ABSTRACT

In this paper, we deal with the valuation of Game Russian option with jumps, which is a contract that the seller and the buyer have both the rights to cancel and to exercise it at any time, respectively. This model can be formulated as a coupled optimal stopping problem. First, we discuss the pricing model with jumps when the stock pays dividends continuously. Secondly, we derive the value function of Game Russian options and investigate properties of optimal boundaries of the buyer. Finally, some numerical results are presented to demonstrate analytical properties of the value function.

Keywords:Stochastic Process; Game Russian Option; Double Exponential Distribution; Optimal Stopping; Optimal Boundaries

1. Introduction

Russian option was introduced by Shepp and Shiryaev [1,2] and it was one of perpetual American lookback options. In Russian option, the buyer has the right to exercise it at any time. On the other hand, in Game Russian option, not only the buyer but also the seller has the right to cancel it at any time. This option is based on Game option introduced by Kifer [3]. Game option frame work can be applied to various American-type options. Therefore, we apply this frame work to Russian option. The valuation of Game Russian option can be formulated as a coupled optimal stopping problem. See Cvitanic and Karatzas [4], Kifer [3].

Kyprianou [5] derived the closed-form solution in the case where the dividend rate is zero. Suzuki and Sawaki [6] gave the pricing formula with positive dividend. Kou and Wang [7] presented the closed-form for the value function of perpetual American put options without dividend and so on. Suzuki and Sawaki [8] studied the pricing formula of Russian option for double exponential jump diffusion processes.

In this paper, we deal with Game Russian options. Game Russian option is a contact that the seller and the buyer have the rights to cancel and to exercise it at any time, respectively. We present the pricing formula of Game Russian options for double exponential jump diffusion processes. The pricing of such an option can be formulated as a coupled optimal stopping problem which is analyzed as Dynkin game. We derive the value function of Game Russian option and its optimal boundaries. Also some numerical results are presented to demonstrate analytical sensitivities of the value function with respect to parameters.

This paper is organized as follows. In Section 2, we introduce a pricing model of Game Russian options by means of a coupled optimal stopping problem given by Kifer [3]. Section 3 presents the value function of Game Russian options for double exponential jump diffusion processes. Section 4 presents numerical examples to verify analytical results. We end the paper with some concluding remarks and future work.

2. Pricing Model

In this section, we consider the pricing model for Game Russian option. Let be the process of the riskless asset price at time defined by, where is the positive interest rate. Let be a standard Brownian motion and be a Poisson process with the intensity. Let denote i.i.d. positive random variables. has a double exponential distribution and its density function is given by

where and such that. Under a risk-neutral probability, the process of the risky asset price at time satisfies the stochastic differential equation

(1)

where and are constants. Define another probability measure as

where is a nonnegative continuous dividend rate of the risky asset,

is the information available at time and

By Girsanov’s theorem, is a Brownian motion with respect to.

We can rewrite (1) as

(2)

Solving (2) gives, where

Let be a function of class. Then the infinitesimal generator of the process is given by

for all.

Next we introduce the four real numbers. Kou and Wang [9] showed that the equation for all has the solutions, where

And the four solutions satisfy the following inequalities

Remark 2.1 When the dividend rate,.

Define the process

Then the value function of Russian option is given by

where the supremum is taken for all stopping times.

Theorem 2.1 (Suzuki and Sawaki [8]) The value function of Russian option with jump is given by

The coefficients are determined by

and

Moreover, the optimal boundary is the solution in to the equation

and the optimal stopping time is given by

3. Game Russian Options

Let denote a cancel time for the seller and an exercise time for the buyer. If the seller cancels the contract, the buyer receives from the seller. We can think of as the penalty cost for the cancellation. On the other hand, if the buyer exercises it, (s)he receives from the seller. Therefore, the payoff function for the buyer is given by

Let denote the set of all stopping times with values in the interval. Then the value function of Game Russian option is defined by

(3)

where

And the function satisfies the inequalities

which provides the lower and the upper bounds for the value function of Game Russian option.

We define two sets and as

and are called the seller’s cancellation region and the buyer’s exercise region, respectively. Then the two optimal stopping times are given by

Then for any, and attain the infimum and supremum in (3), i.e., we have

The pair is the saddle point of.

Remark 3.1 The seller minimizes the payoff function and. From this, it follows that the seller’s optimal cancellation region is.

Lemma 3.1 Suppose that. Then the function is Lipschitz continuous and its Radon-Nikodym derivative satisfies

(4)

Proof. Since and depend on the initial value, we write them as and. Replacing the optimal stopping times by another stopping time, we get the inequalities

Note that for any. For any, we have

where. Therefore, we obtain

This means that is Lipschitz continuous and satisfies (4).

If the penalty is large enough, the seller never cancels. It is of interest to show how much should be large for the seller never to cancel.

Lemma 3.2 Set. If the penalty, the seller never cancels. In other words, Game Russian option is reduced to Russian option.

Proof. Consider the function. Since it holds by Lemma 3.1 and

, we have. Hence, it follows that because the inequality holds.

We assume that and. It means that the jump occurs only upward. This is very useful to analyse stochastic cash management problem for jump diffusion processes (See Sato and Suzuki [10]). Then we can express as

and the equation has three solutions, which satisfy

We introduce the function for

(5)

We set and. In what follows, we determine the coefficients and. In order to determine the coefficients, we prepare the conditions. By value matching condition, we have

and by smooth pasting condition, we have

We can get the last condition by using the infinitesimal generator of the process given by

for all. For, we obtain

From this, we obtain

where. By Lemma 2.1 in Kou and Wang [9], we have. Since

holds, we get the condition

(6)

Lemma 3.3 Solving the following equations

gives the solutions

Since the coefficients depend on, we denote them as and. The number given by (5) satisfies the equation

4. Main Theorem

Theorem 4.1 Let denote the value function of Game Russian option. If, the value function is equal to the one of Russian option, i.e.. If, then is given by

(7)

and the optimal stopping times are given by

The optimal boundary for the buyer is the unique solution to the equation

In order to prove the above theorem, we need the following lemmas.

Lemma 4.1 Assume that a function has the following properties;

1) and, for.

2) It holds and satisfies for.

3) At we have.

Then, is the value function of Game Russian options with dividend, i.e., holds. The optimal exercise region is the interval and the optimal cancellation region is.

Proof. By Ito’s formula, we have

(8)

Set

and. Since a.s. for, we have a.s. Therefore, taking expectation of (8), we have

It holds

Therefore we get.

For any, set. The term of is nonpositive a.s. because. Taking expectation of (8), we get

The above left hand side dominates. Therefore

(9)

Next for any, set. Similarly it holds

Since the left hand side is dominated by, we get

(10)

From (9) and (10), we have.

Lemma 4.2 The function satisfies

Proof. Since for, we have

Hence, we obtain

That is, we obtain.

Lemma 4.3 For the function satisfies and

Proof. The former assertion is known. We shall show the latter one. The second derivative of is nonnegative because and. It follows that is a convex function. Since is a convex function, is increasing. From this, we can see that for. By the boundary conditions and, we have.

Lemma 4.4 Set

(11)

Then the equation has the unique solution in the interval.

Proof. By (11), a direct computation yields

Since and, we have. Furthermore, it holds. Therefore, the equation has the unique solution in.

In the rest of this section, we present some numerical examples to demonstrate theoretical results and some effects of parameters on the price of Game Russian option. We set . Using these parameters, is 0.248.

Figure 1 shows how the optimal exercise boundary increase as the penalty increases from 0.1 up to. From the figure, we can see that the optimal boundary is increasing in the penalty. Figure 2 demonstra-

tes the value function of Game Russian option with jumps. Dashed lines represent the graph of the value in from the bottom, respectively. Real line represents the value in. From Figure 2, we can visually recognize that is convex and increasing in.

5. Conclusion

In this paper, we discussed the valuation of Game Russian option written on dividend paying asset, obtained the value function of it for double exponential jump diffusion processes and also explored some analytical properties of the value function and the optimal boundaries for the seller and buyer, which were useful to provide an approximation of the finite lived Game Russian option. Moreover, we plan to examine convertible bonds with jumps by using Game option frame work. We shall leave it as future work.

REFERENCES

NOTES

^*Corresponding author.