As, for all,

(2.20)

where. And because is a constant, we can obtain that

(2.21)

Focusing on, we implement formula and combine (2.21), for,

(2.22)

is the local martingale and. According to the definition, there are a series of stopping time and as well. Therefore, for every, is a martingale. And according to Lebesgue’s dominated convergence theorem,

more precisely,

,

and in view of (2.21)

.

If we integrate the three formulas above, for all,

,

Now for convenience, is used to replace. Based on Kubilius [9], let, when,

As and are asymptotically independent, when,

(2.23)

which obeys Taylor expansion of the first order and. Here we use Kou’s treatment of for reference and define random walk

and

When,

(2.24)

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

Lemma 2.3. For discrete barrier options with times of monitoring and value of barrier H, there are stopping time, (for convenience, denote), and their corresponding logarithm underlying asset prices, (Formula (2.8)). For (Formula (2.11)), the joint distribution and meet the following equation:

(2.25)

where.

Proof:

On the basis of proposition 2.1 in Kou [1] and Zhang [10], and are asymptotically independent, denote.

When, ,

Therefore,

(2.26)

Then lemma 2.3 can be proved by using corollary 3.3 in Kou [4].

Theorem 2.1. Let denote the price of continuously monitored options with barrier value H, accordingly, is the price of discretely monitored options with the monitoring frequency. Consequently, for an arbitrary discrete barrier option with maturity T, , and when, we obtain the following correction formula:

(2.27)

where, with the Riemann zeta function.

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 [2]’s, there are no restriction of (upward) and (downward) for the striking price K and barrier H.

Cite this paper

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

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