Mathematical Problems in Engineering

Volume 2017, Article ID 3912036, 8 pages

https://doi.org/10.1155/2017/3912036

## Efficient Simulation for Pricing Barrier Options with Two-Factor Stochastic Volatility and Stochastic Interest Rate

School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

Correspondence should be addressed to Zhang Sumei; moc.anis@iemusggnahz

Received 11 August 2017; Accepted 8 October 2017; Published 14 November 2017

Academic Editor: Fazal M. Mahomed

Copyright © 2017 Zhang Sumei and Zhao Jieqiong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an extension of the double Heston stochastic volatility model by combining Hull-White stochastic interest rates. By the change of numeraire and quadratic exponential scheme, this paper develops a new simulation scheme for the extended model. By combining control variates and antithetic variates, this paper provides an efficient Monte Carlo simulation algorithm for pricing barrier options. Based on the differential evolution algorithm the extended model is calibrated to S&P 500 index options to obtain the model parameter values. Numerical results show that the proposed simulation scheme outperforms the Euler scheme, the proposed simulation algorithm is efficient for pricing barrier options, and the extended model is flexible to fit the implied volatility surface.

#### 1. Introduction

A barrier option is a path-dependent option which is exterminated (knocked out) or initiated (knocked in) if the underlying spot price hits the specified barrier level during the life of the option. Because of this supplementary risk, barrier options are cheaper than plain vanilla options and thus are widely traded in exchanges worldwide. One-factor stochastic volatility models can generate “smile," leverage effects, and term structure effects which cannot be explained by the Black-Scholes model [1, 2]. Consequently many papers [3–9] evaluate barrier options under one-factor stochastic volatility models.

However, one-factor stochastic volatility models including the Heston model [10] present the poor performance when fitting the stiff volatility skews [11]. One extension by using multiple stochastic volatility factors has been presented in some literatures [2, 12, 13]. Christoffersen et al. [2] confirm that the double Heston model significantly improves the flexibility of the one-factor stochastic volatility model in capturing the volatility term structure. In addition, stochastic interest rate is crucial for option pricing because it ensures proper discounting of future payoffs. In recent literatures [14–17], Hull-White stochastic interest rate which is analytically tractable has been incorporated into one-factor stochastic volatility model for pricing path-dependent options. Therefore, the model which incorporates multifactor stochastic volatility and stochastic interest rate may be more reasonable for pricing barrier options.

Barrier options with less stochastic factors can be efficiently evaluated by partial differential equation (PDE) methods [3, 4, 18, 19]. However, the evaluation of barrier options with multiple stochastic factors is to solve a high-dimensional PDE which makes PDE methods quite complex and potentially prone to accuracy and stability problems [1]. A more efficient method for pricing barrier options in this case is the Monte Carlo method; see [20, 21]. The main purpose of this paper is to provide a Monte Carlo method for pricing barrier options under a two-factor stochastic volatility and stochastic interest rate model.

The main contributions of this paper are threefold. Firstly, this paper extends the double Heston model to stochastic interest rate. Secondly, this paper provides a new simulation scheme for the extended model. Thirdly, the paper develops an efficient Monte Carlo algorithm for pricing barrier options. The rest of the paper is organized as follows. Section 2 presents the extended model. Section 3 details the simulation scheme for the proposed model. Section 4 develops the simulation algorithm for pricing barrier options. Section 5 provides numerical experiments. Section 6 concludes.

#### 2. The Pricing Model

Let be a complete probability space, where is a risk-neutral bank account measure. Assume that , (), and are all standard Brownian motions which are -adapted. The double Heston model proposed by Christoffersen et al. [2] is defined by the following stochastic partial differential equations: where is constant interest rate, and () are correlated with parameter . , , are the mean reversion speed, long-term volatility level, and volatility of process (), respectively. are supposed to make the processes remain strictly positive. Conditional on , obey times a noncentral distribution with degree of freedom and noncentrality parameter ; that is, where , , and .

We consider the Hull-White stochastic interest rate [22] which is driven by the following mean-reverting process: where positive constants , are the mean reversion speed and volatility of process , respectively. is used to fit the initial term structure of process . Following Brigo and Mercurio [23], the price of a zero-coupon bond maturing at time can be formulated as follows: where

We replace constant in (1) with Hull-White stochastic interest rate (3) and define the double Heston Hull-White (DHHW) model by a four-dimensional system of stochastic differential equations: Suppose , , , and . Assume that any two random processes are uncorrelated with each other except , .

#### 3. Simulation Scheme for the DHHW Model

To reduce the dimension of the Monte Carlo simulation, we change from the measure to the -forward measure by using as numeraire. Set By the Itô formula, we rewrite (6) as follows:

##### 3.1. Variance Simulation

Based on the fact that a noncentral distribution with high noncentrality parameter can be well approximated by a normal distribution, we use quadratic exponential scheme [24] for discrete variance process .

By (2) and simple calculation, we have Set . Provided that and , we approximate (the segment of high value) by where , , and are independent standard normal random variables.

Provided that and , we approximate (the segment of low value) bywhere are independent uniform random numbers.

##### 3.2. Asset Price Simulation

Under the -forward measure , by integrating (11) and applying the Cholesky decomposition, we rewrite asset price process (6) as follows: where are Brownian motions independent of and .

By the drift interpolation method [25], we approximate the integral of the variance process () by

The integral () obeyed normal distribution with mean zero and variance . By (16), we approximate by where are independent standard normal random variables.

By direct calculation, we have Accordingly, we have the integral obeying normal distribution with mean zero and variance ; that is, where is a standard normal random variable.

Substituting (16)–(20) into (15), we have the simulation scheme for the asset price process as follows: where

#### 4. Simulation Algorithm for Pricing Barrier Options

Under the -forward measure , we evaluate barrier options by the following formula: where is a payoff function, is barrier level, and is the first time when barrier is hit. For a down and out call (DOC) option is defined as follows: is given by where is exercise price at maturity time .

##### 4.1. The Basic Algorithm

Based on the simulation scheme for the DHHW model, we evaluate a DOC option by Algorithm 1.