Convergence Analysis of Monte Carlo Calibration of Financial Market Models

Transcription

1 Analysis of Monte Carlo Calibration of Financial Market Models Christoph Käbe Universität Trier Workshop on PDE Constrained Optimization of Certain and Uncertain Processes June 03, 2009

2 Monte Carlo Calibration Fundamentals Focus will be on calibration of European call options Definition 1 (European Call Option) A European call option is the right to buy a predetermined underlying (e.g. stock) at a certain time T (maturity) for a certain price K (strike).

3 Monte Carlo Calibration Fundamentals Focus will be on calibration of European call options Definition 1 (European Call Option) A European call option is the right to buy a predetermined underlying (e.g. stock) at a certain time T (maturity) for a certain price K (strike). Definition 2 (Price of a Call Option) The price of a call option C in t = 0 can be calculated through C = e rt E (max(s T K, 0)) where r is the risk free rate and S T the value of the underlying at future time T.

4 Monte Carlo Calibration Stochastic Differential Equation L-dimensional system of stochastic differential equations (SDE): dy t (x) = a(x, Y t (x))dt + b(x, Y t (x))dw t where x R P Y t = [S t, Yt 2,..., Yt L ] R L W t = (Wt 1,..., Wt L ) R L a : R P R L R L b : R P R L R L R L vector of parameters Solution of SDE Vector of Brownian motions a l (x, Y t (x))dt L ν=1 bl,ν (x, Y t (x))dwt ν, l = 1,..., L

5 Monte Carlo Calibration Least Squares Problem Continuous Optimization Problem (True Problem) min f(x) := I ( C i (x) C i 2 x X obs) i=1 where C i (x) = e rt i E (max(s Ti (x) K i, 0)) s.t. dy t (x) = a(x, Y t (x))dt + b(x, Y t (x))dw t, Y 0 > 0 X R P convex and compact

6 Monte Carlo Calibration Least Squares Problem Continuous Optimization Problem (True Problem) min f(x) := I ( C i (x) C i 2 x X obs) i=1 where C i (x) = e rt i E (max(s Ti (x) K i, 0)) s.t. dy t (x) = a(x, Y t (x))dt + b(x, Y t (x))dw t, Y 0 > 0 X R P convex and compact Discretized Optimization Problem (SAA Problem) min f M, t,ɛ := I ( ) 2 CM, t,ɛ i (x) x X Ci obs i=1 M where C i M, t,ɛ (x) := e rt i 1 M s.t. m=1 ( ) π ɛ (s m N i,ɛ (x) K i) y m n+1,ɛ (x) = ym n,ɛ(x) + a ɛ (x, y m n,ɛ(x)) t n + b ɛ (x, y m n,ɛ(x)) W m n

7 Monte Carlo Calibration Smoothing Non-differentiabilities Consider Heston s Model: ds t,ɛ = (r δ)s t,ɛ dt + v + t,ɛ S t,ɛdw 1 t dv t,ɛ = κ(θ v + t,ɛ )dt + σ v + t,ɛ (ρdw 1 t + 1 ρ 2 dw 2 t )

8 Overview Table of Contents 1 Monte Carlo Calibration 2 Convergence Overview Pathwise Uniqueness Uniform Convergence First Order Optimality Condition 3 Conclusions

9 Overview Convergence True Problem min f(x) := I ( C i (x) C i 2 x X obs) i=1 SAA Problem min f M k, t k,ɛ k := I x X i=1 Increase number of simulations: ( ) 2 CM i k, t k,ɛ k (x) Cobs i M k Decrease discretization step size: t k 0 Decrease smoothing parameter: ɛ k 0 x k X solutions

10 Overview Convergence True Problem min f(x) := I ( C i (x) C i 2 x X obs) i=1 SAA Problem min f M k, t k,ɛ k := I x X i=1 Increase number of simulations: ( ) 2 CM i k, t k,ɛ k (x) Cobs i M k Decrease discretization step size: t k 0 Decrease smoothing parameter: ɛ k 0 X compact x kl x with x in X. x k X solutions

11 Overview Convergence True Problem min f(x) := I ( C i (x) C i 2 x X obs) i=1 SAA Problem min f M k, t k,ɛ k := I x X i=1 Increase number of simulations: ( ) 2 CM i k, t k,ɛ k (x) Cobs i M k Decrease discretization step size: t k 0 Decrease smoothing parameter: ɛ k 0 X compact x kl x with x in X. Question x solution of the true problem? x k X solutions

12 Overview Local Minima min f(x) := x2 x [ 1;1] min f M(x) := x 2 M 1 sin(mx 2 ) x [ 1;1]

13 Overview Local Minima min f(x) := x2 x [ 1;1] min f M(x) := x 2 M 1 sin(mx 2 ) x [ 1;1] Local minima might lead to problems:

14 Overview Literature Review True Problem: min h(x) := E(H(x, ω)) min x X SAA Problem x X h M(x) := 1 M M m=1 H(x, ω m) Shapiro (2000): Convergence if min h(x) produces global minimum Rubinstein & Shapiro (1993): Convergence to a critical first order point under assumption that H(x, ω) is dominated integrable and continuous Bastin et al. (2006): Additionally second order convergence even for stochastic constraints Dependence on three error sources: Monte Carlo, discretization and smoothing!

15 Overview Goal: First Order Optimality Steps to be taken: 1 Pathwise Uniqueness of SDE 2 Uniform Convergence: lim sup f Mk, t k,ɛ k (x) f(x) = 0 k x X lim sup f Mk, t k,ɛ k (x) f(x) = 0 k x X 3 First Order Optimality Condition: f(x ) T (x x ) 0 x X

16 Pathwise Uniqueness Table of Contents 1 Monte Carlo Calibration 2 Convergence Overview Pathwise Uniqueness Uniform Convergence First Order Optimality Condition 3 Conclusions

17 Pathwise Uniqueness Pathwise Uniqueness under Lipschitz Continuity Theorem 3 (Kloeden & Platen) Under the assumptions that There exists a constant K Lip > 0 such that t [0, T ] and y R L a(t, y) a(t, z) + b(t, y) b(t, z) K Lip y z There exists a constant K Grow > 0 such that t [0, T ] and y R L a(t, y) + b(t, y) K Grow (1 + y ) the stochastic differential equation dy t = a(t, Y t )dt + b(t, Y t )dw t, Y 0 (0, ). has a pathwise unique strong solution Y t on [0, T ].

18 Pathwise Uniqueness Problem: Lipschitz Continuity Consider Heston s model ds t,ɛ = (r δ)s t,ɛ dt + π ɛ (v t,ɛ )S t,ɛ dwt 1 dv t,ɛ = κ(θ π ɛ (v t,ɛ ))dt + σ π ɛ (v t,ɛ )(ρdwt ρ 2 dwt 2 ) Lipschitz continuity for ɛ > 0

19 Pathwise Uniqueness Yamada Condition Theorem 4 Let with dy t,ɛ = a ɛ (t, Y t,ɛ )dt + b ɛ (t, Y t,ɛ )dw t. a ɛ (t, Y t,ɛ ) = (a 1 ɛ(t, Y 1 t,ɛ),..., a L ɛ (t, Y L t,ɛ)) T b ɛ (t, Y t,ɛ ) = diag(b 1 ɛ(t, Y 1 t,ɛ),..., b L ɛ (t, Y L t,ɛ)) If there exists a positive increasing function β : [0, ) [0, ) with and b i (t, x) b i (t, y) β( x y ) x, y R, i = 1,..., L δ with an arbitrarily small δ > β 2 (z)dz =.

20 Pathwise Uniqueness Yamada Condition (2)... and a positive increasing concave function α : [0, ) [0, ) such that with a i (t, x) a i (t, y) α( x y ) x, y R, i = 1,..., L δ 0 α 1 (z)dz =. with an arbitrarily small δ > 0, the SDE has a pathwise unique solution. Proof: Yamada (1971)

21 Pathwise Uniqueness Yamada Condition (3) Reconsider Heston s model ds t,ɛ = (r δ)s t,ɛ dt + π ɛ (v t,ɛ )S t,ɛ dwt 1 dv t,ɛ = κ(θ π ɛ (v t,ɛ ))dt + σ π ɛ (v t,ɛ )(ρdwt ρ 2 dwt 2 ) The drift is Lipschitz continuous: a i (t, x) a i (t, y) K Lip x y x, y R, i = 1, 2 and the diffusion is Hölder continuous: b i (t, x) b i (t, y) x y x, y R, i = 1, 2 with δ 0 1 dz = ; K Lip z δ 0 1 dz =. z

22 Pathwise Uniqueness Problem: Independent components required Heston s model: ds t,ɛ = (r δ)s t,ɛ dt + π ɛ (v t,ɛ )S t,ɛ dwt 1 dv t,ɛ = κ(θ π ɛ (v t,ɛ ))dt + σ π ɛ (v t,ɛ )dwt 2

23 Pathwise Uniqueness Problem: Independent components required Heston s model: ds t,ɛ = (r δ)s t,ɛ dt + π ɛ (v t,ɛ )S t,ɛ dwt 1 dv t,ɛ = κ(θ π ɛ (v t,ɛ ))dt + σ π ɛ (v t,ɛ )dwt 2 Solution: Process v t,ɛ has pathwise unique solution following Yamada s Theorem Insert this unique solution in process S t,ɛ Process S t,ɛ has pathwise unique solution following Yamada s Theorem Pathwise unique solution via Yamada s Theorem

24 Uniform Convergence Table of Contents 1 Monte Carlo Calibration 2 Convergence Overview Pathwise Uniqueness Uniform Convergence First Order Optimality Condition 3 Conclusions

25 Uniform Convergence Convergence of the Problem Reconsider: f M, t,ɛ (x) f(x) f M, t,ɛ (x) f t,ɛ (x) (1) + f t,ɛ (x) f ɛ (x) (2) + f ɛ (x) f(x) (3)

26 Uniform Convergence Convergence of the Problem Reconsider: f M, t,ɛ (x) f(x) f M, t,ɛ (x) f t,ɛ (x) (1) + f t,ɛ (x) f ɛ (x) (2) + f ɛ (x) f(x) (3) Assumption: There exists a constant K Grow > 0 such that t [0, T ] and y R L a ɛ (t, y) + b ɛ (t, y) K Grow (1 + y ).

27 Uniform Convergence Convergence of Smoothed and Discretized SDE Theorem 5 Consider the SDE dy t,ɛ = a ɛ (t, Y t,ɛ )dt + b ɛ (t, Y t,ɛ )dw t. and the continuously interpolated process t y t,ɛ = Y t a ɛ (x, y τ(s),ɛ )ds + 0 b ɛ (x, y τ(s),ɛ )dw s where τ(s) = n, s [τ n, τ n+1 ) and n = 0,..., N 1. Assuming that the growth condition holds and the SDE has a pathwise unique solution it holds ( lim sup E y t,ɛ Y T,ɛ 2) = 0. t 0 x X Proof: Kaneko & Nakao (1988)

28 Uniform Convergence Convergence of Smoothed SDE Theorem 6 Assume that the growth condition and the pathwise uniqueness holds for a solution of dy t = a(t, Y t )dt + b(t, Y t )dw t and let Y t,ɛ be a solution of dy t,ɛ = a ɛ (t, Y t,ɛ )dt + b ɛ (t, Y t,ɛ )dw t. If a ɛ and b ɛ converge uniformly to a and b for ɛ 0, i.e. lim sup ɛ 0 t [0,T ] x X sup ( a ɛ (t, x) a(t, x) + b ɛ (t, x) b(t, x) ) = 0. where is a matrix norm, it holds lim sup E ɛ 0 x X Proof: Kaneko & Nakao (1988) ( Y t,ɛ Y t 2) = 0.

29 Uniform Convergence Dominated Integrability & Continuity Lemma 7 Assume that the families {π(s T (x, ω) K), x X} are dominated by a Q-integrable function P (ω). Then there exist t > 0 and ɛ > 0 such that {π ɛ (s N,ɛ (x, ω) K), x X} is dominated by a Q-integrable function for all t [0, t] and ɛ [0, ɛ]. Lemma 8 If the functions π(s T (, ω) K) are continuous on X for Q almost every ω, the functions π ɛ (s N,ɛ (x, ω) K) are continuous on X for 0 < t < and 0 < ɛ <.

30 Uniform Convergence Uniform Convergence Theorem 9 Assume that the families {π(s T (x, ω) K), x X} are dominated by a Q-integrable function P (ω) and furthermore the functions π(s T (, ω) K) are continuous on X for Q almost every ω. If additionally X is compact, then f(x) is continuous on X. Furthermore f M, t,ɛ converges uniformly to f on X, i.e. for given sequences (M k ) k IN, ( t k ) k R + and (ɛ k ) k R + satisfying M k, t k 0, ɛ k 0 it holds lim sup f Mk, t k,ɛ k (x) f(x) = 0. k x X Note that the same can be shown for the gradients!

31 First Order Optimality Condition Table of Contents 1 Monte Carlo Calibration 2 Convergence Overview Pathwise Uniqueness Uniform Convergence First Order Optimality Condition 3 Conclusions

32 First Order Optimality Condition First Order Optimality Condition Theorem 10 Assume that the families {π(s T (x, ω) K), x X} and { x p π(s T (, ω) K), x X}, i = 1,..., I are dominated by a Q-integrable function P (ω) and furthermore the functions π(s T (, ω) K) and x p π(s T (, ω) K), i = 1,..., I are continuous on X for Q almost every ω and additionally that X is compact. Further let (M k ) k N +, ( t k ) k R +, (ɛ k ) k R + and (γ k ) k R + with M k, t k 0, ɛ k 0 and γ k 0 be given sequences and assume that (x k ) k IN X is a sequence of points satisfying f(x k ) T (x x k ) γ k x X. Then every limit point x X of (x k ) k almost surely satisfies the first order optimality condition f(x ) T (x x ) 0 x X.

33 First Order Optimality Condition Convergence: Graphical Illustration

34 First Order Optimality Condition Convergence: Graphical Illustration

35 First Order Optimality Condition Convergence: Graphical Illustration

36 Conclusions Table of Contents 1 Monte Carlo Calibration 2 Convergence Overview Pathwise Uniqueness Uniform Convergence First Order Optimality Condition 3 Conclusions

37 Conclusions Conclusions Set up calibration problem Discretized via Monte Carlo, Euler-Maruyama and smoothing Pathwise Uniqueness for resulting SDE under Yamada Condition Uniform convergence of objectives under unrestrictive assumptions First order optimality condition satisfied for limit point x

38 Conclusions Bibliography Bastin,F., Cirillo,C. and Toint,P.L.: Convergence Theory for Nonconvex Stochastic Programming with an Application to Mixed Logit, Mathematical Programming Series B, Vol. 108, 2006, Rubinstein,R.Y. and Shapiro,A.: Discrete Event Systems, John Wiley, 1993 Shapiro, A.: Stochastic Programming by Monte Carlo Simulation Methods, Stochastic Programming E-Print Series 2000, Kaneko,H. and Nakao,S.: A Note on Approximation for Stochastic Differential Equations, Seminaire de Probabilites, XXII, Lecture Notes in Mathematics, Vol. 1321, 1988, Yamada, T. and Watanabe, S.: On the Uniqueness of Solutions of Stochastic Differential Equations, Journal of Mathematics of Kyoto University, Vol 11, 1971 Kaebe, C., Maruhn, J. and Sachs, E.W.: Adjoint Based Monte Carlo Calibration of Financial Market Models, Journal of Finance and Stochastics (to appear)