5 December 2016 MAA136 Researcher presentation. Anatoliy Malyarenko. Topics for Bachelor and Master Theses. Anatoliy Malyarenko

Transcription

1 5 December 216 MAA136 Researcher presentation 1

2 schemes The main problem of financial engineering: calculate E [f(x t (x))], where {X t (x): t T } is the solution of the system of X t (x) = x + Ṽ (X s (x))ds + V(X s (x))dw s (1) under the risk-neutral measure P. Here Ṽ : R N R N, V: R N R d, W s is a d-dimensional standard Wiener process, and f : R N R. A discretisation scheme is a set of N-dimensional random vectors {X (n) t (x): t = t i } with = t < t 1 < < t n = T. 2

3 Approximation problems Discuss the difference between strong and weak approximations and explain why weak approximation is more important in financial engineering. 3

4 An idea: apply Itô s formula For simplicity, consider the case of N = d = 1. The Itô formula takes the form ( f(x t (x)) = f(x) + f (X s (x))ṽ (X s (x)) + 1 ) 2 f (X s (x))v 2 (X s (x)) ds + f (X s (x))v(x s (x))dw s. The idea: apply Itô s formula again and again to f (X s (x)) and to f (X s (x)). The result is called the stochastic Taylor expansion and has very complicated form, see Kloden and Platen (1992). 4

5 Simplification: use Stratonovich integral instead For the Stratonovich integral, Itô s formula simplifies: f(x t (x)) = f(x) + + f (X s (x))ṽ (X s (x))ds f (X s (x))v(x s (x)) dw s. 5

6 Itô s formula in Stratonovich form Rewrite the system (1) in the form where W s X t (x) = x + = s and where d i= V i (y) = Ṽ i (y) 1 2 is the Stratonovich correction. Then V i (X s (x)) dw i s, (2) d N V k j=1 k=1 j (y) V j i (y) y k f(x t (x)) = f(x) + d i= (V i f)(x s (x)) dw i s, where (V i f)(y) := N j=1 V j i (y) f y j (y). 6

7 The stochastic Taylor expansion in Stratonovich form Let k be an nonnegative integer, and let α be either the empty set if k = or a multi-index α = (α 1,...,α k ) with α i d. Define the number α as k plus the number of zeros among α i s and call this the degree of α. For k 1, define the multiple Stratonovich integral by I(t, α, dw ) := Then k 2 k 1 dw α k t k dw α k 1 t k 1 dw α 1 t 1. f(x t (x)) = α m I(t, α, dw )(V αk V α1 f)(x) + R X m, (3) where the remainder Rm X contains multiple integrals of degrees greater than m. 7

8 Random paths A random path is a measurable map ω : Ω C,BV ([,1];R d+1 ), where the image is the set of all R d+1 -valued continuous functions of bounded variation in [,1] which start at. Along with the system (2), consider the system X t (x) = x + d i= V i (X s (x))dω i s, that is, the system of random ordinary differential. Let X ω t (x) be its solution. Define the time-scaled path ω s [t]: Ω C,BV ([,t];r d+1 ) by { tω, ωs[t] i if i =, s/t = tω i, otherwise. s/t 8

9 Estimating the approximation error We would like to estimate the approximation error E [f(x t (x))] E [f(x ω[t] t (x))]. For f(x ω[t] t (x)) we have an equation similar to (3): f(x ω[t] t (x)) = I(t, α,dω[t])(v αk V α1 f)(x) + Rm, X α m where I(t, α,dω[t]) := k 2 k 1 dω α k t k dω α k 1 t k 1 dω α 1 t 1. 9

10 An idea of cancelling The approximation error becomes E [I(t, α, dw )](V αk V α1 f)(x) + E [Rm] X α m α m E [I(t, α,dω[t])](v αk V α1 f)(x) E [R X m]. Assume the moment matching formula E [I(1, α, dw )] = E [I(1, α,dω)], α m. It is easy to check that the sums cancel each other and the remaining error E [R X m] E [R X m] is small. 1

11 The best case This is when there are functions g i C,BV ([,1];R d+1 ) and positive weights p i, 1 i L with P{ω = g i } = p i and p p L = 1. Then E [R X m] can be calculated exactly. This case is called the cubature on Wiener space, see Lyons and Victoir (24). Otherwise, they must be simulated. Then, we give examples of cubature in Wiener space and moment matching. 11

12 The idea Consider the case with N = d = 1 and f(x t (x)) = f(x 1 (x)). Müller-Gronbach et al (212) provided a deterministic algorithm to approximate E [f(x 1 (x))]. We will analyse and explain this algorithm, then apply it to the simplest case, the Black Scholes model. 12

13 The model Canhanga (216) considered the following market model under the risk-neutral measure: ds = µs dt + V 1 S dw 1 + V 2 S dw 2, dv 1 = 1 ε (θ 1 V 1 )dt + 1 ε ξ 1 ρ 13 V1 dw ε ξ 1 (1 ρ 2 13 )V 1 dw 3, dv 2 = δ(θ 2 V 2 )dt + δξ 2 ρ 24 V2 dw 2 + δξ 2 (1 ρ 2 24 )V 2 dw 4. Here N = 3, d = 4, ε and δ are two small positive numbers, V 1 (resp. V 2 ) is the fast (resp. slowly) changing stochastic volatility. 13

14 The problem In contrast to Canhanga (216), who investigated this model and found the price of a European call by using the Feynman Kac formula, we will use Monte Carlo methods based on cubature on Wiener space. In contrast to the topic for bachelor thesis, we will explain how to construct cubature formulae using algebraic methods introduced in Lyons and Victoir (24). 14

15 For further reading Canhanga, B. Asymptotic option in a market. Mälardalen University Doctoral Dissertations 219 (216). Kloden, P.E., Platen, E. Numerical solution of stochastic differential. Applications of Mathematics (New York), 23. Springer, Berlin (1992). Lyons, T., Victoir, N. Cubature on Wiener space. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 46 (24), no. 241, Müller-Gronbach, T., Ritter, K., Yaroslavtseva, L. Derandomisation of the Euler scheme for scalar stochastic differential. J. Complexity 28 (212),