Quadratic hedging in affine stochastic volatility models

Transcription

1 Quadratic hedging in affine stochastic volatility models Jan Kallsen TU München Pittsburgh, February 20, 2006 (based on joint work with F. Hubalek, L. Krawczyk, A. Pauwels) 1

2 Hedging problem S t = S 0 exp(x t ) H = f(s T ) asset price process option (contingent claim) How to hedge the risk from selling the claim? Hedging error: v + T 0 ϕ tds t H 2

3 Stock price process S t = S 0 exp(x t ) standard market model: X Brownian motion with drift d. h. X continuous, homogeneous in time perfect hedge exists (Black & Scholes, Merton 1973) empirical literature: 1. large daily price changes anormally often (heavy tails) 2. clustering of large daily price changes (volatility clustering) alternative models: 1. Lévy processes with jumps 2. processes with stochastic volatility 3

4 tägliche log Renditen DAX: Oktober 1959 April 2001 tägliche log Renditen Simulation Handelstage

5 Quadratic hedging in affine models S t = S 0 exp(x t ) X t H = f(s T ) martingale component of an affine process option (contingent claim) min v,ϕ E ( (v + T 0 ϕ tds t H ) 2 ) v ϕ variance-optimal initial capital variance-optimal hedging strategy 5

6 Some affine stochastic volatility models Stein & Stein (1991) dx t = (µ + δσ 2 t )dt + σ t dw t, dσ t = (κ λσ t )dt + αdz t Heston (1993) dx t = (µ + δv t )dt + v t dw t, dv t = (κ λv t )dt + σ v t dz t. 6

7 Barndorff-Nielsen & Shephard (2001) dx t = (µ + δv t )dt + v t dw t + ϱdz t, dv t = λv t dt + dz t. dx t = (µ + δv t )dt + v t dw t + v t = ν k=1 α k v (k) t, dv (k) t = λ k v (k) t dt + dzk t. ν k=1 ϱ k dz k t, 7

8 Carr, Geman, Madan, Yor (2003) X t = X 0 + µt + L Vt + ϱ(v t v 0 ), dv t = v t dt, dv t = (κ λv t )dt + σ v t dz t, X t = X 0 + µt + L Vt + ϱz t, dv t = v t dt, dv t = λv t dt + dz t. 8

9 Carr, Wu (2003) dx t = µdt + v 1/α t dl t, dv t = (κ λv t )dt + σ v t dz t. Carr, Wu (2004) X t = X 0 + µt + L Vt, dv t = v t dt, v t = v 0 + κt + Z Vt 9

10 Affine semimartingales Characteristics of semimartingale X in R d : B t = t 0 b sds, C t = t 0 c sds, ν([0, t] G) = t 0 F s(g)ds G B d b t = b (0) + c t = c (0) + d j=1 d j=1 F t (G) = F (0) (G) + X j t b (j) X j t c (j) d j=1 X j t F (j) (G) with given Lévy-Khintchine triplets (b (j), c (j), F (j) ), j = 0,..., d on R d 10

11 Characterization by Duffie, Filipovic, Schachermayer (2003) E (e iλ X s+t Fs ) = exp ( Ψ 0 (t, iλ) + Ψ (1,...,d) (t, iλ) X s ), λ R d, with Ψ (1,...,d) = (Ψ 1,..., Ψ d ) : R + (C m irn ) (C m irn ), Ψ 0 : R + (C m irn ) C solving the following system of generalized Riccati equations: Ψ 0 (0, u) = 0, Ψ (1,...,d) (0, u) = u, d dt Ψj (t, u) = ψ j (Ψ (1,...,d) (t, u)), j = 0,..., d and ψ j denoting the Lévy exponent of (b (j), c (j), F (j) ): ψ j (u) = u b (j) u c (j) u + (e u x 1 u h(x))f (j) (dx) 11

12 General structure of the variance-optimal hedge Cf. Föllmer & Sondermann (1986) Galtchouk-Kunita-Watanabe decomposition: T H = V 0 + ξ tds t + R T, 0 where R martingale, orthogonal to S (i.e. RS martingale) Mean value process of the option: V t := E(H F t ) Variance-optimal hedge: v = V 0, ϕ t = ξ t = d V, S t d S, S t Hedging error: E ( (v + T 0 ϕ t ds t H ) 2 ) = E ( V 0 ϕ t ds t, V ) 0 ϕ t ds t T Problem: How to compute V t, ξ t? 12

13 Integral representation of options Cf. Hubalek & Krawczyk (1998), Carr & Madan (1999), Raible (2000) Assumption: option of the form with some function p(u), H = R+i Su T p(u)du, Example: European call with arbitrary R > 1. H = (S T K) + = 1 2πi R+i Su T K 1 u u(u 1) du 13

14 Integral representation of several options call: (S T K) + = 1 2πi put: (K S T ) + = 1 2πi R+i R+i Su T Su T K 1 u du (R > 1) u(u 1) K 1 u du (R < 0) u(u 1) power call: ((S T K) + ) 2 = 1 2πi R+i Su T 2K 1 u du (R > 2) u(u 1)(u 2) self-quanto call: (S T K) + S T = 1 2πi R+i Su T K 1 u du (R > 2) (u 1)(u 2) digital option: 1 {ST >K} = 1 2πi R+i Su T K u u du (R > 0) log contract: log(s T ) = 1 2πi (R < 0, R > 0) R+i Su T 1 1 u 2du 2πi R +i R i Su T 1 u 2du 14

15 The variance-optimal hedging strategy v = E(H) = ϕ t = R+i R+i V (u) 0 p(u) du, V (u) t ϕ 1 (t, u)v t + ϕ 2 (t, u) p(u)du, S t ϑ 1 v t + ϑ 2 where ψ 1 (q) = iq β 1 2 q αq + ( e iq x 1 iq h(x) ) µ(dx), ψ 2 (q) = iq b 1 2 q aq + ( e iq x 1 iq h(x) ) m(dx), V (u) t := E(exp(u ln S T ) F t ) = exp (u ln S t + Φ 1 (T t, 0, u)v t + Φ 2 (T t, 0, u)), Φ 1, Φ 2 are solutions of generalized Riccati equations, related with ψ 1 and ψ 2, ϕ j (t, u) = ψ j ( iφ 1 (T t, 0, u), i(u + 1)) ψ j ( iφ 1 (T t, 0, u), iu) ψ j (0, i), ϑ j = ψ j (0, 2i) 2ψ j (0, i). 15

16 E ( ( v + = where T ϕ t ds t H 0 R+i R+i T 0 The expected squared hedging error ) 2 ) +D 2 Φ 2 (t, γ 1, u 1 + u 2 )) + l 1ϑ 1 l 2 ϑ 2 ϑ 2 1 e γ 2+(u 1 +u 2 ) ln S 0 ( exp(φ1 (t, γ 1, u 1 + u 2 )v 0 + Φ 2 (t, γ 1, u 1 + u 2 )) ( l 2 ϑ 1 (D 2 Φ 1 (t, γ 1, u 1 + u 2 )v 0 ) + l 0 ϑ 2 1 l 1ϑ 1 ϑ 2 + l 2 ϑ 2 2 ϑ 3 1 exp ( ϑ 2 ϑ 1 γ 1 ) 1 +Φ 1 ( t, ϑ 1 ϑ 2 ln s + γ 1 s, u 1 + u 2 ) v0 + Φ 2 ( t, ϑ 1 ϑ 2 ln s + γ 1 s, u 1 + u 2 )) ds ) p(u1 )p(u 2 )dtdu 1 du 2, 0 ( ϑ1 ϑ 2 + γ 1 s ) exp ( ϑ 2 ϑ 1 γ 1 s l 0 = l 0 (t, u 1, u 2 ) = ϑ 2 λ 2 (t, u 1, u 2 ) ϕ 2 (t, u 1 )ϕ 2 (t, u 2 ), l 1 = l 1 (t, u 1, u 2 ) = ϑ 2 λ 1 (t, u 1, u 2 ) + ϑ 1 λ 2 (t, u 1, u 2 ) ϕ 1 (t, u 1 )ϕ 2 (t, u 2 ) ϕ 1 (t, u 2 )ϕ 2 (t, u 1 ), l 2 = l 2 (t, u 1, u 2 ) = ϑ 1 λ 1 (t, u 1, u 2 ) ϕ 1 (t, u 1 )ϕ 1 (t, u 2 ), γ j = γ j (t, u 1, u 2 ) = Φ j (T t, 0, u 1 ) + Φ j (T t, 0, u 2 ), ϕ j (t, u) = ψ j ( iφ 1 (T t, 0, u), i(u + 1)) ψ j ( iφ 1 (T t, 0, u), iu) ψ j (0, i), λ j (t, u 1, u 2 ) = ψ j ( iγ 1 (t, u 1, u 2 ), i(u 1 + u 2 )) ψ j ( iφ 1 (T t, 0, u 1 ), iu 1 ) ψ j ( iφ 1 (T t, 0, u 2 ), iu 2 ), ϑ j = ψ j (0, 2i) 2ψ j (0, i). 16

17 Numerical illustration expected squared hedging error for an at-the-money call, T = 0.25 years model parameters option price variance of the hedging error optimal hedge no hedge Black-Scholes σ = (5.58) 2 α = NIG β = (3.13) (4.60) 2 δ = a = b = λ = NIG-Γ-OU α = (1.50) (4.42) 2 β = δ = 1 y 0 = κ = η = σ = NIG-CIR α = (1.92) (4.99) 2 β = δ = 1 y 0 = Parameters obtained via calibration by Schoutens (2003) 17

18 25 Variance-optimal initial capital in the Black-Scholes-, NIG-Gamma-OU-, NIG-CIR- and NIG-case for strike = 100 and maturity = 0.25 years Black-Scholes NIG-Gamma-OU NIG-CIR 20 NIG stock price 18

19 1 Variance-optimal initial hedge in the Black-Scholes-, NIG-Gamma-OU-, NIG-CIR- and NIG-case for strike = 100 and maturity = 0.25 years Black-Scholes NIG-Gamma-OU NIG-CIR NIG stock price 19

20 25 20 Variance optimal endowment for NIG(maturity = 3 months, 12 discrete trading dates) S=100, K=100, T=63, R=0.04/252,mu=0, delta=0.003, alpha=108.6, beta= Discrete NIG endowment Continuous NIG endowment NIG Esscher price Black-Scholes price Stock 20

21 1 0.9 Foellmer-Schweizer strategy for NIG (maturity = 3 months, 12 discrete trading dates) S=100, K=100, T=63, R=0.04/252,mu=0, delta=0.003, alpha=108.6, beta= Discrete NIG strategy Continuous NIG strategy Black-Scholes delta Stock 21

22 1.2 Variance of the hedging error (maturity = 3 months) S=100, K=100, T=63, R=0.04/252,mu=0, delta=0.003, alpha=108.6, beta= NIG discrete NIG continuous Black-Scholes discrete Number of trades 22