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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 converges 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.

Since the establishment of Black-Scholes formula for the European call option in 1973, lots of economic models have been reported, such as Heston’s model, jump diffusion model, constant elasticity of variance (CEV) model, Cox-Ingersoll-Ross (CIR) [

The binomial tree method, first proposed by Cox, Ross and Rubinste [

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) [

Assume all the work following is performed in a given risk-neutral probability space ( Ω , F , P ) . Consider following class of stochastic differential equation:

d S t S t = μ d t + σ d W t + d ( ∑ i = 1 N ( t ) ( Y i − 1 ) ) , (1)

where S t denotes the stock price at time t, μ is the expectation yield rate, W t is a standard Brownian motion, N ( t ) is a poisson process with rate λ , and Y i is a sequence of independent identically (iid) nonnegative random variable such that γ = ln ( Y ) has a normal distribution with following density:

f ( y ) = 1 2 π σ y e − ( y − u y ) 2 2 σ y 2 . (2)

According to the Itô formula for the stochastic differential equation with jump diffusion, we can get the solution of the Equation (1) as follows:

S t = S 0 e ( μ − 1 2 σ 2 ) t + σ W ( t ) ∏ i = 1 N ( t ) Y i . (3)

Then, we can furthermore rewrite the Equation (3) as follows:

S t = S 0 e ( μ − 1 2 σ 2 ) + σ W ( t ) + ∑ i = 1 N ( t ) ( ln Y ( i ) ) . (4)

Noting that E ( γ ) = u y , V a r ( γ ) = σ y 2 and J = E ( Y i ) = E ( e γ ) = e u y + 1 2 σ y 2 .

Taking the mathematical expectation on both sides of Equation (4), we have

E ( S t S 0 ) = E ( e ( μ − 1 2 σ 2 ) + σ W ( t ) + ∑ i = 1 N ( t ) ( ln Y ( i ) ) ) . (5)

Because of the independence of W ( t ) , N ( t ) and Y i , one has

E ( S ( t ) S 0 ) = E ( e ( μ − 1 2 σ 2 ) t ) E ( e σ W ( t ) ) E ( e ∑ i = 1 N ( t ) γ i ) , E ( e ( μ − 1 2 σ 2 ) t ) = e ( μ − 1 2 σ 2 ) t , E ( e σ W ( t ) ) = E ( e σ Z t ) = 1 2 π ∫ − ∞ ∞ e σ Z t e − Z 2 2 d Z = e 1 2 σ 2 t , (6)

where Z ∼ N ( 0 , 1 ) , before calculating the mathematical expectation of e ∑ i = 1 N ( t ) γ , we shall introduce following lemma.

Lemma 2.1 If Q ( t ) = ∑ i = 1 N ( t ) Y i and E ( Y i ) = β , then, we have

E ( Q ( t ) ) = ∑ k = 0 ∞ E [ ∑ i = 1 k Y i | N ( t ) = k ] ( P { N ( t ) = k } ) = ∑ k = 0 ∞ β k ( λ t ) k k ! e − λ t = β λ t e − λ t ∑ k = 1 ∞ ( λ t ) k − 1 ( k − 1 ) ! = β λ t . (7)

Let φ Y ( x ) = E ( e x Y i ) be the moment generation function of random variable Y i , then, the moment generation of a compound poisson process is

φ Q ( t ) ( x ) = E ( e x ∑ i = 1 N ( t ) Y i ) = P { N ( t ) = 0 } + ∑ k = 1 ∞ E [ e x ∑ i = 1 k Y i | N ( t ) = k ] P { N ( t ) = k } = P { N ( t ) = 0 } + ∑ k = 1 ∞ E [ e x ∑ i = 1 k Y i ] P { N ( t ) = k } = e − λ t + ∑ k = 1 ∞ E ( e x Y 1 ) E ( e x Y 2 ) ⋯ E ( e x Y k ) ( λ t ) k k ! e − λ t = e λ t ( φ Y ( x ) − 1 ) . (8)

Therefore, we have

E ( e ∑ i = 1 N ( t ) γ ) = e λ t ( φ γ ( x ) − 1 ) , (9)

φ γ ( x ) = E ( e x γ ) = ∫ − ∞ ∞ e x Z 1 2 π σ y e − Z − u y 2 2 σ y 2 d Z = e x u y + x σ y 2 2 . (10)

When x = 1 and φ γ ( x ) = J , we obtain

E ( S ( t ) S ( 0 ) ) = e μ t e λ t ( J − 1 ) = e ( μ + λ J − λ ) t . (11)

Assume the stock price jumps n times in [ 0 , T ] , let P { N ( t ) = k } = e − λ T ( λ T ) k k ! ,

divide the period into m parts. Set Δ t = T / m and t k = k Δ t , ( k = 1 , 2 , ⋯ , m ) . Assume that the stock price will move to two new values S u and S d with probability pand 1 − p . If the initial stock price is S 0 , 0 at the current time t = 0 . The stock price will become S 0 , 0 u or S 0 , 0 d after Δ t . Thus, there will be 3 values S 0 , 0 u 2 , S 0 , 0 u d and S 0 , 0 d 2 . Repeat the above operation and we can achieve the binomial tree shown in

Theorem 3.1 Consider the stock price model as Equation (1), the corresponding pricing formula can be designed as following equation by using the binomial tree method:

C m ( S , 0 ) = ∑ n = 0 ∞ P { N ( t ) = n } C 0 , 0 = e − r f T ∑ n = 0 ∞ e − λ T ( λ T ) n n ! [ S B 1 ( m , k , u , d , p ) − K B 2 ( m , k , p ) ] . (12)

Proof. Let C k , j ( j = 0 , 1 , ⋯ , k ) be the option price at node ( k , j ) after k Δ t , r f is the interest rate. The price can be recursively computed by the backward induction algorithm.

C k , j = e − r f Δ t [ P C k + 1 , j + ( 1 − P ) C k + 1 , j + 1 ] . (13)

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

C = e − r f Δ t [ P C u + ( 1 − P ) C d ]

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:

C 0 , 0 = e − r f T ∑ j = 0 m C m j [ p j ( 1 − p ) m − j max ( S m , j − K , 0 ) ] = e − r f T ∑ j = 0 m C m j [ p j ( 1 − p ) m − j max ( S u j d m − j − K , 0 ) ] = e − r f T ∑ j = k m C m j [ p j ( 1 − p ) m − j ( S u j d m − j − K ) ] = e − r f T ∑ j = k m C m j { S ( p u ) j [ ( 1 − p ) d ] m − j − K p j ( 1 − p ) m − j } = e − r f T [ S B 1 ( m , k , u , d , p ) − K B 2 ( m , k , p ) ] , (14)

where B 1 ( m , k , u , d , p ) = ∑ j = k m C m j ( p u ) j [ ( 1 − p ) d ] m − j , B 2 ( m , k , p ) = ∑ j = k m C m j p j ( 1 − p ) m − j and k = min { j : S u j d m − j − K > 0 , 0 < j < m } . According to the law of the total expectation, the European call option at the initial time should obey the following equation:

C ( S , 0 ) = ∑ n = 0 ∞ P { N ( t ) = n } ( E [ C 0 , 0 | N ( t ) = n ] ) = e − r f T ∑ n = 0 ∞ e − λ T ( λ T ) n n ! [ S B 1 ( m , k , u , d , p ) − K B 2 ( m , k , p ) ] . (15)

To compute the C ( S , 0 ) , we must confirm the parameter p, u and d first. Next, we shall calculate p, u and d by using the moment estimation theory.

E ( S t S t − 1 ) = p u + ( 1 − p ) d , E ( S t S t − 1 ) 2 = p u 2 + ( 1 − p ) d 2 . (16)

In the traditional model, u = 1 / d usually supply and we have following equation:

{ p u + ( 1 − p ) d = e ( μ + λ J − λ ) Δ t , p u 2 + ( 1 − p ) d 2 = e ( 2 μ + σ 2 ) Δ t e λ Δ t ( J 2 e σ y 2 − 1 ) , u = 1 / d . (17)

However, there is a fault in this method, because when σ → 0 , 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:

E ( S t S t − 1 ) 3 = p u 3 + ( 1 − p ) d 3 . (18)

Then, we set up the following equation groups:

{ p u + ( 1 − p ) d = e ( μ + λ J − λ ) Δ t , p u 2 + ( 1 − p ) d 2 = e ( 2 μ + σ 2 ) Δ t e λ Δ t ( J 2 e σ y 2 − 1 ) , p u 3 + ( 1 − p ) d 3 = e 3 ( μ + σ 2 ) Δ t e λ Δ t ( J 3 e 3 σ y 2 − 1 ) , (19)

Thus, we can get the solution of p, u and d as follows:

{ u = X Y − H + ( H − X Y ) 2 − 4 ( X 2 − Y ) ( Y 2 − H X ) 2 ( X 2 − Y ) , d = Y − X u x − u , p = X − d u − d . (20)

where X = e ( μ + λ J − λ ) Δ t , Y = e ( 2 μ + σ 2 ) Δ t e λ Δ t ( J 2 e σ y 2 − 1 ) and

H = e 3 ( μ + σ 2 ) Δ t e λ Δ t ( J 3 e 3 σ y 2 − 1 ) .

Substitute the three parameters into the Equation (14) and the according to option price with a jump diffusion model can be calculated. Now, we want to calculate Europe option with a continuous model and compare the result with binomial tree. While, in the risk-neutral world, the asset grows at the market risk-less instant interest rate r f . Thus, the asset price would satisfy E ( S ( T ) ) = S ( 0 ) e r f T .

Let E ( S ( T ) ) = S ( 0 ) e r f T = S ( 0 ) e ( μ + λ J − λ ) t and μ = r f − λ J + λ .

Then, a new real-valued measure Q is absolutely continuous with respect to P, and can be defined as follows:

d Q d P = ξ with ξ = e ( λ J − λ ) T .

In fact, the new measure Q can be seen as the risk-neutral measure. Consequently, under the risk-neutral measure Q, the price C of European call option with expiry date T and strike price K can be formulated as the discounted expectation of the payoff max [ S ( T ) − K ] .

C = e − r f T E [ max { S ( T ) − K , 0 } ] = e − r f T E [ max ( S 0 e ( r f − λ J + λ ) T − 1 2 σ 2 T + σ z T + ∑ i = 1 k γ i − K , 0 ) ] = e − r f T ∑ k = 0 ∞ P ( N ( t ) = k ) E [ max ( S 0 e ( r f − λ J + λ ) T − 1 2 σ 2 T + σ z T + ∑ i = 1 k γ i − K , 0 ) | N ( t ) = k ] = e − r f T ∑ k = 0 ∞ P ( N ( t ) = k ) E [ max ( S 0 e ( r f − λ J + λ ) T − 1 2 σ 2 T + σ z T + ∑ i = 1 k γ i − K , 0 ) | N ( t ) = k ]

where

ϕ ( z , u 1 , σ 1 2 ) = 1 2 π σ 1 e − ( z − u 1 ) 2 2 σ 1 2 , Z ˜ = σ z T + ∑ i = 1 k γ i ,

u 1 = E [ Z ˜ ] = k u y , σ 1 2 = V ( Z ˜ ) = σ 2 T + k σ y 2

z ( k ) = ln ( K / S ( 0 ) ) − ( r f − λ J − 1 2 σ 2 + λ ) T . .

Meanwhile, we know following equations hold

∫ z ( k ) ∞ e z ϕ ( z , u 1 , σ 1 2 ) d z = ∫ z ( k ) ∞ e z 1 2 π σ 1 e − ( z − u 1 ) 2 2 σ 1 2 d z = e u 1 + 1 2 σ 1 2 Φ ( u 1 + σ 1 2 − z ( k ) σ 1 ) ,

∫ z ( k ) ∞ K e − r f T Φ ( z , u 1 , σ 1 2 ) d z = K e − r f T Φ ( u 1 − z ( k ) σ 1 ) .

Thus, after some simple calculation, we have following theorem.

Theorem 3.2 According to the considered jump-diffusion model (1), the analytical solution of the European call option can be designed as follows:

C = ∑ k = 0 ∞ P ( N ( t ) = k ) [ S ( 0 ) e − λ J T + k u y + 1 2 σ y 2 Φ ( u 1 + σ 1 2 − z ( k ) σ 1 ) − K e − r f T Φ ( u 1 − z ( k ) σ 1 ) ] . (22)

Until now, we have obtained the European call option through two methods. In fact, when m → 0 , the option price given by the binomial tree model will tend to the value aboving because of the Central limit theorem and we will use a numerical analysis to prove it in the next section.

In this section, we aim to demonstrate the effectiveness and applicability of the proposed methods. Assume S 0 = 40 , μ = 0.1 , σ = 0.5 , u y = 0.2 , σ y = 0.25 , λ = 3 .

We also wonder realize the options price changing in maturity [0, T] with stable steps M.

As it shows in

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.

This work was supported in part by the National Nature Science Foundations of China under Grant No. 61673103, 61403248 and the Shanghai Yangfan Program of China under Grant 14YF1409800.

The authors declare no conflicts of interest regarding the publication of this paper.

Zhu, L.K., Kan, X., Shu, H.S. and Wang, Z.F. (2019) A New Binomial Tree Method for European Options under the Jump Diffusion Model. Journal of Applied Mathematics and Physics, 7, 3012-3021. https://doi.org/10.4236/jamp.2019.712211