Slides for DN2281, KTH 1

Transcription

1 Slides for DN2281, KTH 1 January 28, Based on the lecture notes Stochastic and Partial Differential Equations with Adapted Numerics, by J. Carlsson, K.-S. Moon, A. Szepessy, R. Tempone, G. Zouraris.

2 DN2281, Computational Methods for Stochastic Differential Equations. Spring Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Weak and strong approximation, efficient numerical methods and error estimates, variance reduction techniques. Prerequisite: Familiarity with stochastic processes, ordinary differential equations, numerical methods. Instructor: Erik von Schwerin

3 January 22, Class contents: 1. Course Introduction, Admin details 2. Motivating examples

4 Admin details list 15 lectures, link to schedule on the course web page Reading: Lecture notes on home page. Please print as you go, not all at once! Examination: 5 sets of homeworks, written reports, one group presents 1 final project/paper presentation Final written exam, mostly consisting of questions randomly selected from a list of study questions Homeworks, presentations, to be done in groups (of 2?). First homework due in 2 weeks, form groups next lecture The group that makes the presentation is encouraged to hand in a draft of their solution a couple of days in advance. office hours, for now me at schwerin@kth.se to set a time to meet

5 The goal of this course is to give useful understanding for solving problems formulated by stochastic differential equations models in science, engineering, and mathematical finance.

6 Motivating examples (Chapter 1)

7 e art ch. Analysis Info s Blogs Events Board stics gs ors ip lders ransactions oster Finance Search Inc. Noisy (GOOG) Evolution of Stock Values N GOOG Sun, Aug 24, 2008, 9:26PM ET - U.S. Markets Closed. On Aug 22: (0.83%) e ECN NEW! l Prices Enter name(s) or symbol(s) COMPARE TECHNICAL INDICATORS CHART SETTINGS RESET y nts Coverage pinion stimates Reports lysts GOOGLE from 2006 to 2008 ls tatement Sheet w HEADLINES Basic Chart Full Screen Print Share Send Feedback

8 S&P 500 over a nearly 50 years period. Above: linear. Below: logarithmic

9 Noisy Evolution of Stock Values Denote stock value by S(t). Assume that S(t) satisfies the differential equation which has the solution ds dt = a(t)s(t), S(t) = e t 0 a(u)du S(0). Since we do not know precisely how S(t) evolves we would like to generalize the model to a stochastic setting a(t) = r(t) + noise.

10 For instance, ds(t) = r(t)s(t)dt + σs(t)dw (t), (1) where dw (t) will introduce noise in the evolution. What is the meaning of (1)? The answer is not as direct as in the deterministic ODE case. One way to give meaning to (1) is to use the Forward Euler discretization, S n+1 S n = r n S n t n + σ n S n W n. (2) Here W n are independent normally distributed random variables...

11 ... with zero mean and variance t n, i.e. E[ W n ] = 0 and Var[ W n ] = t n = t n+1 t n. Then (1) is understood as a limit of (2) when max t 0.

12 Noisy Evolution of Stock Values 1.25 N = S log(s) time time One realization with N = 64 steps, σ = 0.15 and r = 0.05

13 Noisy Evolution of Stock Values 1.25 N = S log(s) time time One realization with N = 128 steps, σ = 0.15 and r = 0.05

14 Noisy Evolution of Stock Values 1.25 N = S log(s) time time One realization with N = 256 steps, σ = 0.15 and r = 0.05

15 Applications to Option pricing European call option: is a contract signed at time t which gives the right, but not the obligation, to buy a stock (or other financial instrument) for a fixed price K at a fixed future time T > t. At time t the buyer pays the seller the amount f (s, t; T ) for the option contract. What is a fair price for f (s, t; T )?

16 The Black-Scholes model for the value f : (0, T ) (0, ) R of a European call option is the partial differential equation t f + rs s f + σ2 s s f = rf, 0 < t < T, f (s, T ) = max(s K, 0), (3) where the constants r and σ denote the riskless interest rate and the volatility, respectively.

17 Stochastic representation of f (s, t) The Feynmann-Kač formula gives the alternative probability representation of the option price f (s, t) = E[e r(t t) max(s(t ) K, 0)) S(t) = s], (4) where the underlying stock value S is modeled by the stochastic differential equation (1) satisfying S(t) = s. Thus, f (s, t) is both the solution of a PDE (3) and the expected value of the solution of a SDE (4)! Which one should we choose to discretize?

18 g(x) p(x) time t x Sample paths for the approximation of a put option. x 0

19 Stochastic Particle Simulations Molecular dynamics simulation of particles, with positions X t in a potential, V (X ). Standard method to simulate MD in the microcanonical ensemble of constant number of particles, volume, and energy, (N,V,E), is to solve Newton s equations (deterministic) dx t i Mdv t i = vi t dt, = Xi V (X t ) dt (5)

20 Stochastic Particle Simulations To simulate a system with constant temperature instead of energy, (N,V,T), one often simulate (5) but add a regular rescaling of the kinetic energy to keep T constant, thermostats. An alternative is to simulate the Langevin dynamics dx t i Mdv t i = v t i dt, = Xi V (X t ) dt v i t τ dt + 2γ τ dw t i, (6) where W i are independent Brownian motions, τ is a relaxation time parameter, and γ := k B T.

21 Stochastic Particle Simulations Under some assumptions on the potential V, one can sample the same invariant measure, e V (X )/γ dx R 3N e V (X )/γ dx, (7) using overdamped Langevin dynamics: Smoluchowski Dynamics at constant temperature, T, dx s i = Xi V (X s )dt + 2γ dw s i. (8)

22 Optimal Control of Investments Suppose that we invest in a risky asset, whose value S(t) evolves according to the stochastic differential equation ds(t) = µs(t)dt + σs(t)dw (t), and in a riskless asset Q(t) that evolves with dq(t) = rq(t)dt. It is reasonable to assume r < µ, why? Our total wealth is then X (t) = Q(t) + S(t). Goal: determine an optimal instantaneous policy of investment to maximize the expected value of our wealth at a given final time T.

23 Let the time dependent proportion, be defined by so that α(t) [0, 1], α(t)x (t) = S(t), (1 α(t))x (t) = Q(t). Then our optimal control problem can be stated as max E[g(X (T )) X (t) = x] u(t, x), (9) α A where g is a given function. How can we determine α?

24 The solution to (9) can be obtained by means of a Hamilton Jacobi equation, which is in general a nonlinear partial differential equation satisfied by u(t, x) of the form u t + H(u, u x, u xx ) = 0. Part of our work is to study the theory of Hamilton Jacobi equations and numerical methods for control problems to determine the Hamiltonian H and the control α.

25 Towards a definition of SDEs: Ito Integrals