The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

Abstract. We develop and apply a numerical scheme for pricing options for the stochastic volatility model proposed by Barndorff-Nielsen and Shephard. This non-Gaussian OrnsteinUhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To price numerically options with respect to this risk neutral measure, one needs to consider a Black & Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black & Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.

On the explicit evaluation of the Geometric Asian options in stochastic volatility models with jumps

“…The minimal entropy martingale measure for BNS models without and with leverage are studied in [25][26][27][28], Esscher transforms and other equivalent martingale measures are studied in [29]. Portfolio optimization has been studied in [30][31][32].…”

Derivative-free Greeks for the Barndorff-Nielsen and Shephard stochastic volatility model

“…The mean-reverting volatility process includes jumps given by a subordinator, a Lévy process with strictly non-negative increments. At the same time, as the model is able to generate realistic asset prices, it is analytically tractable enough for derivative pricing and portfolio optimization, see [3,20,21]. For the BNS model, the density of the price distribution is not known explicitly.…”

Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models

“…In [BK05,BMB05,BG05,RS06] the minimal entropy martingale measure is investigated. The papers [BKR03,Lin06] address the portfolio optimization problem.…”