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This article introduces a hyper-exponential jump diffusion process based on the continuity correction for discrete barrier options under the standard B-S model, using measure transformation and stopping time theory to prove the correction, thus broadening the conditions of the continuity correction of Kou.

In 2003, S. G. Kou [

This article uses Kou’s theoretical derivation method for the first passage time and C. D. Fuh’s thought of proving the correction formula based on double exponential jump diffusion model for reference. In order to generalize the model into the pricing of both the discrete single and double barrier options on the basis of the hyper-exponential jump diffusion process, we combine the correction formula of discrete single barrier options raised by Kou with D. Jun’s correction of discrete double barrier options under the B-S model. Although formally, the correction formula in this article seems to be same as Kou’s and D. Jun’s, there are no restrictions of strike price K and barrier value H in the correction formula in this passage, which means that the scope of application of Kou’s and D. Jun’s correction are widened. Additionally, compared with the double exponential jump diffusion model of C. D. Fuh, the hyper-exponential jump diffusion model in this article is more general.

In 1997, Steven Kou put forward the concept of continuity correction in his paper, which combined the continuity barrier options with the discrete barrier options through the continuity correction formula. Denote the stopping time by

or

Here H stands for the barrier level and

Kou has pointed out in his article that when m is large enough, which means

If

Here,

According to the correction formula, we could find that the barrier-crossing probability will be lower after the adjustment of discretely monitored barrier. The amount of the adjustment is

We assume that under the risk neutrality measure Q, the asset price

Here

rate, dividend rate and risk volatility, respectively.

Using Ito lemma and theories of calculation of stochastic partial differential equations, the solution of model (2.4) under continuously monitored situation is:

where

It’s obvious that, when we assume the discretely monitored time interval is

where

where

Now we try to find the relationship between this two joint distribution densities,

Let

For convenience, we use

Proposition 2.1. [

where

In this section, based on the discrete model (2.8), firstly, we make an adjustment on the discrete model

where

The preparation has been done, however, two preparatory lemmas will be introduced before finally exhibit the main conclusion.

Lemma 2.2. For continuously monitored stopping time

Proof: Assume (2.14) is a true statement, in view of (2.12) and (2.13), when

From proposition 2.1, the equation above will be true if the following equation is proved:

where

In order to obtain Equation (2.15), here, we solve this problem by adjusting Kou’s method of the first passage time under the double exponential jump diffusion process. Noticing that

where L stands for the infinitesimal generator

Here,

As

where

Focusing on

is the local martingale and

more precisely,

and in view of (2.21)

If we integrate the three formulas above, for all

Now for convenience,

As

which obeys Taylor expansion of the first order and

When

Lemma 2.2 can be proved by substituting (2.24) and (2.16) into (2.15).

Lemma 2.3. For discrete barrier options with

where

Proof:

On the basis of proposition 2.1 in Kou [

When

Therefore,

Then lemma 2.3 can be proved by using corollary 3.3 in Kou [

Theorem 2.1. Let

where

The proof of the theorem can be derived directly from Lemma 2.2 and 2.3. What we need to illustrate is that although the conclusion here seems to be similar to Kou [

Ting Liu,Chang Feng,Yanqiong Lu,Bei Yao, (2015) A Note on the Kou’s Continuity Correction Formula. Open Journal of Social Sciences,03,28-34. doi: 10.4236/jss.2015.311005