Option Pricing Model with Transaction Costs and Jumps in Illiquid Markets

Option pricing model is a wildly interested topic in an area of financial Mathematics. The pioneer model was introduced by Fischer Black and Myron Scholes which is known as the Black-Scholes model. This model was derived under various assumptions such as liquidity and no transaction costs for which a underlying asset price in stock market might not be satisfied. With this fact, the underlying asset price models were remodeled, in order to determine an option value. This research aims to extend the Black-Scholes model by relaxing the assumption of no transaction costs in illiquid markets. Also, jumps of asset price are considered in this work. To do this, a differential form of asset price with transaction costs and jumps in illiquid markets is introduced and then used to construct the extended option pricing model. Furthermore, a numerical result of a call option price under a new situation is provided.


Introduction
Derivatives are financial instruments that give the right to buy or sell an underlying asset in the future. These contacts, such as future, forward, swap and option, were used for speculating and managing risks in an investment. For an option, it is a financial contract that gives option holders the right to buy or sell an underlying asset from option writers by a specified date and price. The contract, giving option holders the right to buy an underlying asset, is called a call option while the contract that gives option holders the right to sell an underlying asset is called a put option.
In 1973, Fischer Black and Myron Scholes [1] constructed the Black-Scholes model for determining prices of options. However, their model required various assumptions such as constant volatility, no transaction costs and perfect liquidity. However, purchasing on some stocks may be illiquid. Also, random jumps of prices of some underlying assets have occurred. With these reasons, the Black-Scholes assumptions may not fulfill the real financial market situations. Therefore, many researchers tried to extend the result of Black and Scholes by reducing some of the above assumptions (see, [2] [3] and [4]).
In illiquid market, the investor's trading in the stock market affects the stock price. This impact is called a price impact. The price impact is referred to the correlation between trading and subsequent price change. This may be a result from a bid trader who is able to move the price by his/her actions.
In 2005, Hong Liu and Jiongmin Yong [5] examined the effects of price impact in an illiquid market in replicating a European option. They investigated a generalized Black-Scholes pricing model in illiquid market. Moreover, the presence of the price impact has been studied and analysed in several researches. For example, Kristoffer Glover, Peter Duck, and David Newton [6] consider the effects of illiquidity on the Black-Scholes model. Traian Pirvu and Ahmadreza Yazdanian [7] investigated the effects of price impact in imperfect liquidity on the replication of a European Spread option. In this paper, we combined the idea of [8] and [9] by introducing a differential form of an asset price process related to transaction costs and jump diffusion term in an illiquid market. Also, we provide a European option pricing model with transaction costs and jumps in illiquid markets. This model extends results in [1] [5] [8] and [9] by reducing more assumptions. Moreover, numerical simulations of an option price are shown by using the Monte Carlo simulation.
The contents of this research are organized into four sections. In Section 2, the differential form of assets price with transaction costs and jumps in illiquid markets is introduced. Also, an asset price process is investigated and a simulation example of option price is given in this section. In Section 3, a model of option pricing associated to the propose differential form is provided. Finally, con-cluding remarks are given in Section 4.

Differential Form of Assets Price with Transaction Costs and Jumps in Illiquid Market
In this section, we introduce the differential form of assets price with transactions costs and jumps for illiquid market. We consider a financial market having two types of assets; a risk-free asset and a risky asset. For 0 t > , let t A and t S be risk-free asset and risky asset prices at time t, respectively, T t − be the time to maturity date, K be strike price and where r is the risk-free interest rate and assumed that the price of risky asset sa- where µ and σ are the constant drift and constant volatility, respectively, t W is a standard one-dimensional Brownian motion. In 2005, Hong Liu and Jiongmin Yong [5] extended the result of Black and Scholes [1]. They derived a generalized Black-Scholes pricing model in illiquid market. In [5], Hong Liu and Jiongmin Yong assumed that the price of risk-free asset and the price of risky asset follow where ( ) , t r t S is the interest rate, the drift and the volatility, respectively, depending on time t and t S , ( ) , t t S λ is price impact function of the trader (non-negative) and t θ is the number of shares. They also assumed that where t η and t ζ are adapted process to a filtration ( ) 0 t t >  generated by the Brownian motion. After that, in 2013, Youssef El-Khatib and Abdulnasser Hatemi-J [8] applied a jump diffusion model to price process in Liu and Yong model [5]. The price of risky asset is assumed as where a is a real constant and − is the compensated Poisson process where t N is a Poisson process with deterministic intensity ρ . They assumed further that where b is a real constant. In 2016, Francis Agana and et al. [9] added the term of transaction costs to price process in Liu and Yong model [5]. They assumed that the price of risky asset satisfies is the transaction costs. In this work, we combined the idea of [8] and [9] to construct a model of option pricing. We assume that the price of the risky asset is generated by the following stochastic differential equation: and t θ satisfies Thus, by (9) and (10), the price process of the risky asset satisfies the following differential form: In solving Equation (11), we apply the Ito lemma in [8] with We obtain Next, a Monte Carlo simulation for a call option price is presented. This computation is obtained as a special case when the coefficients , , , , , , a b µ λ κ η ζ and ρ are constants. By Equation (14)     However, the average of the asset price lies between 39 and 41. Although the simulated price at expiration date is less than the strike price which is 44, the simulated option price is still positive due to the definition of pay-off. This can be explained that if an investor plans to buy and then hold an option, which an underlying asset price follows Equation (9), to its expiration date, the option fair price at current time is positive because of positive jumps of the underlying asset price.

Option Pricing Model with Transaction Costs and Jumps in Illiquid Markets
In this section, we construct a partial differential equation for option pricing by using a arbitrage pricing technique [10] consisting of the following steps: 1) Constructing a self-financing portfolio with the risk-free asset and the risky asset.
2) Providing a differential form of option price by applying Ito's lemma to option price function depending on t and t S .
3) Comparing the coefficients in the above differential form by using arbitrage pricing technique. This is, comparing the coefficients in random and non random parts in the replicating portfolio from step 1.
, let t V is the wealth process and t ψ is the number of units invested in the risk-free asset. The value of the portfolio t V satisfies Assume that the trading portfolio is self-financing. Then, and we have the following Proposition. Proposition 1. If the portfolio is self-financing, then the wealth process in (18) follows the stochastic differential equation: Proof. By (3), (11), (17) and (18), we have Therefore, the proposition is proved. □ The following theorem gives the partial differential equation for option pricing with transaction costs and jumps in illiquid markets. In this theorem, we combine the step 2 and step 3 in arbitrage pricing technique.
, t f t S be the price of the European call option at time and t S satisfies the Equation (11). Then the partial differential equation of the option price is given by with the terminal condition ( ) ( ) Proof. G t X f t S = . We obtain the differential form of option price satisfies the following stochastic differential equation: Thus, we can compare the coefficients for d t W in Equations (19) and (21), we Consider Equations (22) and (23), we can see that the t θ in Equation (22) has no jumping variable but that in Equation (23) (24) is reduced to the equation obtained in [9]. Similarly, If the no transaction cost assumption is assumed (i.e.