: Interest (Short) Rate Modeling The well-known Vasicek interest rate model (Oldrich Vasicek, Wells Fargo, 1977) describes the interest rate X t as an Ornstein-Uhlenbeck process. Haijun Li Math 416/516: Stochastic Simulation Week 13 21 / 28

Remark: Interest Rate Modeling The basic Vasicek interest rate model was extended to the mean-reverting Itō diffusion. Haijun Li Math 416/516: Stochastic Simulation Week 13 22 / 28

Remark: Interest Rate Modeling The Cox-Ingersoll-Ross model (John Cox, Jonathan Ingersoll and Stephen Ross, 1985; also see Lin Chen, US Federal Reserve, 1994) describes the evolution of interest rates, as an extension of the well-known Vasicek model. Haijun Li Math 416/516: Stochastic Simulation Week 13 23 / 28

Monte Carlo vs Numerical Methods Once sample paths (or scenarios) of the solution of a SDE are obtained, they can be used to estimate the distributional quantities (expectations, variances, higher-order moments) of the solution. Since derivative prices are often written as expectations of underlying asset values, which are the solutions of SDEs, Monte Carlo method becomes an essential tool in the pricing of derivative securities and in risk management. Monte Carlo is generally not a competitive method for calculating univariate expectation. For example, the error in a trapezoidal rule for the integral of a d-dimensional twice continuously differentiable function is O(n 2/d ), which is in contrast to the standard error O(n 1/2 ) of the Monte Carlo method for the same problem. The performance degradation with increasing dimension is a characteristic of all deterministic integration methods, and thus Monte Carlo methods are attractive in high dimension. Haijun Li Math 416/516: Stochastic Simulation Week 13 24 / 28

Black-Scholes Model Assume that the stock price follows a geometric Brownian motion: that is, S t = S 0 e (c σ2 /2)t+σB t, t 0, S t LogN(log S 0 + (c σ 2 /2)t, σ 2 t), where the drift c and volatility σ are the (physical) parameters of the stock price process. The risk-free interest rate is given by r. Under the risk-neutral probability measure, S t LogN(log S 0 + (r σ 2 /2)t, σ 2 t). Fundamental Theorem of Arbitrage-Free Pricing Estimate value at t = 0 with payoff X at maturity T via MC methods: v = E [e rt X] Haijun Li Math 416/516: Stochastic Simulation Week 13 25 / 28

Typical Example: Discrete-time Asian Options Consider the payoff v = (X K ) +, where X = ( m j=1 X t j )/m for a fixed set of monitoring dates 0 = t 0 < t 1 < < t m = T. Goal: To find the value v = E(e rt (X K ) + ) where X tj = X tj 1 e (r 1 2 σ2 )(t j t j 1 )+σ t j t j 1 Z j, 1 j m. Typical Monte Carlo Algorithm for i = 1,..., n for j = 1,..., m generate the standard normal Z ij set X i (j) = X i (j 1)e (r 1 2 σ2 )(t j t j 1 )+σ t j t j 1 Z ij set X i = (X i (1) + + X i (m))/m set C i = e rt (X i K ) + set Ĉn = (C 1 + + C n )/n. Haijun Li Math 416/516: Stochastic Simulation Week 13 26 / 28

Efficiency of Simulation Estimators Ĉ n from above illustrative example is unbiased and asymptotically normal. More precisely, let s denote our computational budget, and τ denote the computational time needed for C i, then s[ Ĉ s/τ C] d N(0, σ 2 C τ), as s. In comparing unbiased estimators, we should prefer the one for which σc 2 τ is smallest. For finite but at least moderately large n, we can supplement the point estimate Ĉn with a (1 α)100% confidence interval Ĉ n ± t α/2,n 1 s C n, where s C is the sample standard deviation, and t α/2,n 1 is the upper 100(α/2)th percentage point of a t distribution with n 1 degrees of freedom. Haijun Li Math 416/516: Stochastic Simulation Week 13 27 / 28

Be Aware of Your Estimation Biases! Bias frequently occurs in estimation via MC methods. For example, the bias can arise due to the following errors. 1 Model discretization error: For many models, exact sampling of the continuous-time dynamics is infeasible, some discretization approximation has to be used, resulting a bias. 2 Payoff discretization error: Discretization has to be used for the payoffs that are functionals of the underlying asset processes. 3 Nonlinear functions of means: In a compound option, the price of the first option depends on the price of the second option..., but these prices can only be estimated, resulting a bias. Haijun Li Math 416/516: Stochastic Simulation Week 13 28 / 28