Cross hedging with stochastic correlation

Transcription

1 Gregor Heyne (joint work with Stefan Ankirchner) Humboldt Universität Berlin, QPL March 20th, 2009 Finance and Insurance, Jena

2 Motivation Consider a call option on the DAX. How to hedge? Maybe with Futures which have the DAX as underlying?

3 Motivation DAX FDAX 7500 Dax FDax Days to maturity

4 Related literature Henderson, 2002, Valuation of claims on nontraded assets using utility maximization; Musiela, Zariphopoulou, 2004, An example of indifference prices under exponential preferences; Davis, 2006, Optimal hedging with basis risk; Hulley, McWalter, 2008, Quadratic hedging of basis risk

5 Motivation Correlation between the logreturns of the DAX and DAX Futures Correlation Days to maturity

6 Modelling correlation Direct modelling with Jacobi process: dρ t = κ(ϑ ρ t )dt + α (1 (ρ t ) 2Ŵ t, with α,κ > 0, ϑ ( 1,1) Indirect modelling with OU process: Set ρ t = b 1 (U t ), with du t = a(ϑ U t )dt + σ U dŵt, where a > 0, ϑ R, σ U > 0 and b : ( 1,1) R continuous bijection.

7 The Model We want to hedge the contingent claim H = h(i T ) by trading the asset X. dx t = X t ((µ X r)dt + σ X dw 1 t ) di t = I t ((µ I r)dt + σ I (ρ t dw 1 t + dρ t = a(ρ t )dt + g(ρ t )dŵt 1 ρ 2 t dw 2 t )) Optimality criterion: Local risk minimization

8 FS decomposition via BSDEs El Karoui, Peng, Quenez, 1997: The FS decomposition of H = h(i T ) is given by T Z 1 T T s h(i T ) = Y 0 + dx s + Zs 2 dws 2 + Zs 3 dws 3, 0 σ X X s 0 0 where Y and Z are determined by T T Y t = h(i T ) Z s dw s Zs 1 t t µ X r ds, σ X for 0 t T.

9 Explicit solution of linear BSDE The solution of T T Y t = h(i T ) Z s dw s Zs 1 t t µ X r ds, σ X is given by and where Y t = IE[h(I T ) F t ] ( ) Z t = σ(t,i t,ρ t ) x ψ(t,i t,ρ t ), v ψ(t,i t,ρ t ) ψ(t,x,v) = IE[h(I t,x,v T )].

10 Technical details ψ(t,x,v) = IE[h(I t,x,v T )] 1 Under which conditions is ψ(t,x,v) differentiable with respect to x and v? 2 Under which conditions are I t,x,v and ρ t,v differentiable with respect to x and v? 3 Existence of representation of Z?

11 Differentiability of I and ρ Denote d dv ρt,v s and d dv It,x,v s by ρ t,v s and Īt,x,v s. Then s ρ s t,v = 1+ t s Īs t,x,v = t s a (ρu t,v ) ρ u t,v du + t g (ρ t,v u ) ρ u t,v dŵu, Īu t,x,v ((µ I r)du + σ I (ρu t,v dwu (ρu t,v ) 2 dwu 2 )) s + t Iu t,x,v σ I ( ρ t,v u dwu 1 ρu t,v 1 (ρ t,v u ρ u t,v dw 2 ) 2 u ).

12 Differentiability of I and ρ Set τ = inf{s 0 : ρ s { 1,1}}. (H1) τ > T, IP a.s. Moreover T t ( ρ s t,v ) 2 1 (ρs t,v du <, IP a.s. ) 2

13 Differentiability of ψ(t,x,v) = IE[h(I t,x,v T )] (H2) There exists p > 1 such that for every v 0 ( 1,1) there exists an open intervall U ( 1,1) of v 0 such that T ( ρ s t,v ) 2 supie[ v U t 1 (ρ t,v s ) 2 p ds] <. Lemma Let h be Lipschitz such that the weak derivative h is Lebesgue-almost everywhere continuous. Under the Conditions (H1) and (H2) the partial derivative v ψ(t,x,v) exists and is given by v ψ(t,x,v) = IE[h (I t,x,v T )Īt,x,v T ].

14 Representation of Z We want to justify Z t,x,v s ( = σ(s,is t,x,v,ρs t,v ) x ψ(s,i t,x,v s v ψ(s,i t,x,v s,ρs t,v ),ρs t,v ) Problem: ψ(s,x,v) = IE[h(I t,x,v )] is only locally Lipschitz. T ). We use di n t = I n t (µ I r)dt + σ I I n t ( (1 1 n )ρ tdw 1 t + 1 (1 1 n )2 ρ 2 t dw 2 t ) and in approximation proof. ψ n (t,x,v) = IE[h n (I n,t,x,v T )]

15 Representation of Z Lemma Assume that (H1) and (H2) hold, and that a and g are continuously differentiable with bounded derivatives. Let h be Lipschitz such that the weak derivative h is Lebesgue-almost everywhere continuous. Then Z t,x,v s ( = σ(s,is t,x,v,ρs t,v ) x ψ(s,i t,x,v s v ψ(s,i t,x,v s,ρs t,v ),ρs t,v ) ). (1)

16 Representation of Z Lemma Assume that (H1) and (H2) hold, and that a and g are continuously differentiable with bounded derivatives. Let h be Lipschitz such that the weak derivative h is Lebesgue-almost everywhere continuous. Then Z t,x,v s ( = σ(s,is t,x,v,ρs t,v ) x ψ(s,i t,x,v s v ψ(s,i t,x,v s,ρs t,v ),ρs t,v ) ). (1) By an argument of Imkeller, Reveillac, Richter one can show (1) without the use of Malliavin calculus! This allows us to drop the bounded derivatives assumption.

17 Main results Theorem Suppose that the coefficients a and g in the dynamics of ρ are continuously differentiable. Assume furthermore that Conditions (H1) and (H2) are satisfied. Let h be Lipschitz such that the weak derivative h is Lebesgue-almost everywhere continuous. Then, there exists a locally risk minimizing strategy ϕ = (ξ,η) for the derivative h(i T ). ξ is given by ξ t = ξ(t,i t,x t,ρ t ), where, with P denoting the minimal martingale measure of X, ξ(t,x,y,v) = v xσ I yσ X IE[h (I t,x,v T )I t,1,v T ]+ g(v)γ IE[h (I t,x,v yσ T X )Īt,x,v T ].

18 Main results Problem: (H2) There exists p > 1 such that for every v 0 ( 1,1) there exists an open intervall U ( 1,1) of v 0 such that T ( ρ s t,v ) 2 supie[ v U t 1 (ρ t,v s ) 2 Very technical, unintuitive assumption. BUT: Theorem p ds] <. Let a and g be bounded with bounded derivatives. We assume that g( 1) = g(1) = 0, and that g does not have any roots in ( 1,1). If 2a(x)(1 x) 2a(x)(1+x) lim sup x 1 g 2 < 0 and liminf (x) x 1 g 2 > 0, (x) then both Conditions (H1) and (H2) are satisfied.

19 Examples dρ t = κ(ϑ ρ t )dt + α(1 (ρ t ) 2 )Ŵt, with α,κ > 0, ϑ ( 1,1)

20 Examples dρ t = κ(ϑ ρ t )dt + α(1 (ρ t ) 2 )Ŵt, with α,κ > 0, ϑ ( 1,1) Set ρ t = b 1 (U t ), with du t = a(ϑ U t )dt + σ U dŵt, where a > 0, ϑ R, σ U > 0 and b(x) = x 1 x 2.

21 Examples dρ t = κ(ϑ ρ t )dt + α(1 (ρ t ) 2 )Ŵt, with α,κ > 0, ϑ ( 1,1) Set ρ t = b 1 (U t ), with du t = a(ϑ U t )dt + σ U dŵt, where a > 0, ϑ R, σ U > 0 and b(x) = x 1 x 2. dρ t = κ(ϑ ρ t )dt + α (1 (ρ t ) 2Ŵ t, with κ α2 1 ± ϑ and 1 < ϑ ϑ 2 ± α2 2κ < 1, also fulfills Conditions (H1) and (H2)!

22 Conclusions Correlation is random. Explicit hedge formula under the assumption (H2) There exists p > 1 such that for every v 0 ( 1,1) there exists an open intervall U ( 1,1) of v 0 such that T ( ρ s t,v ) 2 supie[ v U t 1 (ρ t,v s ) 2 p ds] <. (H2) is fulfilled by a large class of models for correlation dynamics (including Jacobi and OU processes).

23 Thank you for your attention!