Callable Russian Options and Their Optimal Boundaries

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 𝐵_𝑡 be the bond price at time 𝑡 which is given bywhere 𝑟 is the riskless interest rate. Let 𝑆_𝑡 be the stock price at time 𝑡 which satisfies the stochastic differential equationwhere 𝑑 and 𝜅>0 are constants, 𝑑 is dividend rate, and 𝑊_𝑡 is a standard Brownian motion on a probability space (Ω,ℱ,𝑃). Solving (2.2) with the initial condition 𝑆₀=𝑥 givesDefine another probability measure 𝑃 byLetwhere 𝑊_𝑡 is a standard Brownian motion with respect to 𝑃. Substituting (2.5) into (2.2), we getSolving the above equation, we obtain

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 𝜎 be a cancel time for the seller and 𝜏 be an exercise time for the buyer. We setWhen the buyer exercises the contract, the seller pay Ψ_𝜏(𝜓) to the buyer. When the seller cancels it, the buyer receives Ψ_𝜎(𝜓)+𝛿. We assume that seller's right precedes buyer's one when 𝜎=𝜏. The payoff function of the callable Russian option is given bywhere 𝛿 is the penalty cost for the cancel and a positive constant.

Let 𝒯_0,∞ be the set of stopping times with respect to filtration ℱ defined on the nonnegative interval. Letting 𝛼 and 𝜓 be some given parameters satisfying 𝛼>0 and 𝜓≥1, the value function of the callable Russian option 𝑉(𝜓) is defined byThe infimum and supremum are taken over all stopping times 𝜎 and 𝜏, respectively.

We define two sets 𝐴 and 𝐵 as𝐴 and 𝐵 are called the seller's cancel region and the buyer's exercise region, respectively. Let 𝜎^𝜓𝐴 and 𝜏^𝜓𝐵 be the first hitting times that the process Ψ_𝑡(𝜓) is in the region 𝐴 and 𝐵, that is,Lemma 2.1. Assume that 𝑑−(1/2)𝜅²−2𝑟<0. Then, one has Proof. First, suppose that max(𝜓𝑥,sup𝑆_𝑢)=𝜓𝑥. Then, it holdsNext, suppose that max(𝜓𝑥,sup𝑆_𝑢)=sup𝑆_𝑢. By the same argument as Karatzas and Shreve [1, page 65], we obtain where 𝑊^∗ is the standard Brownian motion which attains the supremum in (2.15). Therefore, it follows thatThe proof is complete.

By this lemma, we may apply Proposition 3.3 in Kifer [7]. Therefore, we can see that the stopping times 𝜎^𝜓=𝜎^𝜓𝐴 and 𝜏̂^𝜓=𝜏^𝜓𝐵 attain the infimum and the supremum in (2.10). Then, we haveAnd 𝑉(𝜓) satisfies the inequalitieswhich provides the lower and the upper bounds for the value function of the callable Russian option. Let 𝑉_𝑅(𝜓) be the value function of Russian option. And we know 𝑉(𝜓)≤𝑉_𝑅(𝜓) because the seller as a minimizer has the right to cancel the option. Moreover, it is clear that 𝑉(𝜓) is increasing in 𝜓 and 𝑥.

Should the penalty cost 𝛿 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 𝛿^∗=𝑉(1)−1. If 𝛿≥𝛿^∗, the seller never cancels. Therefore, callable Russian options are reduced to Russian options.Proof. We set ℎ(𝜓)=𝑉(𝜓)−𝜓−𝛿. ℎ^(𝜓)=𝑉^(𝜓)−1<0. Because we know ℎ(1)=𝑉(1)−1−𝛿=𝛿^∗−𝛿<0 by the condition 𝛿≥𝛿^∗, we have ℎ(𝜓)<0, that is, 𝑉(𝜓)<𝜓+𝛿 holds. By using the relation 𝑉(𝜓)≤𝑉_𝑅(𝜓), we obtain 𝑉(𝜓)<𝜓+𝛿, that is, it is optimal for the seller not to cancel. Therefore, the seller never cancels the contract for 𝛿≥𝛿^∗.Lemma 2.3. Suppose 𝑟>𝑑. Then, the function 𝑉(𝜓) is Lipschitz continuous in 𝜓. And it holds Proof. SetReplacing the optimal stopping times 𝜎^𝜙 and 𝜏̂^𝜓 from the nonoptimal stopping times 𝜎^𝜓 and 𝜏̂^𝜙, we haverespectively. Note that 𝑧⁺¹−𝑧⁺²≤(𝑧₁−𝑧₂)⁺. For any 𝜙>𝜓, we havewhere 𝐻_𝑡=exp{(𝑟−𝑑+(1/2)𝜅²)𝑡+𝜅𝑊_𝑡}. Since the above expectation is less than 1, we haveThis means that 𝑉 is Lipschitz continuous in 𝜓, 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 𝐵={𝜓∈𝐑⁺∣𝑉_𝑅(𝜓)=𝜓} be the exercise region of Russian option. By the inequality 𝑉(𝜓)≤𝑉_𝑅(𝜓), it holds 𝐵⊃𝐵≠∅. Consequently, we can see that the exercise region 𝐵 is the interval [𝑙_∗,∞). On the other hand, the seller minimizes 𝑅(𝜎,𝜏) and it holds Ψ_𝑡(𝜓)≥Ψ₀(𝜓)=𝜓≥1. From this, it follows that the seller's optimal boundary 𝐴 is a point {1}. The function 𝑉(𝜓) is represented bywhereIn order to calculate (2.25), we prepare the following lemma.Lemma 2.4. Let 𝜎^𝑥𝑎 and 𝜏^𝑥𝑏 be the first hitting times of the process 𝑆_𝑡(𝑥) to the points {𝑎} and {𝑏}. Set 𝜈=(𝑟−𝑑)/𝜅−(1/2)𝜅,𝜂₁=(1/𝜅)(√𝜈²+2𝛼+𝜈), and 𝜂₂=(1/𝜅)(√𝜈²+2𝛼−𝜈). Then for 𝑎<𝑥<𝑏, one has Proof. First, we prove (2.26). DefineWe define 𝑑𝑃=𝐿_𝑇𝑑𝑃 as 𝑊_𝑡≡𝑊_𝑡+𝜈𝑡. By Girsanov's theorem, 𝑃 is a standard Brownian motion under the probability measure 𝑇_𝜌1. Let 𝑇_𝜌2 and 𝑊_𝑡 be the first time that the process 𝜌₁ hits 𝜌₂ or 𝑇𝑇_𝜌1𝜌2=inf𝑡>0∣𝑊_𝑡=𝜌₁,=inf𝑡>0∣𝑊_𝑡=𝜌₂.(2.29), respectively, that is,Since we obtain 𝑆_𝑡(𝑥)=𝑥exp(𝜅𝑊_𝑡) from 𝜎^𝑥𝑎=𝑇_𝜌1,a.s.,𝜌₁=1𝜏_𝑥𝑏𝜅log𝑥𝑎,=𝑇_𝜌2,a.s.,𝜌₂=1𝐿_{𝑇−1𝜌1}𝜅log𝑥𝑏,=exp12𝜈²=exp−1𝑇_𝜌1+𝜈𝑊_𝑇𝜌12𝜈²=exp−1𝑇_𝜌1+𝜈𝑊_𝑇𝜌12𝜈²𝑇_𝜌1+𝜈𝜌₁.(2.30), we haveTherefore, we haveFrom Karatzas and Shreve [8, Exercise 8.11, page 100], we can see thatTherefore, we obtainWe omit the proof of (2.27) since it is similar to that of (2.26).

We study the boundary point 1<𝜓<𝑙<∞ of the exercise region for the buyer. For 𝑉(𝜓,𝑙), we consider the function 𝑉(𝜓,𝑙)=𝑉^𝜓𝜓,(𝑙),1𝜓≤≥𝜓𝑙.≤𝑙,(2.34). It is represented byThe family of the functions 𝑉(𝜓)=𝑉𝜓,𝑙_∗=sup_1<𝜓<𝑙𝑉(𝜓,𝑙).(2.35) satisfiesTo get an optimal boundary point 𝑉(𝜓,𝑙), we compute the partial derivative of 𝑙 with respect to 1<𝜓<𝑙, which is given by the following lemma.Lemma 2.5. For any 𝜕𝑉𝜕𝑙(𝜓,𝑙)=𝜓^𝜂2−𝜓^−𝜂1𝑙(𝑙^𝜂1−𝑙^−𝜂2)²𝑙^𝜂1𝑙^−𝜂21−𝜂₂𝑙^𝜂1+1−1+𝜂₁𝑙^−𝜂2+1+(1+𝛿)𝜂₁+𝜂₂.(2.36), one has Proof. First, the derivative of the first term isNext, the derivative of the second term iswhere the last equality follows from the relationAfter multiplying (2.37) by 𝜓^𝜂2−𝜓^−𝜂1 and (2.38) by 𝑓(𝑙)=1−𝜂₂𝑙^𝜂1+1−1+𝜂₁𝑙^−𝜂2+1+(1+𝛿)𝜂₁+𝜂₂.(2.40), we obtain (2.36).

We setSince 𝑓(∞)=−∞ and 𝑓(𝑙)=0, the equation (1,∞) has at least one solution in the interval 1<𝑙_𝑛<𝑙_𝑛−1<⋯<𝑙₁<∞. We label all real solutions as 𝜕𝑉𝜕𝑙(𝜓,𝑙)||^{𝑙=𝑙𝑖}=0,𝑖=1,…,𝑛∀𝜓.(2.41). Then, we haveThen 𝑉(𝜓,𝑙) attains the supremum of 𝑉(𝜓). In the following, we will show that the function 𝑉(𝜓) is convex and satisfies smooth-pasting condition.Lemma 2.6. 𝜓 is a convex function in 𝑉.Proof. From (2.50), 12𝜅²𝜓²𝑑²𝑉𝑑𝜓²=−(𝑟−𝑑)𝜓𝑑𝑉𝑑𝜓+𝛼𝑉(𝜓).(2.42) satisfiesIf 𝑑²𝑉/𝑑𝜓²>0, we get 𝑟>𝑑. Next assume that 𝑉(𝜓)=𝑉(−𝜓). We consider function 𝜓<0 for 12𝜅²𝜓²𝑑²𝑉𝑑𝜓²−(𝑟−𝑑)𝜓𝑑𝑉𝑑𝜓−𝑟𝑉=12𝜅²𝜓²𝑑²𝑉𝑑𝜓²+(𝑟−𝑑)𝜓𝑑𝑉𝑑𝜓−𝑟𝑉=0.(2.43). Then,Since we find that 𝑉 from the above equation, 𝑉 is a convex function. It follows from this the fact that 𝑉(𝜓) is a convex function.Lemma 2.7. 𝑑𝑉𝑑𝜓𝑙^−∗=𝑑𝑑𝑉𝜓𝑙^+∗=1.(2.44) satisfies Proof. Since 𝜓>𝑙_∗ for (𝑑𝑉/𝑑𝜓)(𝑙^+∗)=1, it holds 1≤𝜓<𝑙_∗. For 𝑑𝑉𝑑𝜓=𝑙^𝜂2−𝑙𝑙^−𝜂1𝜂₂𝜓^𝜂2−1+𝜂₁𝑙^−𝜂1−1+1+𝛿𝑙^𝜂1−𝑙^−𝜂2−𝜂¹𝜓𝑙^𝜂11𝜓−𝜂²𝜓𝑙^𝜂21=1𝜓𝜓(𝑙^𝜂1−𝑙^−𝜂2)𝑙^{𝜂1−𝜂2+1}𝜂₂𝜓^𝜂2+𝜂₁𝜓^−𝜂1−(1+𝛿)𝜂₁𝑙𝜓^𝜂1+𝜂₂𝜓=1𝑙^𝜂2𝜓(𝑙^𝜂1−𝑙^−𝜂2)𝜂²𝜓𝑙^𝜂2𝑙^𝜂1+1+𝜂₁𝑙𝜓^𝜂1𝑙^−𝜂2+1−(1+𝛿)𝜂₁𝑙𝜓^𝜂1+𝜂₂𝜓𝑙^𝜂2.(2.45), we derivative (2.47):Therefore, we getThis completes the proof.

Therefore, we obtain the following theorem.Theorem 2.8. The value function of callable Russian option 𝑉(𝜓)=⎧⎪⎨⎪⎩(1+𝛿)𝑙^∗/𝜓^𝜂1−𝜓/𝑙_∗^𝜂2𝑙^𝜂1_∗−𝑙^−𝜂2_∗+𝑙_∗𝜓^𝜂2−𝜓^−𝜂1𝑙^𝜂2_∗−𝑙^−𝜂1_∗𝜓,,1𝜓≤≥𝜓𝑙_∗≤𝑙_∗,.(2.47) is given by And the optimal stopping times are The optimal boundary for the buyer (1,∞) is the solution in 𝑓(𝑙)=0 to 𝑓(𝑙)=1−𝜂₂𝑙^𝜂1+1−1+𝜂₁𝑙^−𝜂2+1+(1+𝛿)𝜂₁+𝜂₂.(2.49), where

We can get (2.47) by another method. For 𝑉(𝜓), the function 12𝜅²𝜓²𝑑²𝑉𝑑𝜓²+(𝑟−𝑑)𝜓𝑑𝑉𝑑𝜓−𝛼𝑉(𝜓)=0.(2.50) satisfies the differential equationAlso, we have the boundary conditions as follows:The general solution to (2.50) is represented bywhere 𝜆₂ and 12𝜅²𝜆²+𝑟−𝑑−12𝜅²𝜆−𝛼=0.(2.55) are constants. Here, 𝜆₁,𝜆₂ and 𝜆_1,2=±√𝜈²+2𝛼−𝜈𝜅.(2.56) are the roots ofTherefore, 1−𝜂₂𝑙^𝜂1+1−1+𝜂₁𝑙^−𝜂2+1+(1+𝛿)𝜂₁+𝜂₂=0.(2.58) are

From conditions (2.51) and (2.52), we getAnd from (2.57) and (2.53), we haveSubstituting (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: 𝜓.

Figure 1 shows an optimal boundary for the buyer as a function of penalty costs 𝑉(𝜓), which is increasing in 𝛿. 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 𝛿. Furthermore, we know that 𝑟 is increasing in 𝑑. 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.

Table 1 

Penalty 𝜅, interest rate 𝛼, dividend rate 𝑙_∗, volatility 𝛿, discount factor 𝑟, and the optimal boundary for the buyer 𝑑.

Figure 1 

Optimal boundary for the buyer.

Figure 2 

The value function𝛿=0.01,0.02,0.03 (𝑑).

Figure 3 

The value function𝑑 ().

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.