Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway In collaboration with Thilo Meyer-Brandis and Martin Groth (both CMA) Workshop in Quantitative Finance at the Isaac Newton Institute, Cambridge, July 8, 2005
Motivation The indifference price of a claim C: Investor indifferent between issuing a claim and investing the proceeds, or investing without issuing any claim Hodges and Neuberger (1989) Zero risk aversion for exponential utility gives a price Λ(t) = e r(t t) E QME [C F t ] Q is the minimal entropy martingal measure (MEMM) [ ( )] dq dq H(Q, P) = E dp ln, Q P dp Analyze the Barndorff-Nielsen and Shephard (2001) model
Overview of the talk The non-gaussian stochastic volatility model of Barndorff-Nielsen and Shephard Derivation of the MEMM for this model via dynamic programming A Black & Scholes integro-pde for options Numerical examples
Model proposed by Barndorff-Nielsen and Shephard (2001) Asset price dynamics ds(t) S(t) = (µ + βy (t)) dt + Y (t) db(t) Volatility (squared) is a non-gaussian Ornstein-Uhlenbeck process with a stationary distribution dy (t) = λy (t) dt + dl(λt) L is a subordinator, e.g. Lévy process without continuous martingale part and only positive jumps
An example: Suppose NIG(µ, β, α, δ)-logreturns Mean-variance mixture model R σ 2 N(µ + βσ 2, σ 2 ), σ 2 IG(α, δ) Choose L(t) to be a IG-subordinator
Optimization without claim issued Optimization with claim issued MEMM density process Wealth X (t) when the cash amount π is invested in risky asset (assuming risk-free return is zero): dx (t) = π(t) (µ + βy (t)) dt + π(t) Y (t) db(t) Optimal terminal wealth with exponential utility 1 exp( γx) V 0 (t, x, y) = sup π A t E [ 1 exp ( γx t,x,y (T ) )]. A t : set of admissible controls (integrability condition)
Optimization without claim issued Optimization with claim issued MEMM density process HJB-equation for V 0 V 0 t + max π { (µ + βy)πv 0 x + 1 2 yπ2 V 0 xx Terminal condition V 0 (T, x, y) = 1 exp( γx) Integro-operator L Y V 0 = λyv 0 y +λ Logarithmic transform of V 0 0 } + L Y V 0 = 0 { V 0 (t, x, y + z) V 0 (t, x, y) } ν(dz). V 0 (t, x, y) = 1 exp( γx)h(t, y)
Optimization without claim issued Optimization with claim issued MEMM density process The function H H turns out to rescale the jumps of L under MEMM Integro-pde for H H t (µ + βy)2 H + L Y H = 0 2y Terminal data H(T, y) = 1
Optimization without claim issued Optimization with claim issued MEMM density process Feynman-Kac representation of H: [ ( H(t, y) = E exp 1 T 2 t (µ + βy t,y (u)) 2 Y t,y (u) H C 1,1, thus strong solution of the integro-pde b If µ = 0, H(t, y) = exp ( b(t)y + c(t)) )] du A verification theorem validates the solution
Optimization without claim issued Optimization with claim issued MEMM density process Optimal terminal portfolio with a claim issued (payoff g(s)) V (t, x, s, y) = sup π A t E [ 1 exp ( γ(x t,x,y (T ) g(s t,s (T )) )]. HJB-equation V t +max {(µ + βy)πv x + 12 } π yπ2 V xx + yπsv xs +L S V +L Y V = 0 Transformation of value function V (t, x, y, s) = 1 exp ( γx + γλ (γ) (t, s, y) ) H(t, y)
Optimization without claim issued Optimization with claim issued MEMM density process Λ (γ) is the indifference price of the claim g V 0 (t, x, y) = V (t, x + Λ (γ) (t, s, y), s, y). A Black & Scholes equation for this price 0 = Λ (γ) λ γ t 0 + 1 2 ys2 Λ (γ) ss { exp λyλ (γ) y + ( ( γ Λ (γ) (y + z) Λ (γ) (y) )) } H(t, y + z) 1 ν(dz) H(t, y) Derivations leading to this are all informal
Optimization without claim issued Optimization with claim issued MEMM density process MEMM price From general theory, γ 0 gives the MEMM-price of the claim Informal limit leads to the Black & Scholes integro-pde Λ t + 1 2 σ2 (y)s 2 Λ ss λyλ y + λ 0 (Λ(t, y + z, s) Λ(t, y, s)) Terminal condition Λ(T, s, y) = g(s). Use this equation to: H(t, y + z) ν(dz) = 0. H(t, y) Read off the density process for the MEMM Q ME Price options (numerically)
Optimization without claim issued Optimization with claim issued MEMM density process The density process Consider the state dynamics d S(t) = Ỹ (t) S(t) d B(t), dỹ (t) = λỹ (t) dt + d L(λt), B(t) is Brownian motion and L(t) is a pure jump Markov process with the predictable compensating measure ν(ω, dz, dt) = H(t, Ỹ (t, ω) + z) ν(dz) dt. H(t, Ỹ (t, ω))
Optimization without claim issued Optimization with claim issued MEMM density process Define Z(t) := Z B (t) Z L (t) where ( t Z B µ + βy (s) (t) = exp db(s) 1 ) t (µ + βy (s)) 2 ds Y (s) 2 0 Y (s) 0 Z L (t) = exp ( t 0 + 0 t 0 H(s, Y (s) + z) ln N(dz, dt) H(s, Y (s)) ( 1 0 H(s, Y (s) + z) H(s, Y (s)) ) ) ν(dz)dt
Claim: Z(t) is the density process for the MEMM, dq ME = Z(T ) dp The derivation of Q ME informal Needs to verify that it indeed is the MEMM Use the results of Rheinländer (2003) Gives sufficient conditions for a MEMM
Theorem: Suppose we have { ( ) } β 2 exp (1 exp( λt )) z 1 ν(dz) <, λ 0 Then Z(t) is the density process of the minimal entropy martingale measure Q ME. Proof: Goes in two parts: 1. Z(t) is a martingale which can be represented as ( t Z(t) = c exp 0 ) µ + βy (s) S 1 (s)ds(s) Y (s) 2. Q ME induced by Z(T ) has finite entropy
1 holds by using Ito s Formula and the integro-pde for H(t, y) Recall that H C 1,1. Exponential integrability gives martingale property from the Novikov condition. 2 follows from direct estimation using the exponential integrability condition
A Feynman-Kac solution of the Black & Scholes integro-pde Λ(t, s, y) = E QME [ g ( S t,s,y (T ) )] This verifies that Λ solving the integro-pde is the MEMM price of g Λ t + 1 2 σ2 (y)s 2 Λ ss λyλ y + λ 0 (Λ(t, y + z, s) Λ(t, y, s)) H(t, y + z) ν(dz) = 0. H(t, y) We discuss a numerical solution of this integro-pde
Three practical problems for putting up a scheme: 1. Asymptotics in y and s when considering a finite domain? 2. Integral operator outside the finite domain? 3. y = 0, e.g., zero initial volatility? Note in addition: To solve for Λ, we need to solve for H as well
Example: A call option Strike K = 200 at exercise time T = 1, zero interest rate. Using properties of H, we can show that when y Ỹ (t) ye λt Hence, S becomes asymptotically a geometric Brownian motion with squared volatility y exp( λt) Λ(t, s, y) C(t, s, σ 2 = y exp( λt)) When s, we find Λ(t, s, y) s K
We can collect asymptotics in s and y into Λ(t, s, y) C(t, s, σ 2 = y exp( λt)) Yields values for Λ outside the finite domain to be used in the integral term For zero volatility, y = 0, we put on a gradient condition yielded by the integro-pde for Λ Asymptotics for H when y : H(t, y) exp( b(t)y + c(t)) For H, we consider the integro-pde defined for all real y s, thus avoiding imposing a boundary condition for y = 0
Implemented a Lax-Wendroff scheme for both H and Λ Central differences Integral evaluated using a trapezoid method Suppose L is inverse Gaussian subordinator, IG(11.98, 0.0872) About 8.5% in average yearly volatility Parameters collected from Nicolato and Venardos (2002) Consider the parameters µ = 0.05, β = 0.5
Numerical solution of H
Numerical solution of Λ
Difference with Black & Scholes price Black & Scholes with expected Y (t) as squared volatility About 8.5% in volatility
Volatility smile Implied volatility from Black & Scholes formula based on the MEMM prices
Further work Convergence analysis of the numerical scheme Pricing of Asian options Influence of autocorrelation structure in BNS-model on price