On Option Pricing in Illiquid Markets with Jumps

Abstract

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 (𝑆_𝑡)_{𝑡∈[0,𝑇]}⁻ 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 𝑇 (the expiration date) for an amount 𝐾 (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 ℎ(𝑆_𝑇)=Max(𝑆_𝑇−𝐾,0)=(𝑆_𝑇−𝐾)⁺. 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 European option on the underlying asset price. They obtain a generalization of the Black-Scholes pricing partial differential equation (PDE) as the following: 𝜕𝑣()+𝜎𝜕𝑡𝑆,𝑡²𝑆²21−𝜆(𝑆,𝑡)𝑆𝜕²𝑣/𝜕𝑆²(𝑆,𝑡)²𝜕²𝑣𝜕𝑆²(𝑆,𝑡)+𝑟𝜕𝑣()𝜕𝑆𝑆,𝑡−𝑟_𝑡𝑣()()∈][×]],𝑣()(𝑆)𝑆,𝑡=0,for𝑆,𝑡0,+∞0,𝑇𝑆,𝑇=𝑓,0<𝑆<∞, (1)(2) where 𝜆(𝑆,𝑡) is the price impact function of the trader. The classical Black-Scholes PDE is a special case of (2) when 𝜆(𝑆,𝑡)=0.

There are also several other papers that have studied the financial markets with jumps (among others are Merton [6], Dritschel and Protter [7], El-Khatib and Privault [8], and El-Khatib and Al-Mdallal [9]). However, none of the previous studies based on the jump-diffusion approach deals with illiquid markets, to the authors’ best knowledge. This paper extends the model of Liu and Yong [5] by including a jump-diffusion structure in the underlying option pricing model. This appears to be an important issue because the model that is suggested in this paper allows for the possibility to account for sudden and random significant changes in the market that might not be captured by the existing models in the literature such as the continuous model suggested by Liu and Yong [5]. Hence, the approach that is developed in this paper is expected to be more useful in financial risk management, especially in the cases in which the financial markets are under stress.

The disposition of the rest of the paper is the following. Section 2 introduces the jump-diffusion model for an illiquid market. Section 3 deals with the pricing problem of an option within the context of a jump-diffusion model along with the proof for the suggested solution. Section 4 concludes the paper.

2. A Jump-Diffusion Model for Illiquid Markets

We start with presenting some necessary denotations. Let (𝑁_𝑡)_{𝑡∈[0,𝑇]} be a Poisson process with deterministic intensity 𝜌. Let also 𝑀_𝑡=𝑁_𝑡−𝜌𝑡 be its associated compensated process. The process (𝑊_𝑡)_{𝑡∈[0,𝑇]} denotes a Brownian motion. The probability space of interest is (Ω,ℱ,𝑃) with (𝑀_𝑡)_{𝑡∈[0,𝑇]} and (𝑊_𝑡)_{𝑡∈[0,𝑇]} being independent. Let (ℱ_𝑡)_{𝑡∈[0,𝑇]} signify the filtration generated by (𝑁_𝑡)_{𝑡∈[0,𝑇]} and (𝑊_𝑡)_{𝑡∈[0,𝑇]}. The market is assumed to have two assets: a risky asset (𝑆_𝑡)_{𝑡∈[0,𝑇]} and a risk-free asset denoted by (𝐴_𝑡)_{𝑡∈[0,𝑇]}. The maturity is 𝑇, the strike is 𝐾, and the payoff is ℎ(𝑆_𝑇)=(𝑆_𝑇−𝐾)⁺≡Max{𝑆_𝑇−𝐾,0}. As in Liu and Yong [5], the return on the risk-free asset indirectly depends on 𝑆_𝑡, and the option trader’s trading in the stock market has a direct impact on the stock price. This price impact, which an investor can cause by trading on an asset, functions in such way that it increases the price when buying the asset and it decreases the price when selling the asset. The price of the risk-free asset is given by 𝑑𝐴_𝑡=𝑟𝑡,𝑆_𝑡𝐴_𝑡𝑑𝑡,𝑡∈[0,𝑇], (3) where 𝑟>0 denotes the interest rate. The price of the risky asset is generated by the following stochastic differential equation: 𝑑𝑆_𝑡𝑆_𝑡=𝜇𝑡,𝑆_𝑡𝑑𝑡+𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝑎𝑑𝑀_𝑡+𝜆𝑡,𝑆_𝑡𝑑𝜃_𝑡,𝑡∈[0,𝑇],𝑆₀=𝑥>0, (4) where 𝜇 and 𝜎 represent the expected return and volatility, respectively, the term 𝑎 is a real constant, and 𝜆(𝑆,𝑡) denotes the price impact factor created by the trader via selling or buying the underlying asset. 𝜃_𝑡 is the number of shares that the trader has in the stock at time 𝑡. Hence, 𝜆(𝑆,𝑡)𝑑𝜃_𝑡 captures the price impact of trading. Before dealing with the pricing of a European option in a jump-diffusion illiquid market, we need to observe the following remark.

Remark 1. The parameter 𝑎 in (4) determines the direction of the jumps (it also affects the jumps size). In fact, the following can be stated. (i)If 𝑎<0, then the jumps are pushing the stock price down; that is, the stock price is decreasing at each jump. (ii)If 𝑎=0, then there are no jumps, and therefore model (4) is reduced to the model in Liu and Yong [5]. (iii)If 𝑎>0, then the jumps are pushing the stock up; that is, the stock price is increasing at each jump.

3. Pricing of a European Option in Jump-Diffusion Illiquid Markets

Let (𝑉_𝑡)_{𝑡∈[0,𝑇]} be the wealth process for the trader. Let also (𝜓_𝑡)_{𝑡∈[0,𝑇]} denote the number of units invested in the risk-free asset. Then, the value of the portfolio is given by 𝑉_𝑡=𝜓_𝑡𝐴_𝑡+𝜃_𝑡𝑆_𝑡,𝑡∈[0,𝑇]. (5) Assume that the number of shares of the risky asset satisfies the following condition: 𝑑𝜃_𝑡=𝜂_𝑡𝑑𝑡+𝜁_𝑡𝑑𝑊_𝑡+𝑏𝑑𝑀_𝑡,𝑡∈[0,𝑇]. (6) Let us consider a European call option with the payoff defined as ℎ(𝑆_𝑇)∶=(𝑆_𝑇−𝐾)⁺. In order to replicate the option for a perfect hedge, we search for a strategy (𝜓_𝑡,𝜃_𝑡)_{𝑡∈[0,𝑇]} which, at the expiration date of the option, leads to having a value of the underlying wealth to be equal to the payoff; that is, 𝑉_𝑇=ℎ(𝑆_𝑇). Then we can state the following Proposition.

Proposition 2. The wealth process for the trader of the jump-diffusion model (4) satisfies the following stochastic differential equation: 𝑑𝑉_𝑡=𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝜃_𝑡𝑆_𝑡𝑑𝑡+𝜃_𝑡𝑆_𝑡𝜆𝑡,𝑆_𝑡𝜁_𝑡+𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝜃_𝑡𝑆_𝑡𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑑𝑀_𝑡. (7)

Proof. By using (3), (4), (5), and (6), we have the following: 𝑑𝑉_𝑡=𝜓_𝑡𝑑𝐴_𝑡+𝜃_𝑡𝑑𝑆_𝑡=𝑉_𝑡−𝜃_𝑡𝑆_𝑡𝐴_𝑡𝑑𝐴_𝑡+𝜃_𝑡𝑆_𝑡×𝜇𝑡,𝑆_𝑡𝑑𝑡+𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝑎𝑑𝑀_𝑡+𝜆𝑡,𝑆_𝑡𝑑𝜃_𝑡=𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡𝜃_𝑡𝑆_𝑡𝑑𝑡+𝜃_𝑡𝑆_𝑡𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝑎𝑑𝑀_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝑑𝑡+𝜁_𝑡𝑑𝑊_𝑡+𝑏𝑑𝑀_𝑡=𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝜃_𝑡𝑆_𝑡𝑑𝑡+𝜃_𝑡𝑆_𝑡𝜆𝑡,𝑆_𝑡𝜁_𝑡+𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝜃_𝑡𝑆_𝑡𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑑𝑀_𝑡, (8) which ends the proof.

Our aim in this paper is to price the European option with payoff ℎ(𝑆_𝑇) where 𝑆_𝑇 is given by (4). We replicate the European option by searching a wealth (𝑉_𝑡)_{𝑡∈[0,𝑇]} which leads to the terminal value 𝑉_𝑇=ℎ(𝑆_𝑇). Thus, as in Liu and Yong [5], we need to solve the following system of stochastic differential equations: 𝑑𝜃_𝑡=𝜂_𝑡𝑑𝑡+𝜁_𝑡𝑑𝑊_𝑡+𝑏𝑑𝑀_𝑡,𝑑𝑆_𝑡𝑆_𝑡=𝜇𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡+𝜎𝑑𝑡𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑑𝑊_𝑡+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑑𝑀_𝑡,𝑑𝑉_𝑡=𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝜃_𝑡𝑆_𝑡𝑑𝑡+𝜃_𝑡𝑆_𝑡𝜆𝑡,𝑆_𝑡𝜁_𝑡+𝜎𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝜃_𝑡𝑆_𝑡𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑑𝑀_𝑡,𝜃₀>0,𝑆₀>0,𝑉_𝑇=ℎ𝑆_𝑇. (9) The above system is called FBSDEs (forward-backward stochastic differential equations) system. In order to derive the PDE for the European option price, we need the Itô formula which is given by the following lemma (see Protter [10]).

Lemma 3. Let 𝑔, 𝑙, and 𝑘 be three adapted processes such that ^𝑡₀𝑔_𝑠𝑑𝑠<∞,_𝑡0𝑙_𝑠²𝑑𝑠<∞,_𝑡0𝜌𝑘_𝑠𝑑𝑠<∞. (10) Let 𝑋=(𝑋_𝑡)_{𝑡∈[0,𝑇]} be the process defined by 𝑑𝑋_𝑡=𝑔_𝑡𝑑𝑡+𝑙_𝑡𝑑𝑊_𝑡+𝑘_𝑡𝑑𝑀_𝑡. (11) For any function 𝐺∈𝒞^1,2([0,𝑇]×]−∞,∞[), one has 𝐺𝑡,𝑋_𝑡=𝐺0,𝑋₀+^𝑡₀𝜕_𝑠𝐺𝑠,𝑋_𝑠+𝑔_𝑠−𝑘_𝑠𝜌𝜕_𝑥𝐺𝑠,𝑋_𝑠−+12𝑙²_𝑠𝜕²_𝑥𝑥𝐺𝑠,𝑋_𝑠−+𝑑𝑠_𝑡0𝑙_𝑠𝜕_𝑥𝐺𝑠,𝑋_𝑠−𝑑𝑊_𝑠+_𝑠≤𝑡𝐺𝑠,𝑋_𝑠−𝐺𝑠,𝑋_𝑠−. (12)

Equation (12) can be written in the following format:𝐺𝑡,𝑋_𝑡=𝐺0,𝑋₀+^𝑡₀𝜕_𝑠𝐺𝑠,𝑋_𝑠+𝑔_𝑠−𝑘_𝑠𝜌𝜕_𝑥𝐺𝑠,𝑋_𝑠−+12𝑙²_𝑠𝜕²_𝑥𝑥𝐺𝑠,𝑋_𝑠−+𝜌𝐺𝑠,𝑋_𝑠−+𝑘_𝑠−𝐺𝑠,𝑋_𝑠−+𝑑𝑠_𝑡0𝐺𝑠,𝑋_𝑠−+𝑘_𝑠−𝐺𝑠,𝑋_𝑠−𝑑𝑀_𝑠+^𝑡₀𝑙_𝑠𝜕_𝑥𝐺𝑠,𝑋_𝑠−𝑑𝑊_𝑠. (13) The following proposition provides the PDE for the price of the European option in the jump-diffusion illiquid market presented in Section 2.

Proposition 4. Let 𝑓(𝑡,𝑆_𝑡) denote the price of the European option at time 𝑡∈[0,𝑇] for the model presented in Section 2. Then the corresponding P.D.E. for the underlying option price is given by 𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝜃_𝑡𝑆_𝑡=𝜕_𝑡𝑓𝑡,𝑆_𝑡+𝜇𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡−𝜌𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑆_𝑡𝜕_𝑆𝑓𝑡,𝑆_𝑡+12𝜎𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡²𝑆²_𝑡𝜕²_𝑆𝑆𝑓𝑡,𝑆_𝑡+𝜌𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓𝑡,𝑆_𝑡−, (14) with the terminal condition 𝑓(𝑇,𝑆_𝑇)=ℎ(𝑆_𝑇). Moreover, the market is incomplete, and there is no strategy leading to the terminal wealth 𝑉_𝑇=ℎ(𝑆_𝑇)∶=𝑓(𝑇,𝑆_𝑇). However, the number of shares 𝜃 that minimizes the variance is given by 𝜃_𝑡=(𝜎+𝜆𝜁)²𝑆²𝜕_𝑆𝑓(𝜎+𝜆𝜁)²𝑆²+𝜌𝑆²(𝑎𝜎+𝑏𝜆𝜁)²+𝜌𝑆(𝑎𝜎+𝑏𝜆𝜁)𝑓𝑡,𝑆_𝑡−(1+𝑎𝜎+𝑏𝜆𝜁)−𝑓(𝜎+𝜆𝜁)²𝑆²+𝜌𝑆²(𝑎𝜎+𝑏𝜆𝜁)². (15)

Proof. Let (𝜃,𝑆,𝑉) be an adapted solution of the FBSDE (9), and assume that there exists a smooth function 𝑓∈𝒞^3,1(]−∞,∞[×[0,𝑇]) such that 𝑓(𝑡,𝑆_𝑡) represents the price of the European option at time 𝑡∈[0,𝑇]. Since the price of the option at maturity is equal to the payoff, then 𝑓(𝑇,S_𝑇)=ℎ(𝑆_𝑇). Now, using the Itô formula (13), we obtain 𝑑𝑓𝑡,𝑆_𝑡=𝜇𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡−𝜌𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑆_𝑡𝜕_𝑆𝑓𝑡,𝑆_𝑡+12𝜎𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡²×𝑆²_𝑡𝜕²_𝑆𝑆𝑓𝑡,𝑆_𝑡+𝜕_𝑡𝑓𝑡,𝑆_𝑡+𝜌𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,S_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓𝑡,𝑆_𝑡−+𝜎𝑑𝑡𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑆_𝑡𝜕_𝑆𝑓𝑡,𝑆_𝑡𝑑𝑊_𝑡+𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓𝑡,𝑆_𝑡−𝑑𝑀_𝑡. (16) By comparing (7) and (16), one can deduce that it is impossible to find a strategy (𝜂_𝑡,𝜁_𝑡)_{𝑡∈[0,𝑇]} that results in the terminal wealth 𝑉_𝑇=ℎ(𝑆_𝑇)∶=𝑓(𝑇,𝑆_𝑇). Thus, we put the term belonging to 𝑑𝑡 equations (7) and (16) equal to each other, which gives the PDE of the option price, and then we minimize the distance between the wealth 𝑉_𝑇 and the price 𝑓(𝑇,𝑆_𝑇)=ℎ(𝑆_𝑇) over the number of shares of the underlying asset, that is, 𝜃_𝑡. The PDE of the option price in this case is 𝑟𝑡,𝑆_𝑡𝑉_𝑡+𝜇𝑡,𝑆_𝑡−𝑟𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡𝜃_𝑡𝑆_𝑡=𝜕_𝑡𝑓𝑡,𝑆_𝑡+𝜇𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜂_𝑡−𝜌𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡𝑆_𝑡𝜕_𝑆𝑓𝑡,𝑆_𝑡+12𝜎𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡²𝑆²_𝑡𝜕²_𝑆𝑆𝑓𝑡,𝑆_𝑡+𝜌𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓𝑡,𝑆_𝑡−, (17) with the terminal condition 𝑓𝑇,𝑆_𝑇=ℎS_𝑇. (18) To find the number of shares 𝜃_𝑡 invested in 𝑆_𝑡, we need to solve the following problem: Minimize_𝜃𝐸Π²(𝜃), (19) where Π(𝜃)∶=(ℎ(𝑆_𝑇)−𝑉_𝑇). By using (7), (16), and (17), we have 𝐸Π²(𝜃)=𝐸^𝑇₀𝜎𝑡,𝑆_𝑡+𝜆𝑡,𝑆_𝑡𝜁_𝑡×𝑆_𝑡𝜕_𝑆𝑓𝑡,𝑆_𝑡−𝜃_𝑡𝑑𝑊_𝑡²+𝐸^𝑇₀𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓𝑡,𝑆_𝑡−−𝜃_𝑡𝑆_𝑡𝑎𝜎(𝑡,𝑆_𝑡)+𝑏𝜆(𝑡,𝑆_𝑡)𝜁_𝑡𝑑𝑀_𝑡²=𝐸^𝑇₀[𝜎(𝑡,𝑆_𝑡)+𝜆(𝑡,𝑆_𝑡)𝜁_𝑡]𝑆_𝑡(𝜕_𝑆𝑓(𝑡,𝑆_𝑡)−𝜃_𝑡)²𝑑𝑡+𝐸^𝑇₀𝜌𝑓𝑡,𝑆_𝑡−1+𝑎𝜎𝑡,𝑆_𝑡+𝑏𝜆𝑡,𝑆_𝑡𝜁_𝑡−𝑓(𝑡,𝑆_𝑡−)−𝜃_𝑡𝑆_𝑡[𝑎𝜎(𝑡,𝑆_𝑡)+𝑏𝜆(𝑡,𝑆_𝑡)𝜁_𝑡]²𝑑𝑡=𝐸^𝑇₀𝑙𝜃_𝑡𝑑𝑡, (20) where 𝑙(𝑥)=(𝜎+𝜆𝜁)²𝑆²𝜕_𝑆𝑓−𝑥²+𝜌𝑓𝑡,𝑆_𝑡−(1+𝑎𝜎+𝑏𝜆𝜁)−𝑓−𝑥𝑆(𝑎𝜎+𝑏𝜆𝜁)². (21) The minimum is obtained at 𝑙^′(𝑥)=0, which yields the following result: 2(𝜎+𝜆𝜁)²𝑆²𝜕_𝑆()𝜌𝑓𝑓−𝑥−2𝑆𝑎𝜎+𝑏𝜆𝜁𝑡,𝑆_𝑡−(𝜃1+𝑎𝜎+𝑏𝜆𝜁)−𝑓−𝑥𝑆𝑎𝜎+𝑏𝜆𝜁=0,_𝑡=(𝜎+𝜆𝜁)²𝑆²𝜕_𝑆𝑓(𝜎+𝜆𝜁)²𝑆²+𝜌𝑆²(𝑎𝜎+𝑏𝜆𝜁)²+𝜌𝑆(𝑎𝜎+𝑏𝜆𝜁)𝑓𝑡,𝑆_𝑡−(1+𝑎𝜎+𝑏𝜆𝜁)−𝑓(𝜎+𝜆𝜁)²𝑆²+𝜌𝑆²(𝑎𝜎+𝑏𝜆𝜁)², (22) which ends the proof.

It is worth mentioning that, in the case where there are no jumps, that is, when 𝑎=𝑏=0, 𝜃=𝜕_𝑆𝑓, and the PDE in the previous proposition is reduced to the PDE that is obtained in Liu and Yong [5], assuming that there are no dividends.

4. Conclusions

Option pricing is an integral part of modern risk management in increasingly globalized financial markets. The classical Black-Scholes model is regularly used for this purpose. However, one of the main pillars that makes this model operational is the underlying assumption that the markets are perfectly liquid. This assumption is, nonetheless, not fulfilled in reality since perfectly liquid markets do not exist. In our opinion, the question should not be whether the markets are illiquid or not; the question should be about the degree of illiquidity. Thus, taking into account the fact that markets are illiquid can improve the precision of the underlying option pricing.

This paper is the first attempt, to our best knowledge, that extends the existing literature on option pricing by introducing a jump-diffusion model for illiquid markets. This seems to be a more realistic approach to deal with a market that is incomplete. A solution for the option pricing within this context is provided along with the underlying proof. The suggested solution might be useful to investors in order to determine the optimal value of an option in a market that is characterized by illiquidity.