# Callable Russian Options and  Their Optimal Boundaries

Authors: Atsuo Suzuki; Katsushige Sawaki
Journal: Journal of Applied Mathematics and Decision Sciences
 (2009)

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## 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.


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## Body

## 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 Section2, 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. LetBt be the bond
price at time t which is given
by(2.1)dBt=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 equation(2.2)dSt=(r−d)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,ℱ,P˜). Solving (2.2) with the initial condition S0=x gives(2.3)St=xexp{(r−d−12κ2)t+κW˜t}.Define another probability
measure P^ by(2.4)dP^dP˜=exp(κW˜t−12κ2t).Let(2.5)W^t=W˜t−κt,where W^t is a standard
Brownian motion with respect to P^. Substituting (2.5) into (2.2), we
get(2.6)dSt=(r−d+κ2)Stdt+κStdW^t.Solving the above equation, we
obtain(2.7)St(x)=xexp{(r−d+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 set(2.8)Ψt(ψ)≡max(ψx,sup0≤u≤tSu)St,ψ≥1.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 by(2.9)R(σ,τ)=(Ψσ(ψ)+δ)1{σ<τ}+Ψτ(ψ)1{τ≤σ},where δ is the penalty
cost for the cancel and a positive constant.Let𝒯0,∞ 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
by(2.10)V(ψ)=infσ∈𝒯0,∞supτ∈𝒯0,∞E^[e−α(σ⋀τ)R(σ,τ)],α>0.The infimum and
supremum are taken over all stopping times σ and τ, respectively.We define two
setsA and B as(2.11)A={ψ∈R+∣V(ψ)=ψ+δ},B={ψ∈R+∣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,(2.12)σAψ=inf{t>0∣Ψt(ψ)∈A},τBψ=inf{t>0∣Ψt(ψ)∈B}.Lemma 2.1.
Assume thatd−(1/2)κ2−2r<0. Then, one has
(2.13)limt→∞e−rtΨt(ψ)=0.Proof.
First, suppose thatmax(ψx,supSu)=ψx. Then, it holds(2.14)limt→∞e−rtSt−1=limt→∞exp{−κW˜t+(d+12κ2−2r)t}=0.Next, suppose that max(ψx,supSu)=supSu. By the same argument as Karatzas and Shreve [1, page 65], we
obtain(2.15)limt→∞supSu=xexp{κ·sup0<u<∞(W˜u+r−dκ−12κ2)}=xexp{κW∗}, where W∗ is the standard
Brownian motion which attains the supremum in (2.15). Therefore, it follows that(2.16)limt→∞e−rtsupSuSt=0.The 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=σAψ and τ^ψ=τBψ attain the
infimum and the supremum in (2.10). Then, we have(2.17)V(ψ)=E^[e−α(σ^ψ⋀τ^ψ)R(σ^ψ,τ^ψ)].And V(ψ) satisfies the
inequalities(2.18)ψ≤V(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 costδ 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.
Setδ∗=V(1)−1. If δ≥δ∗, the seller never cancels. Therefore, callable
Russian options are reduced to Russian options.Proof.
We seth(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)≤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 δ≥δ∗.Lemma 2.3.
Supposer>d. Then, the function V(ψ) is Lipschitz
continuous in ψ. And it holds(2.19)0≤dV(ψ)dψ≤1.Proof.
Set(2.20)Jψ(σ^ψ,τ^ϕ)=E^[e−α(σ^ψ⋀τ^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 have(2.21)V(ψ)≥Jψ(σ^ψ,τ^ϕ),V(ϕ)≤Jϕ(σ^ψ,τ^ϕ),respectively. Note that z1+−z2+≤(z1−z2)+. For any ϕ>ψ, we have(2.22)0≤V(ϕ)−V(ψ)≤Jϕ(σ^ψ,τ^ϕ)−Jψ(σ^ψ,τ^ϕ)=E^[e−α(σ^ψ⋀τ^ϕ)(Ψσ^ψ⋀τ^ϕ(ϕ)−Ψσ^ψ⋀τ^ϕ(ψ))]=E^[e−α(σ^ψ⋀τ^ϕ)Hσ^ψ⋀τ^ϕ−1((ϕ−supHu)+−(ψ−supHu)+)]≤(ϕ−ψ)E^[e−α(σ^ψ⋀τ^ϕ)Hσ^ψ⋀τ^ϕ−1],where Ht=exp{(r−d+(1/2)κ2)t+κW^t}. Since the above expectation is less than 1, we
have(2.23)0≤V(ϕ)−V(ψ)≤ϕ−ψ.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. LetB˜={ψ∈R+∣VR(ψ)=ψ} be the exercise
region of Russian option. By the inequality V(ψ)≤VR(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 Ψt(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
by(2.24)V(ψ)={Vψ(l∗),1≤ψ≤l∗,ψ,ψ≥l∗,where(2.25)Vψ(l∗)=(1+δ)E^[e−ασ1ψ1{σ1ψ<τ[l∗,∞)ψ}]+lE^[e−ατ[l∗,∞)ψ1{τ[l∗,∞)ψ≤σ1ψ}].In order to calculate (2.25), we
prepare the following lemma.Lemma 2.4.
Letσax and τbx be the first
hitting times of the process St(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(2.26)E˜[e−ασax1{σax<τbx}]=(b/x)η1−(x/b)η2(b/a)η1−(a/b)η2,(2.27)E˜[e−ατbx1{τbx<σax}]=(a/x)η1−(x/a)η2(a/b)η1−(b/a)η2.Proof.
First, we prove (2.26). Define(2.28)Lt=exp(−12ν2t−νW˜t).We define P^ as dP^=LTdP˜. By Girsanov's theorem, W^t≡W˜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,(2.29)Tρ1=inf{t>0∣W^t=ρ1},Tρ2=inf{t>0∣W^t=ρ2}.Since we obtain logSt(x)=logx+κW^t from St(x)=xexp(κW^t), we have(2.30)σax=Tρ1,a.s.,ρ1=1κlogax,τbx=Tρ2,a.s.,ρ2=1κlogbx,LTρ1−1=exp(12ν2Tρ1+νW˜Tρ1)=exp(−12ν2Tρ1+νW^Tρ1)=exp(−12ν2Tρ1+νρ1).Therefore, we
have(2.31)E˜[e−ασax1{σax<τbx}]=E˜[e−αTρ11{Tρ1<Tρ2}]=E^[exp(−12ν2Tρ1+νρ1)e−αTρ11{Tρ1<Tρ2}]=eνρ1E^[exp{−(α+12ν2)Tρ1}1{Tρ1<Tρ2}].From Karatzas and Shreve
[8, Exercise 8.11,
page 100], we can see that(2.32)E^[exp{−(α+12ν2)Tρ1}1{Tρ1<Tρ2}]=sinhρ2ν2+2αsinh(ρ2−ρ1)ν2+2α.Therefore, we
obtain(2.33)E˜[e−ασax1{σax<τbx}]=sinhρ2ν2+2αsinh(ρ2−ρ1)ν2+2αeνρ2eν(ρ2−ρ1)=eκρ2γ1−e−κρ2γ2eκ(ρ2−ρ1)γ1−e−κ(ρ2−ρ1)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 pointl∗ of the exercise
region for the buyer. For 1<ψ<l<∞, we consider the function V(ψ,l). It is represented by(2.34)V(ψ,l)={Vψ(l),1≤ψ≤l,ψ,ψ≥l.The family of the functions {V(ψ,l),1<ψ<l} satisfies(2.35)V(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 any1<ψ<l, one has(2.36)∂V∂l(ψ,l)=ψη2−ψ−η1l(lη1−l−η2)2lη1l−η2{(1−η2)lη1+1−(1+η1)l−η2+1+(1+δ)(η1+η2)}.Proof.
First, the derivative of the first
term is(2.37)∂∂l((l/ψ)η1−(ψ/l)η2lη1−l−η2)=1(lη1−l−η2)2{(η1(lψ)η1−11ψ+η2(ψl)η21l)(lη1−l−η2)−((lψ)η1−(ψl)η2)(η1lη1−1+η2l−η2−1)}=1l(lη1−l−η2)2(η1+η2){lη1(ψl)η2−l−η2(lψ)η1}=1l(lη1−l−η2)2(η1+η2)lη1l−η2(ψη2−ψ−η1).Next, the derivative of the
second term is(2.38)∂∂l(llη2−l−η1)=(1−η2)lη2−(1+η1)l−η1(lη2−l−η1)2=(1−η2)lη1−(1+η1)l−η2(lη1−l−η2)2lη1−η2,where the last equality follows
from the relation(2.39)(lη2−l−η1)lη1−1l−η2+1=lη1−l−η2.After multiplying (2.37) by (1+δ) and (2.38) by ψη2−ψ−η1, we obtain (2.36).We set(2.40)f(l)=(1−η2)lη1+1−(1+η1)l−η2+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<ln−1<⋯<l1<∞. Then, we have(2.41)∂V∂l(ψ,l)|l=li=0,i=1,…,n∀ψ.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 satisfies(2.42)12κ2ψ2d2Vdψ2=−(r−d)ψdVdψ+αV(ψ).If r≤d, we get d2V/dψ2>0. Next assume that r>d. We consider function V˜(ψ)=V(−ψ) for ψ<0. Then,(2.43)12κ2ψ2d2V˜dψ2−(r−d)ψdV˜dψ−rV˜=12κ2ψ2d2Vdψ2+(r−d)ψdVdψ−rV=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) satisfies(2.44)dVdψ(l∗−)=dVdψ(l∗+)=1.Proof.
SinceV(ψ)=ψ for ψ>l∗, it holds (dV/dψ)(l∗+)=1. For 1≤ψ<l∗, we derivative (2.47):(2.45)dVdψ=llη2−l−η1(η2ψη2−1+η1l−η1−1)+1+δlη1−l−η2(−η1(lψ)η11ψ−η2(ψl)η21ψ)=1ψ(lη1−l−η2){lη1−η2+1(η2ψη2+η1ψ−η1)−(1+δ)(η1(lψ)η1+η2(ψl)η2)}=1ψ(lη1−l−η2){η2(ψl)η2lη1+1+η1(lψ)η1l−η2+1−(1+δ)(η1(lψ)η1+η2(ψl)η2)}.Therefore, we
get(2.46)dVdψ(l∗)−1=1(l∗η1+1−l∗1−η2){η2l∗η1+1+η1l∗−η2+1−(1+δ)(η1+η2)−(l∗η1+1−l∗−η2+1)}=1(l∗η1−l∗−η2){(η2−1)l∗η1+1+(η1+1)l∗−η2+1−(1+δ)(η1+η2)}=1(l∗η1−l∗−η2)f(l∗)=0.This completes the proof.Therefore, we obtain the following theorem.Theorem 2.8.
The value function of callable Russian optionV(ψ) is given
by(2.47)V(ψ)={(1+δ)(l∗/ψ)η1−(ψ/l∗)η2l∗η1−l∗−η2+l∗ψη2−ψ−η1l∗η2−l∗−η1,1≤ψ≤l∗,ψ,ψ≥l∗.And the optimal stopping times
are(2.48)σ^ψ=inf{t>0∣Ψt(ψ)=1},τ^ψ=inf{t>0∣Ψt(ψ)≥l∗}.The optimal boundary for the
buyer l∗ is the solution
in (1,∞) to f(l)=0, where(2.49)f(l)=(1−η2)lη1+1−(1+η1)l−η2+1+(1+δ)(η1+η2).We can get (2.47) by
another method. For 1<ψ<l, the function V(ψ) satisfies the
differential equation(2.50)12κ2ψ2d2Vdψ2+(r−d)ψdVdψ−αV(ψ)=0.Also, we have the boundary
conditions as follows:(2.51)V(1)=C1+C2=1+δ,(2.52)V(l)=C1lλ1+C2lλ2=l,(2.53)V′(l)=C1λ1lλ1−1+C2λ2lλ2−1=1.The general solution to (2.50)
is represented by(2.54)V(ψ)=C1ψλ1+C2ψλ2,where C1 and C2 are constants.
Here, λ1 and λ2 are the roots
of(2.55)12κ2λ2+(r−d−12κ2)λ−α=0.Therefore, λ1,λ2 are(2.56)λ1,2=±ν2+2α−νκ.From conditions (2.51) and (2.52), we
get(2.57)C1=l−(δ+1)lλ2lλ1−lλ2,C2=(δ+1)lλ1−llλ1−lλ2.And from (2.57) and (2.53), we
have(2.58)(1−η2)lη1+1−(1+η1)l−η2+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.Figure1 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.Table 1
Penaltyδ, interest rate r, dividend rate d, volatility κ, discount factor α, and the optimal boundary for the buyer l∗.
δrdκαl∗0.010.10.090.30.51.043370.020.10.090.30.51.06160.030.10.090.30.51.075680.030.20.090.30.51.082460.030.30.090.30.51.092280.030.40.090.30.51.108420.030.50.090.30.51.143670.030.10.010.30.51.080920.030.10.050.30.51.078130.030.10.10.30.51.075110.030.10.30.30.51.066330.030.10.50.30.51.060610.030.10.090.10.51.024680.030.10.090.20.51.049970.030.10.090.40.51.10180.030.10.090.50.51.128330.030.10.090.30.11.181660.030.10.090.30.21.123120.030.10.090.30.31.099010.030.10.090.30.41.08505Figure 1
Optimal boundary for the buyer.Figure 2
The value functionV(ψ) (δ=0.03).Figure 3
The value functionV(ψ) (δ=0.01,0.02,0.03).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 valued 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.



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