Game Russian Options for Double Exponential Jump Diffusion Processes ()
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-
Figure 1. Optimal boundary for the buyer.
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.
NOTES
*Corresponding author.