JAMDSJournal of Applied Mathematics and Decision Sciences1532-76121173-9126Hindawi Publishing Corporation59398610.1155/2009/593986593986Research ArticleCallable Russian Options and Their Optimal BoundariesSuzukiAtsuo1SawakiKatsushige2YuLean1Faculty of Urban ScienceMeijo University4-3-3 NijigaokaKaniGifu 509-0261Japanmeijo-u.ac.jp2Nanzan Business SchoolNanzan University18 Yamazato-choShowa-kuNagoya 466-8673Japannanzan-u.ac.jp200924052009200928112008100220092009Copyright © 2009This 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.

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 ) 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 ). 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  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 . 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 Bt be the bond price at time t which is given bydBt=rBtdt,B0>0,r>0,where r is the riskless interest rate. Let St be the stock price at time t which satisfies the stochastic differential equationdSt=(rd)Stdt+κStdW˜t,S0=x,where d and κ>0 are constants, d is dividend rate, and W˜t is a standard Brownian motion on a probability space (Ω,,P˜). Solving (2.2) with the initial condition S0=x givesSt=xexp{(rd12κ2)t+κW˜t}.Define another probability measure P^ bydP^dP˜=exp(κW˜t12κ2t).LetW^t=W˜tκt,where W^t is a standard Brownian motion with respect to P^. Substituting (2.5) into (2.2), we getdSt=(rd+κ2)Stdt+κStdW^t.Solving the above equation, we obtainSt(x)=xexp{(rd+12κ2)t+κW^t}.

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 setΨt(ψ)max(ψx,sup0utSu)St,ψ1.When the buyer exercises the contract, the seller pay Ψτ(ψ) to the buyer. When the seller cancels it, the buyer receives Ψσ(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 byR(σ,τ)=(Ψσ(ψ)+δ)1{σ<τ}+Ψτ(ψ)1{τσ},where δ is the penalty cost for the cancel and a positive constant.

Let 𝒯0, be the set of stopping times with respect to filtration defined on the nonnegative interval. Letting α and ψ be some given parameters satisfying α>0 and ψ1, the value function of the callable Russian option V(ψ) is defined byV(ψ)=infσ𝒯0,supτ𝒯0,E^[eα(στ)R(σ,τ)],α>0.The infimum and supremum are taken over all stopping times σ and τ, respectively.

We define two sets A and B asA={ψR+V(ψ)=ψ+δ},B={ψR+V(ψ)=ψ}.A and B are called the seller's cancel region and the buyer's exercise region, respectively. Let σAψ and τBψ be the first hitting times that the process Ψt(ψ) is in the region A and B, that is,σAψ=inf{t>0Ψt(ψ)A},τBψ=inf{t>0Ψt(ψ)B}.Lemma 2.1.

Assume that d(1/2)κ22r<0. Then, one has limtertΨt(ψ)=0.

Proof.

First, suppose that max(ψx,supSu)=ψx. Then, it holdslimtertSt1=limtexp{κW˜t+(d+12κ22r)t}=0.Next, suppose that max(ψx,supSu)=supSu. By the same argument as Karatzas and Shreve [1, page 65], we obtainlimtsupSu=xexp{κ·sup0<u<(W˜u+rdκ12κ2)}=xexp{κW}, where W is the standard Brownian motion which attains the supremum in (2.15). Therefore, it follows thatlimtertsupSuSt=0.The proof is complete.

By this lemma, we may apply Proposition 3.3 in Kifer . Therefore, we can see that the stopping times σ^ψ=σAψ and τ^ψ=τBψ attain the infimum and the supremum in (2.10). Then, we haveV(ψ)=E^[eα(σ^ψτ^ψ)R(σ^ψ,τ^ψ)].And V(ψ) satisfies the inequalitiesψV(ψ)ψ+δ,which provides the lower and the upper bounds for the value function of the callable Russian option. Let VR(ψ) be the value function of Russian option. And we know V(ψ)VR(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 δδ, the seller never cancels. Therefore, callable Russian options are reduced to Russian options.

Proof.

We set h(ψ)=V(ψ)ψδ. h(ψ)=V(ψ)1<0. Because we know h(1)=V(1)1δ=δδ<0 by the condition δδ, we have h(ψ)<0, that is, V(ψ)<ψ+δ holds. By using the relation V(ψ)VR(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 δδ.

Lemma 2.3.

Suppose r>d. Then, the function V(ψ) is Lipschitz continuous in ψ. And it holds0dV(ψ)dψ1.

Proof.

SetJψ(σ^ψ,τ^ϕ)=E^[eα(σ^ψτ^ψ)R(σ^ψ,τ^ψ)].Replacing the optimal stopping times σ^ϕ and τ^ψ from the nonoptimal stopping times σ^ψ and τ^ϕ, we haveV(ψ)Jψ(σ^ψ,τ^ϕ),V(ϕ)Jϕ(σ^ψ,τ^ϕ),respectively. Note that z1+z2+(z1z2)+. For any ϕ>ψ, we have0V(ϕ)V(ψ)Jϕ(σ^ψ,τ^ϕ)Jψ(σ^ψ,τ^ϕ)=E^[eα(σ^ψτ^ϕ)(Ψσ^ψτ^ϕ(ϕ)Ψσ^ψτ^ϕ(ψ))]=E^[eα(σ^ψτ^ϕ)Hσ^ψτ^ϕ1((ϕsupHu)+(ψsupHu)+)](ϕψ)E^[eα(σ^ψτ^ϕ)Hσ^ψτ^ϕ1],where Ht=exp{(rd+(1/2)κ2)t+κW^t}. Since the above expectation is less than 1, we have0V(ϕ)V(ψ)ϕψ.This 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˜={ψR+VR(ψ)=ψ} be the exercise region of Russian option. By the inequality V(ψ)VR(ψ), it holds BB˜. 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 Ψt(ψ)Ψ0(ψ)=ψ1. From this, it follows that the seller's optimal boundary A is a point {1}. The function V(ψ) is represented byV(ψ)={Vψ(l),1ψl,ψ,ψl,whereVψ(l)=(1+δ)E^[eασ1ψ1{σ1ψ<τ[l,)ψ}]+lE^[eατ[l,)ψ1{τ[l,)ψσ1ψ}].In order to calculate (2.25), we prepare the following lemma.Lemma 2.4.

Let σax and τbx be the first hitting times of the process St(x) to the points {a} and {b}. Set ν=(rd)/κ(1/2)κ,η1=(1/κ)(ν2+2α+ν), and η2=(1/κ)(ν2+2αν). Then for a<x<b, one hasE˜[eασax1{σax<τbx}]=(b/x)η1(x/b)η2(b/a)η1(a/b)η2,E˜[eατbx1{τbx<σax}]=(a/x)η1(x/a)η2(a/b)η1(b/a)η2.

Proof.

First, we prove (2.26). DefineLt=exp(12ν2tνW˜t).We define P^ as dP^=LTdP˜. By Girsanov's theorem, W^tW˜t+νt is a standard Brownian motion under the probability measure P^. Let Tρ1 and Tρ2 be the first time that the process W^t hits ρ1 or ρ2, respectively, that is,Tρ1=inf{t>0W^t=ρ1},Tρ2=inf{t>0W^t=ρ2}.Since we obtain logSt(x)=logx+κW^t from St(x)=xexp(κW^t), we haveσax=Tρ1,a.s.,ρ1=1κlogax,τbx=Tρ2,a.s.,ρ2=1κlogbx,LTρ11=exp(12ν2Tρ1+νW˜Tρ1)=exp(12ν2Tρ1+νW^Tρ1)=exp(12ν2Tρ1+νρ1).Therefore, we haveE˜[eασax1{σax<τbx}]=E˜[eαTρ11{Tρ1<Tρ2}]=E^[exp(12ν2Tρ1+νρ1)eαTρ11{Tρ1<Tρ2}]=eνρ1E^[exp{(α+12ν2)Tρ1}1{Tρ1<Tρ2}].From Karatzas and Shreve [8, Exercise 8.11, page 100], we can see thatE^[exp{(α+12ν2)Tρ1}1{Tρ1<Tρ2}]=sinhρ2ν2+2αsinh(ρ2ρ1)ν2+2α.Therefore, we obtainE˜[eασax1{σax<τbx}]=sinhρ2ν2+2αsinh(ρ2ρ1)ν2+2αeνρ2eν(ρ2ρ1)=eκρ2γ1eκρ2γ2eκ(ρ2ρ1)γ1eκ(ρ2ρ1)γ2=(b/x)γ1(x/b)γ2(b/a)γ1(a/b)γ2.We omit the proof of (2.27) since it is similar to that of (2.26).

We study the boundary point l of the exercise region for the buyer. For 1<ψ<l<, we consider the function V(ψ,l). It is represented byV(ψ,l)={Vψ(l),1ψl,ψ,ψl.The family of the functions {V(ψ,l),1<ψ<l} satisfiesV(ψ)=V(ψ,l)=sup1<ψ<lV(ψ,l).To get an optimal boundary point l, we compute the partial derivative of V(ψ,l) with respect to l, which is given by the following lemma.Lemma 2.5.

For any 1<ψ<l, one hasVl(ψ,l)=ψη2ψη1l(lη1lη2)2lη1lη2{(1η2)lη1+1(1+η1)lη2+1+(1+δ)(η1+η2)}.

Proof.

First, the derivative of the first term isl((l/ψ)η1(ψ/l)η2lη1lη2)=1(lη1lη2)2{(η1(lψ)η111ψ+η2(ψl)η21l)(lη1lη2)((lψ)η1(ψl)η2)(η1lη11+η2lη21)}=1l(lη1lη2)2(η1+η2){lη1(ψl)η2lη2(lψ)η1}=1l(lη1lη2)2(η1+η2)lη1lη2(ψη2ψη1).Next, the derivative of the second term isl(llη2lη1)=(1η2)lη2(1+η1)lη1(lη2lη1)2=(1η2)lη1(1+η1)lη2(lη1lη2)2lη1η2,where the last equality follows from the relation(lη2lη1)lη11lη2+1=lη1lη2.After multiplying (2.37) by (1+δ) and (2.38) by ψη2ψη1, we obtain (2.36).

We setf(l)=(1η2)lη1+1(1+η1)lη2+1+(1+δ)(η1+η2).Since f(1)=δ(η1+η2)>0 and f()=, the equation f(l)=0 has at least one solution in the interval (1,). We label all real solutions as 1<ln<ln1<<l1<. Then, we haveVl(ψ,l)|l=li=0,i=1,,nψ.Then l=l1 attains the supremum of V(ψ,l). In the following, we will show that the function V(ψ) is convex and satisfies smooth-pasting condition.Lemma 2.6.

V(ψ) is a convex function in ψ.

Proof.

From (2.50), V satisfies12κ2ψ2d2Vdψ2=(rd)ψdVdψ+αV(ψ).If rd, we get d2V/dψ2>0. Next assume that r>d. We consider function V˜(ψ)=V(ψ) for ψ<0. Then,12κ2ψ2d2V˜dψ2(rd)ψdV˜dψrV˜=12κ2ψ2d2Vdψ2+(rd)ψdVdψrV=0.Since we find that d2V˜/dψ2>0 from the above equation, V˜ is a convex function. It follows from this the fact that V is a convex function.

Lemma 2.7.

V(ψ) satisfiesdVdψ(l)=dVdψ(l+)=1.

Proof.

Since V(ψ)=ψ for ψ>l, it holds (dV/dψ)(l+)=1. For 1ψ<l, we derivative (2.47):dVdψ=llη2lη1(η2ψη21+η1lη11)+1+δlη1lη2(η1(lψ)η11ψη2(ψl)η21ψ)=1ψ(lη1lη2){lη1η2+1(η2ψη2+η1ψη1)(1+δ)(η1(lψ)η1+η2(ψl)η2)}=1ψ(lη1lη2){η2(ψl)η2lη1+1+η1(lψ)η1lη2+1(1+δ)(η1(lψ)η1+η2(ψl)η2)}.Therefore, we getdVdψ(l)1=1(lη1+1l1η2){η2lη1+1+η1lη2+1(1+δ)(η1+η2)(lη1+1lη2+1)}=1(lη1lη2){(η21)lη1+1+(η1+1)lη2+1(1+δ)(η1+η2)}=1(lη1lη2)f(l)=0.This completes the proof.

Therefore, we obtain the following theorem.Theorem 2.8.

The value function of callable Russian option V(ψ) is given byV(ψ)={(1+δ)(l/ψ)η1(ψ/l)η2lη1lη2+lψη2ψη1lη2lη1,1ψl,ψ,ψl.And the optimal stopping times areσ^ψ=inf{t>0Ψt(ψ)=1},τ^ψ=inf{t>0Ψt(ψ)l}.The optimal boundary for the buyer l is the solution in (1,) to f(l)=0, wheref(l)=(1η2)lη1+1(1+η1)lη2+1+(1+δ)(η1+η2).

We can get (2.47) by another method. For 1<ψ<l, the function V(ψ) satisfies the differential equation12κ2ψ2d2Vdψ2+(rd)ψdVdψαV(ψ)=0.Also, we have the boundary conditions as follows:V(1)=C1+C2=1+δ,V(l)=C1lλ1+C2lλ2=l,V(l)=C1λ1lλ11+C2λ2lλ21=1.The general solution to (2.50) is represented byV(ψ)=C1ψλ1+C2ψλ2,where C1 and C2 are constants. Here, λ1 and λ2 are the roots of12κ2λ2+(rd12κ2)λα=0.Therefore, λ1,λ2 areλ1,2=±ν2+2ανκ.

From conditions (2.51) and (2.52), we getC1=l(δ+1)lλ2lλ1lλ2,C2=(δ+1)lλ1llλ1lλ2.And from (2.57) and (2.53), we have(1η2)lη1+1(1+η1)lη2+1+(1+δ)(η1+η2)=0.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: α=0.5,r=0.1,d=0.09,κ=0.3,δ=0.03.

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

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

δrdκαl
0.010.10.090.30.51.04337
0.020.10.090.30.51.0616
0.030.10.090.30.51.07568

0.030.20.090.30.51.08246
0.030.30.090.30.51.09228
0.030.40.090.30.51.10842
0.030.50.090.30.51.14367

0.030.10.010.30.51.08092
0.030.10.050.30.51.07813
0.030.10.10.30.51.07511
0.030.10.30.30.51.06633
0.030.10.50.30.51.06061

0.030.10.090.10.51.02468
0.030.10.090.20.51.04997
0.030.10.090.40.51.1018
0.030.10.090.50.51.12833

0.030.10.090.30.11.18166
0.030.10.090.30.21.12312
0.030.10.090.30.31.09901
0.030.10.090.30.41.08505

Optimal boundary for the buyer.

The value functionV(ψ) (δ=0.03).

The value functionV(ψ) (δ=0.01,0.02,0.03).

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 d 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 d 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.

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