Option pricing in the model with stochastic volatility driven by Ornstein–Uhlenbeck process. Simulation

Sergii Kuchuk-Iatsenko,Yuliya Mishura

doi:10.15559/15-vmsta43

Abstract

We consider a discrete-time approximation of paths of an Ornstein--Uhlenbeck process as a mean for estimation of a price of European call option in the model of financial market with stochastic volatility. The Euler--Maruyama approximation scheme is implemented. We determine the estimates for the option price for predetermined sets of parameters. The rate of convergence of the price and an average volatility when discretization intervals tighten are determined. Discretization precision is analyzed for the case where the exact value of the price can be derived.

Highlights

We consider a discrete-time approximation for the price of European call option in the model of financial market with stochastic volatility driven by the Ornstein–Uhlenbeck process
The problem of construction of discrete-time analogues for stochastic volatility models of financial markets is studied in a series of works including [5, 7, 2, 16, 1, 6, 18]
The simplest discrete-time approximation is the stochastic generalization of Euler approximation for deterministic differential equations proposed in [11], which is referred to as the Euler– Maruyama scheme

Summary

Introduction

We consider a discrete-time approximation for the price of European call option in the model of financial market with stochastic volatility driven by the Ornstein–Uhlenbeck process. The simplest discrete-time approximation is the stochastic generalization of Euler approximation for deterministic differential equations proposed in [11], which is referred to as the Euler– Maruyama scheme. Exact simulation provides more precision compared to the Euler approximation, in this paper, we use the latter This is motivated by the fact that the Euler approximation is cheaper in terms of computation time and by our desire to assess the rate of convergence of conditional option prices when the volatility is discretized using the Euler scheme. Appendix A contains definitions and auxiliary results on discretization schemes and orders of their convergence mostly coming from [8]

The model and discrete approximation of volatility process

The price of European call option

Numeric examples

Checking approximation precision in the case of deterministic volatility