Option pricing under jump-diffusion models with mean-reverting bivariate jumps

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Abstract

We propose a jump-diffusion model where the bivariate jumps are serially correlated with a mean-reverting structure. Mathematical analysis of the jump accumulation process is given, and the European call option price is derived in analytical form. The model and analysis are further extended to allow for more general jump sizes. Numerical examples are provided to investigate the effects of mean-reversion in jumps on the risk-neutral return distributions, option prices, hedging parameters, and implied volatility smiles.

Introduction

In options pricing, the jump-diffusion model is a popular extension from the diffusion model for the underlying stock price process. Earlier work can be dated back to Merton’s classical model  [7], and there have been a few variations proposed more recently, such as  [1], [5], [6]. In these models, the risk-neutral dynamics of the stock price St is assumed to followdStSt=(rqλk)dt+σdWt+dJt, where r,q are risk-free rate and dividend yield, Wt is a Brownian motion, Jt=j=1N(t)(Yj1) is a compound Poisson process with i.i.d. random jump ratio Yj and k=E[Yj]1 is the mean jump size. The Poisson process N(t) with jump arrival rate λ records the number of jumps happening in the time interval [0,t]. The stock price at time t can also expressed as St=S0e(rqσ22λk)t+σWt+Xt, where Xt=j=1N(t)lnYj is another compound Poisson process (as opposed to Jt). The variations in the above-mentioned studies differ in their assumptions on the distribution of lnYj. For example, in Merton’s model  [7], lnYj follows a normal distribution. In the model considered by Amin  [1] and Gukhal  [5], lnYj follows a bivariate distribution meaning that it is either +δ or δ. Kou  [6] proposed to assume that lnYj follows a double exponential distribution to reflect its leptokurtic feature. In these main stream jump-diffusion models, the jump process Jt or Xt is time homogeneous and has no serial correlation because both the underlying Poisson process and the jump size distribution are time homogeneous (with a fixed arrival rate λ and a fixed distribution in lnYj).

In view of this lack, this paper intends to consider an extension of the bivariate jump-diffusion (BJD) model of  [1], [5] such that the jump process is serially correlated. The serial correlation is introduced by assuming that it is less likely for the process to see the next jump going toward the same direction as the previous jump. If a positive (negative) jump has happened, then the probability of seeing a further positive (negative) jump will become smaller. This is motivated by the empirical observations that markets tend to overreact to unexpected events and the effect from a jump-causing event tends to become weaker as time goes by (see the empirical studies such as  [3], [2], [4]). This makes the probability of seeing a sequence of positive (negative) jumps smaller than the corresponding probability in a time homogeneous model where the successive jumps follow i.i.d. distributions.

To this end, we construct a model termed mean-reverting bivariate jump-diffusion (MR-BJD) model which nests the BJD model as a special case. The mean-reverting property can be seen in the jump accumulation process Xt in that it tends to be pulled back to its mean value when it moves away. We investigate the mathematical properties of the jump accumulation process in the MR-BJD model, and link it to the discrete OU process  [8] so that we may use the results of the latter process to conduct our analysis.

The analysis of the jump accumulation process Xt yields some analytical results that are useful in the subsequent derivations of the option pricing formulas for the proposed MR-BJD model. These formulas are obtained in analytical form which makes their calculations a simple task. To incorporate more general jump size distribution, we consider an extended version of the MR-BJD model, termed the MR-BNJD model, and provide its full analysis. A series of numerical examples are provided to examine the effects from the mean-reverting property on the risk-neutral distributions, option prices and hedging parameters. In addition, our numerical examples also demonstrate how the mean-reversion in jumps affects the shape of volatility smiles.

Section snippets

Modeling mean-reverting bivariate jumps

In this section we generalize the bivariate jump-diffusion (BJD) model such that the jumps are serially correlated with a mean-reverting structure. In the original BJD model as proposed in  [1], [5], the successive jump sizes are assumed to follow a bivariate distribution as belowlnYj={+δ,with probability  p,δ,with probability  1p. The mean jump size (mean return caused by a jump) is k=E[Y]1=peδ+(1p)eδ1. The compound Poisson process Xt=j=1N(t)lnYj measures the accumulated contribution

Option pricing formula under the MR-BJD model

In this section we derive the analytical formula for the European call option price under the proposed MR-BJD model. Let t denote the maturity time of the call option. Generalized from (2), the stock price under the risk-neutral measure is assumed to followSt=S0e(rqσ22)tη+σWt+Xt where Xt=j=1N(t)lnYj is the jump accumulation process as defined in Section  2. The constant η in (12) is introduced for this MR-BJD model to be free of arbitrage, i.e.  η is chosen such that the martingale

An extension to the MR-BNJD Model

The bivariate distribution of jump size is somewhat limited because it only takes two values. In this section we extend the mean-reverting jumps to a more general jump size distribution. Based on (4), we assume lnYj follows a bivariate normal distribution, i.e. If  lnYjN(±δ,ν2), then  lnYj+1{N(+δ,ν2),with probability  pξ;N(δ,ν2),with probability  1p±ξ. Here N(a,b) stands for a normal distribution with mean a and variance b. A jump-diffusion model with the above jump size structure is

Numerical examples

This section provides numerical examples to investigate the effects from the mean-reverting jumps on the return distributions, option prices, and volatility smiles. The results are plotted under different values of parameter ξ including the special case ξ=0 which corresponds to the original BJD.

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