On Option Pricing in Illiquid Markets with Jumps

One of the shortcomings of the Black-Scholes model on option pricing is the assumption that trading of the underlying asset does not affect the price of that asset. This assumption can be fulfilled only in perfectly liquid markets. Since most markets are illiquid, this assumption might be too restrictive. Thus, taking into account the price impact on option pricing is an important issue. This issue has been dealt with, to some extent, for illiquid markets by assuming a continuous process, mainly based on the Brownian motion. However, the recent financial crisis and its effects on the global stock markets have propagated the urgent need for more realistic models where the stochastic process describing the price trajectories involves random jumps. Nonetheless, works related to markets with jumps are scant compared to the continuous ones. In addition, those previous studies do not deal with illiquid markets. The contribution of this paper is to tackle the pricing problem for options in illiquid markets with jumps as well as the hedging strategy within this context, which is the first of its kind to the authors’ best knowledge. 1. Introduction Financial derivatives are important tools for dealing with financial risks. An option is an example of such derivatives, which gives the right, but not the obligation, to engage in a future transaction on some underlying financial asset. For instance, a European call option on an asset with the price is a contract between two agents (buyer and seller), which gives the holder the right to buy the asset at a pre-specified future time (the expiration date) for an amount (called the strike). The buyer of the option is not obliged to exercise the option. When the contract is, issued the buyer of the option needs to pay a certain amount of money called the premium. The payoff for this option is defined as . The writer of the option receives a premium that is invested in the combination of the risky and risk-free assets. The pricing problem is then to determine the premium, that is, the price that the seller should charge for this option. The pricing problem has been solved in the pioneer work of Black and Scholes [1]. One of the shortcomings of the Black-Scholes model is the assumption that an option trader cannot affect the underlying asset price. However, it is well known that, in a market with imperfect liquidity, trading does affect the underlying asset price (see, e.g., Chan and Lakonishok [2], Keim and Madhavan [3], and Sharpe et al. [4]). In Liu and Yong [5], the authors study the effect of the replication of a