Journal of Applied Mathematics and Decision Sciences
Volume 2009 (2009), Article ID 593986, 13 pages
doi:10.1155/2009/593986
Research Article

Callable Russian Options and Their Optimal Boundaries

Atsuo Suzuki1 and Katsushige Sawaki2

1Faculty of Urban Science, Meijo University, 4-3-3 Nijigaoka, Kani, Gifu 509-0261, Japan
2Nanzan Business School, Nanzan University, 18 Yamazato-cho, Showa-ku, Nagoya 466-8673, Japan

Received 28 November 2008; Accepted 10 February 2009

Academic Editor: Lean Yu

Copyright © 2009 Atsuo Suzuki and Katsushige Sawaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We deal with the pricing of callable Russian options. A callable Russian option is a contract in which both of the seller and the buyer have the rights to cancel and to exercise at any time, respectively. The pricing of such an option can be formulated as an optimal stopping problem between the seller and the buyer, and is analyzed as Dynkin game. We derive the value function of callable Russian options and their optimal boundaries.

1. Introduction

For the last two decades there have been numerous papers (see [1]) on valuing American-style options with finite lived maturity. The valuation of such American-style options may often be able to be formulated as optimal stopping or free boundary problems which provide us partial differential equations with specific conditions. One of the difficult problems with pricing such options is finding a closed form solution of the option price. However, there are shortcuts that make it easy to calculate the closed form solution to that option (see [2–4]). Perpetuities can provide us such a shortcut because free boundaries of optimal exercise policies no longer depend on the time.

In this paper, we consider the pricing of Russian options with call provision where the issuer (seller) has the right to call back the option as well as the investor (buyer) has the right to exercise it. The incorporation of call provision provides the issuer with option to retire the obligation whenever the investor exercises his/her option. In their pioneering theoretical studies on Russian options, Shepp and Shiryaev [5, 6] gave an analytical formula for pricing the noncallable Russian option which is one of perpetual American lookback options. The result of this paper is to provide the closed formed solution and optimal boundaries of the callable Russian option with continuous dividend, which is different from the pioneering theoretical paper Kyprianou [2] in the sense that our model has dividend payment.

The paper is organized as follows. In Section 2, we introduce a pricing model of callable Russian options by means of a coupled optimal stopping problem given by Kifer [7]. Section 3 represents the value function of callable Russian options with dividend. Section 4 presents numerical examples to verify analytical results. We end the paper with some concluding remarks and future work.

2. Model

We consider the Black-Scholes economy consisting of two securities, that is, the riskless bond and the stock. Let be the bond price at time which is given by(2.1)where is the riskless interest rate. Let be the stock price at time which satisfies the stochastic differential equation(2.2)where and are constants, is dividend rate, and is a standard Brownian motion on a probability space . Solving (2.2) with the initial condition gives(2.3)Define another probability measure by(2.4)Let(2.5)where is a standard Brownian motion with respect to . Substituting (2.5) into (2.2), we get(2.6)Solving the above equation, we obtain(2.7)

Russian option was introduced by Shepp and Shiryaev [5, 6] and is the contract that only the buyer has the right to exercise it. On the other hand, a callable Russian option is the contract that the seller and the buyer have both the rights to cancel and to exercise it at any time, respectively. Let be a cancel time for the seller and be an exercise time for the buyer. We set(2.8)When the buyer exercises the contract, the seller pay to the buyer. When the seller cancels it, the buyer receives . We assume that seller's right precedes buyer's one when . The payoff function of the callable Russian option is given by(2.9)where is the penalty cost for the cancel and a positive constant.

Let be the set of stopping times with respect to filtration defined on the nonnegative interval. Letting and be some given parameters satisfying and , the value function of the callable Russian option is defined by(2.10)The infimum and supremum are taken over all stopping times and , respectively.

We define two sets and as(2.11) and are called the seller's cancel region and the buyer's exercise region, respectively. Let and be the first hitting times that the process is in the region and , that is,(2.12)Lemma 2.1. Assume that . Then, one has (2.13)Proof. First, suppose that . Then, it holds(2.14)Next, suppose that . By the same argument as Karatzas and Shreve [1, page 65], we obtain(2.15) where is the standard Brownian motion which attains the supremum in (2.15). Therefore, it follows that(2.16)The proof is complete.

By this lemma, we may apply Proposition 3.3 in Kifer [7]. Therefore, we can see that the stopping times and attain the infimum and the supremum in (2.10). Then, we have(2.17)And satisfies the inequalities(2.18)which provides the lower and the upper bounds for the value function of the callable Russian option. Let be the value function of Russian option. And we know because the seller as a minimizer has the right to cancel the option. Moreover, it is clear that is increasing in and .

Should the penalty cost be large enough, it is optimal for the seller not to cancel the option. This raises a question how large such a penalty cost should be. The following lemma is to answer the question.Lemma 2.2. Set . If , the seller never cancels. Therefore, callable Russian options are reduced to Russian options.Proof. We set . Because we know by the condition , we have , that is, holds. By using the relation , we obtain , that is, it is optimal for the seller not to cancel. Therefore, the seller never cancels the contract for .Lemma 2.3. Suppose . Then, the function is Lipschitz continuous in . And it holds (2.19)Proof. Set(2.20)Replacing the optimal stopping times and from the nonoptimal stopping times and , we have(2.21)respectively. Note that . For any , we have(2.22)where . Since the above expectation is less than 1, we have(2.23)This means that is Lipschitz continuous in , and (2.19) holds.

By regarding callable Russian options as a perpetual double barrier option, the optimal stopping problem can be transformed into a constant boundary problem with lower and upper boundaries. Let be the exercise region of Russian option. By the inequality , it holds . Consequently, we can see that the exercise region is the interval . On the other hand, the seller minimizes and it holds . From this, it follows that the seller's optimal boundary is a point . The function is represented by(2.24)where(2.25)In order to calculate (2.25), we prepare the following lemma.Lemma 2.4. Let and be the first hitting times of the process to the points and . Set , and . Then for , one has (2.26)(2.27)Proof. First, we prove (2.26). Define(2.28)We define as . By Girsanov's theorem, is a standard Brownian motion under the probability measure . Let and be the first time that the process hits or , respectively, that is,(2.29)Since we obtain from , we have(2.30)Therefore, we have(2.31)From Karatzas and Shreve [8, Exercise 8.11, page 100], we can see that(2.32)Therefore, we obtain(2.33)We omit the proof of (2.27) since it is similar to that of (2.26).

We study the boundary point of the exercise region for the buyer. For , we consider the function . It is represented by(2.34)The family of the functions satisfies(2.35)To get an optimal boundary point , we compute the partial derivative of with respect to , which is given by the following lemma.Lemma 2.5. For any , one has (2.36)Proof. First, the derivative of the first term is(2.37)Next, the derivative of the second term is(2.38)where the last equality follows from the relation(2.39)After multiplying (2.37) by and (2.38) by , we obtain (2.36).

We set(2.40)Since and , the equation has at least one solution in the interval . We label all real solutions as . Then, we have(2.41)Then attains the supremum of . In the following, we will show that the function is convex and satisfies smooth-pasting condition.Lemma 2.6. is a convex function in .Proof. From (2.50), satisfies(2.42)If , we get . Next assume that . We consider function for . Then,(2.43)Since we find that from the above equation, is a convex function. It follows from this the fact that is a convex function.Lemma 2.7. satisfies (2.44)Proof. Since for , it holds . For , we derivative (2.47):(2.45)Therefore, we get(2.46)This completes the proof.

Therefore, we obtain the following theorem.Theorem 2.8. The value function of callable Russian option is given by (2.47)And the optimal stopping times are (2.48)The optimal boundary for the buyer is the solution in to , where (2.49)

We can get (2.47) by another method. For , the function satisfies the differential equation(2.50)Also, we have the boundary conditions as follows:(2.51)(2.52)(2.53)The general solution to (2.50) is represented by(2.54)where and are constants. Here, and are the roots of(2.55)Therefore, are(2.56)

From conditions (2.51) and (2.52), we get(2.57)And from (2.57) and (2.53), we have(2.58)Substituting (2.57) into (2.54), we can obtain (2.47).

3. Numerical Examples

In this section, we present some numerical examples which show that theoretical results are varied and some effects of the parameters on the price of the callable Russian option. We use the values of the parameters as follows: .

Figure 1 shows an optimal boundary for the buyer as a function of penalty costs , which is increasing in . Figures 2 and 3 show that the price of the callable Russian option has the low and upper bounds and is increasing and convex in . Furthermore, we know that is increasing in . Figure 4 demonstrates that the price of the callable Russian option with dividend is equal to or less than the one without dividend. Table 1 presents the values of the optimal boundaries for several combinations of the parameters.

tab1
Table 1: Penalty , interest rate , dividend rate , volatility , discount factor , and the optimal boundary for the buyer .
593986.fig.001
Figure 1: Optimal boundary for the buyer.
593986.fig.002
Figure 2: The value function ().
593986.fig.003
Figure 3: The value function ().
593986.fig.004
Figure 4: Real line with dividend; dash line without dividend.

4. Concluding Remarks

In this paper, we considered the pricing model of callable Russian options, where the stock pays continuously dividend. We derived the closed-form solution of such a Russian option as well as the optimal boundaries for the seller and the buyer, respectively. It is of interest to note that the price of the callable Russian option with dividend is not equal to the one as dividend value goes to zero. This implicitly insist that the price of the callable Russian option without dividend is not merely the limit value of the one as if dividend vanishes as goes to zero. We leave the rigorous proof for this question to future research. Further research is left for future work. For example, can the price of callable Russian options be decomposed into the sum of the prices of the noncallable Russian option and the callable discount? If the callable Russian option is finite lived, it is an interesting problem to evaluate the price of callable Russian option as the difference between the existing price formula and the premium value of the call provision.

Acknowledgment

This work was supported by Grant-in-Aid for Scientific Research (A) 20241037, (B) 20330068, and a Grant-in-Aid for Young Scientists (Start-up) 20810038.

References

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  3. A. Suzuki and K. Sawaki, “The pricing of callable perpetual American options,” Transactions of the Operations Research Society of Japan, vol. 49, pp. 19–31, 2006 (Japanese).
  4. A. Suzuki and K. Sawaki, “The pricing of perpetual game put options and optimal boundaries,” in Recent Advances in Stochastic Operations Research, pp. 175–188, World Scientific, River Edge, NJ, USA, 2007.
  5. L. A. Shepp and A. N. Shiryaev, “The Russian option: reduced regret,” The Annals of Applied Probability, vol. 3, no. 3, pp. 631–640, 1993.
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