An Introduction to Stochastic Calculus

Transcription

1 An Introduction to Stochastic Calculus Haijun Li Department of Mathematics Washington State University Week 13 Haijun Li An Introduction to Stochastic Calculus Week 13 1 / 17

2 Outline 1 Numerical Solutions The Euler Approximation The Milstein Approximation Monte Carlo Methods in Financial Engineering References Haijun Li An Introduction to Stochastic Calculus Week 13 2 / 17

3 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 and option pricing. Haijun Li An Introduction to Stochastic Calculus Week 13 3 / 17

4 The Euler Approximation Scheme For illustration, 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 n : 0 = t 0 < t 1 < < t n 1 < t n = T, with i = t i, 1 i n, and mesh(τ n ) = max 1 i n i. Let i B = B ti B ti 1, 1 i n. 2 Define recursively, 1 i n, with X (n) 0 = X 0 X (n) t i = X (n) + µ(x (n) ) i + σ(x (n) ) i B, Haijun Li An Introduction to Stochastic Calculus Week 13 4 / 17

5 Idea: The First-Order Approximation 1 Consider, 1 i n, X ti = X ti 1 + ti µ(x s )ds + ti σ(x s )db s. 2 The Euler approximation is based on a discretization of the integrals ti µ(x s )ds µ(x ti 1 ) i, 3 That is, for 1 i n, ti σ(x s )db s σ(x ti 1 ) i B. X ti X ti 1 + µ(x ti 1 ) i + σ(x ti 1 ) i B. In practice one usually chooses equi-distant points t i such that mesh(τ n ) = T /n, and X (n) it /n = X (n) (n) (i 1)T /n + µ(x (i 1)T /n ) i + σ(x (n) (i 1)T /n ) ib, 1 i n. Haijun Li An Introduction to Stochastic Calculus Week 13 5 / 17

6 Strong Numerical Solution Strong Marginal Convergence 1 The numerical solution X (n) converges strongly to X with order γ > 0 if there exists a constant c > 0 such that E X T X (n) T c mesh(τ n) γ, n 1. 2 X (n) is a strong numerical solution of the SDE if E X T X (n) T 0, as mesh(τ n) 0. One could use E sup 0 t T X t X (n) t as a more appropriate criteria to describe the pathwise closeness of X and X (n). But this quantity is more difficult to deal with theoretically. The Euler Approximation The equidistant Euler approximation converges strongly with order 0.5. Haijun Li An Introduction to Stochastic Calculus Week 13 6 / 17

7 160 CHAPTER 3. Numerical Solutions (dashed lines) VS Exact Solution LT ,! I 8,,,, t Figure : dx t = 0.01X t dt X t db t, X 0 = 1. Haijun Li An Introduction to Stochastic Calculus Week 13 7 / 17

8 Weak Numerical Solution In contrast to a strong numerical solution, a weak numerical solution aims at the approximation of the moments of the solution X. Let f be chosen from a class of smooth functions, e.g., certain polynomials or functions with a specific polynomial growth. Weak Marginal Convergence 1 The numerical solution X (n) converges weakly to X with order γ > 0 if there exists a constant c > 0 such that Ef (X T ) Ef (X (n) T ) c mesh(τ n) γ, n 1. 2 X (n) is a weak numerical solution of the SDE if Ef (X T ) Ef (X (n) T ) 0, as mesh(τ n) 0. The Euler Approximation The equidistant Euler approximation converges weakly with order 1.0 for a class of functions f with appropriate polynomial growth. Haijun Li An Introduction to Stochastic Calculus Week 13 8 / 17

9 The Milstein Approximation Scheme In contrast to the first order approximation, the Milstein approximation exploits a so-called Taylor-Itô expansion that incorporates high order approximation. Heuristics: Apply the Itô lemma to the integrands µ(x s ) and σ(x s ) at each point of discretization, and then estimate the higher order terms using the fact that (db s ) 2 = ds. Taylor-Itô expansions involve multiple stochastic integrals. Their rigorous treatment requires a more advanced theory of the stochastic calculus. The Milstein Approximation Define recursively for 1 i n, = X (n) +µ(x (n) ) i +σ(x (n) ) i B X (n) t i with X (n) 0 = X 0. σ(x (n) )σ (X (n) )[( i B) 2 i ], The equidistant Milstein approximation converges strongly with order 1.0. Haijun Li An Introduction to Stochastic Calculus Week 13 9 / 17

10 Euler (left 3.4. NUMERICAL column) SOLUTION VS Milstein (right column) 165 I I I 1C I n I Figure : dx t = 0.01X t dt + 2X t db t, X 0 = 1. Haijun Li An Introduction to Stochastic Calculus Week / 17

11 Monte Carlo vs Numerical Methods Once sample paths (or scenarios) of the solution of an Itô SDE are obtained, they can be used to estimate the distributional quantities (expectations, variances, covariance and 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 evaluating integrals in high dimension. Haijun Li An Introduction to Stochastic Calculus Week / 17

12 Illustrative Example: European Call Option The price of one share of a risky asset (stock) is described by dx t = cx t dt + σx t db t, t [0, T ]. The price of a riskless asset (bond) is described by dβ t = rβdβ t, t [0, T ]. At time of maturity T, V T = (X T K ) +. Using the Fundamental Theorem of Arbitrage-Free Pricing, we have C := V 0 = E(e rt (X T K ) + ). with X T = X 0 e (r 1 2 σ2 )T +σb T. Although this formula can be written explicitly in terms of the normal distribution (the Black-Scholes formula), we can also estimate C using Monte Carlo method. Haijun Li An Introduction to Stochastic Calculus Week / 17

13 MC Estimate of European Call Options Algorithm for i = 1,..., n generate the standard normal Z i set X i (T ) = X 0 e (r 1 2 σ2 )T +σ T Z i set C i = e rt (X i (T ) K ) + set Ĉn = (C C n )/n. The estimator Ĉn is unbiased and strongly consistent. For finite but at least moderately large n, we can supplement the point estimate Ĉn with a (1 α)100% confidence interval s Ĉ n ± t C α/2,n 1 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 An Introduction to Stochastic Calculus Week / 17

14 Another Illustrative Example: Asian Options Consider the payoff V T = ( X K ) +, where X = ( m j=1 X t j )/m for a fixed set of dates 0 = t 0 < t 1 < < t m = T. Again, the Fundamental Theorem of Arbitrage-Free Pricing implies that C := V 0 = E(e rt ( X K ) + ) where X tj+1 = X tj e (r 1 2 σ2 )(t j+1 t j )+σ t j+1 t j Z j+1. 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 C n )/n. Haijun Li An Introduction to Stochastic Calculus Week / 17

15 Efficiency of Simulation Estimators Ĉ n from above two examples 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. 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 An Introduction to Stochastic Calculus Week / 17

16 Some References and Further Reading This lecture notes are written using the books Elementary Stochastic Calculus (World Scientific, 2002) by Thomas Mikosch, and Introductory Stochastic Analysis for Finance and Insurance (Wiley, 2006) by Sheldon Lin. A Standard Advanced Textbook on Itô Integrals: Brownian Motion and Stochastic Calculus (Springer 1991) by I. Karatzas and S. E. Shreve. Stochastic Integrals and SDEs Driven by Lévy Processes: Lévy Processes and Stochastic Calculus (Cambridge 2009) by D. Applebaum. Stochastic Finance: Stochastic Calculus for Finance I, II (Springer 2004) by S. E. Shreve. SDE Application in Actuarial Science: Introductory Stochastic Analysis for Finance and Insurance (Wiley, 2006) by Sheldon Lin, and Stochastic Control in Insurance (Springer 2008) by H. Schmidli. Haijun Li An Introduction to Stochastic Calculus Week / 17

17 More References Numerical Analysis on SDEs: Numerical Solution of Stochastic Deferential Equations (Springer 1995) by P. Kloeden and E. Platen. Monte Carlo Simulation: Monte Carlo Methods in Financial Engineering (Springer 2004) by Paul Glasserman. Lévy Matters: Financial Modelling with Jump Processes (Chapman & Hall 2004) by Rama Cont and Peter Tankov. Financial Times Series (GARCH, univariate and multivariate): Statistics of Financial Markets (Springer 2008) by J. Franke, C. M. Hafner and W. K. Hardle. Haijun Li An Introduction to Stochastic Calculus Week / 17