A New Binomial Tree Method for European Options under the Jump Diffusion Model

In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model con-verges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.

tions, interest rate derivatives, and path-dependent options. Boyarchenko and Levendorskii have proposed the expected present value (EPV) pricing model in [4] according to the Lévy process, and studied the mean return of stochastic volatility.
The binomial tree method, first proposed by Cox, Ross and Rubinste [5] in 1979, is one of the most popular approaches to price options in diffusion models. Nowadays, the binomial tree pricing method has been well studied in the past few years. For example, Amin [6] first generalized Cox, Ross and Rubinstein's binomial tree method to jump diffusion models for vanilla options. Alfredo Ibonez [7] discussed the algorithm of American put options and obtained the optimal execution boundary by Newton interpolation method. Hou and Zhou [8] analyzed the currently used option pricing binary tree by using random error, the correction method promoted a binary tree parameter model, but it is limited to the analysis of European options and lacks practical examples. Zhang and Yue [9] discussed the no-arbitrage conditions of binary tree option pricing, and obtained a single time period and European call option pricing formula for multi-period market. More relevant studies on the various kinds option pricing and binary tree methods were investigated in [10] [11] [12] [13] and [14] and Liu [15], Song [16] and Lian [17].
In view of the above discussion, although the previous studies have their own characteristics, there are still some shortcomings: 1) The price of the underlying asset is subject to the majority of the Black-Scholes model, which does not fully reflect the characteristics of the market price. 2) Simply consider Binary tree pricing, without combining the binary tree pricing with the analytical pricing of the model. 3) When calculating the binary tree parameters, most literatures default the condition ud = 1 or p = 1/2. In this paper, following the idea of Merton (1976) [2] and Zhang (2000) [18], we try to use the binary tree method to analyze the pricing problem of European options based on a class of jump diffusion model in this paper. The rest of the paper is organized as follows. In Section 2, the jump-diffusion model is introduced and the problem under consideration is formulated. In Section 3, the binomial tree is constructed to design the pricing formula for European options under the jump-diffusion model. In Section 4, a simulation example is given to demonstrate the main results obtained. Finally, we conclude the paper in Section 5.

Model Formulation and Preliminaries
Assume all the work following is performed in a given risk-neutral probability space ( ) , ,P Ω  . Consider following class of stochastic differential equation: where t S denotes the stock price at time t, µ is the expectation yield rate, t W is a standard Brownian motion, ( ) N t is a poisson process with rate λ , and i Y is a sequence of independent identically (iid) nonnegative random variable such that  According to the Itô formula for the stochastic differential equation with jump diffusion, we can get the solution of the Equation (1) as follows: Then, we can furthermore rewrite the Equation (3) as follows: Noting that Taking the mathematical expectation on both sides of Equation (4), we have Because of the independence of ( ) ( ) , we shall introduce following lemma.
be the moment generation function of random variable i Y , then, the moment generation of a compound poisson process is .

Binomial Tree Model
Assume the stock price jumps n times in [ ] Assume that the stock price will move to two new values Su and Sd with probability pand 1 p − . If the initial stock price is 0,0 S at the current time 0 t = . The stock price will become 0,0 S u or 0,0 S d after t ∆ . Thus, there will be 3 values ( ) The proof of (13) can be easily given, we assume at anytime the price of option is C and at the next step. The price of option will become Cu with probability P or Cd with probability 1 -P like Figure 1. So we have By the method of induction, at the any node in a binary tree, the formula of (13) is always correct.
Therefore, the price of European options at the initial time can be given as follows:  To compute the ( ) ,0 C S , we must confirm the parameter p, u and d first. Next, we shall calculate p, u and d by using the moment estimation theory.  However, there is a fault in this method, because when 0 σ → , p may be zero or a negative number which obviously contradicts with the reality. In this paper, we restructure the formula to calculate the parameter of the binomial tree by introducing the third moment: Then, we set up the following equation groups: Thus, we can get the solution of p, u and d as follows:   We also wonder realize the options price changing in maturity [0, T] with stable steps M.
As it shows in Figure 3 and Figure 4, at the same maturity, when binary tree steps M is becoming greater. The difference between analytical solution and our binary tree methods get small at time T. It is consistent with the real finance market and also proves the correction of our methods.

Conclusion
In this paper, we have dealt with the pricing problem for European option based on a class of jump diffusion model. A new binomial tree has been constructed and the corresponding pricing scheme for European option has been proposed. We have adopted the third moment to calculate the parameter of binomial tree. Compared with the analytic solutions, a numerical example has shown that the proposed binomial tree model with jump-diffusion can approximate the Merton model, and can be proved that the established binomial tree method works better than traditional binomial tree. When nodes m tends to be infinite, both of them are the same as analytic solutions. Furthermore, the model can be extended to the pricing of exotic options such as American options, lookback options and butterfly options based on the jump diffusion process.