The Valuation of Discrete-Time Contingent Claims with Upper and Lower Bounds; Revisited with Refinements (Financial Modeling and Analysis)

The

Valuation of Discrete-Time Contingent

Claims

with

Upper

and Lower

Bounds;

Revisited with

Refinements

南山大学大学院ビジネス研究科 澤木 勝茂 (Katsushige SAWAKI)

Graduate School of Business Administration,

Nanzan University

Abstract

In thispaper wedeal with thevaluation ofcallable-putable contingentclaims with general

payoff functions under thesetting ofan optimalstoppingproblembetween theseller and the buyer. The seller can cancel the claim issued by him/her as well as the buyer can exercise the right. Such claimsmayprovidethe upper boundof the lossto theseller andthe lower of the payoffto the buyer, respectively. We derivethe valuationformula of the callable-putable claims where the asset price follows a random walk. Some analytical properties ofoptimal

stopping rules and the valuefunction are investigated in more detail.

1

Introduction

We consider a financial market consisting of a riskless asset and of a risky asset over the

discrete time horizon $t=0,1,2\cdots,$$T$. Suppose that a new callable contingent claim (hereafter

abbreviated by CC) has been issued by the firm into the market. The callable CC enable the

seller to cancel by paying an extra penalty to the buyer. On the other hand, the buyer can

exercise the right at any time up to the maturity. The game option introduced by Kifer [11] is

one of such securities. Callable convertible bonds, liquid yield option notes and callable stock

options are examples of such financial derivatives (see [1], [8], [14], [19]and [20]).

In this paperwe deal with avaluation model of such callable CC where the payoff functions

are more general and different from the payoff if both of the buyer and seller do not exercise their

right before the maturity. The decision making related to callable CC consists of the selection

of the cancellation time by the seller and the exercise time by the buyer, that is, a pair oftwo

stopping times. When the seller stops at a time before the buyer does, the seller must pay to

the buyer more than when the buyer stops before the seller does. When either ofthem do not

stop before the maturity, then the payoff would turn out to be intermediate.

This paper is organized as follows. Section 2 sets up a discrete time valuation model for

callable CC whose payoff functionsare more general. Insection 3 we derive optimal policies and

investigate their analytical properties by using contraction mappings. In section 4 we discuss

a special

case

of binominal price processes to derive the specific stop and continue regions. In

section 5, concluding remarks are given together with some directions for the future research.

2

Pricing Model

We consider the discrete time case where the capital market consists of riskless bond $B_{t}$ with

interest rate $r_{t}$ at time $t$, so that

and ofa risky asset whose price $S_{t}$ at time $t$ equals

$S_{t}=S_{0}\Pi_{k=1}^{t}(1+\rho_{k})=S_{t-1}(1+\rho_{t})$ (2.2)

where $\rho_{k}(\omega)=\frac{1}{2}(d+u_{k}+\omega_{k}(u_{k}-d_{k}))$, $=\omega(\omega_{1}, \omega_{2}, \cdots, \omega_{T})\in\Omega\{1, -1\}^{T}$ which is the sample

space of finite sequences $\omega$ with the product probability $P=\{(p_{k}, 1-p_{k})\}^{T}$.

To exclude an arbitrage opportunity

as

usual, we

assume

for each $k$

$-1<d_{k}<r_{k}<u_{k},$ $0<p_{k}<1$

.

(2.3)

The equivalent martingale probability $p^{*}$ with respect to_$p$ is given by

$p_{k}^{*}= \frac{r_{k}-d}{u-d},$ $q_{k}^{*}=1-p_{k}^{*}$

.

It is clear that $E^{*}(\rho_{k})=r_{k}$

Given aninitial wealth$w_{0}$, an investment strategy is a sequence ofportfolios$\pi=(\pi_{1}, \pi_{2}, \cdots, \pi_{T})$

at each time where a portfolio $\pi_{t}$ is a pair of $(\alpha_{t}, \beta_{t})$ with $\alpha_{t}$ and $\beta_{t}$ representing the amount of

risky asset and of riskless bond at time$t$, respectively. The wealth formed by the portfolio $\pi$ at

time $t$ is given by

$W_{t}^{\pi}=\alpha_{t}S_{t}+\beta_{t}B_{t},$ $t\geq 1$ (2.4)

with $W_{0}=w$ is given.

An investment strategy $\pi$ is called self-financing if

$\alpha_{1}S_{0}+\beta_{1}B_{0}=w$

and

St-l

$(\alpha_{t}-\alpha_{t-1})+B_{t-1}(\beta_{t}-\beta_{t-1})=0$, $t>1$

which means no cash-in and no cash-out from or to the external sources.

Let$\hat{W}_{t}^{\pi}=B_{t}^{-1}W_{t}^{\pi}$. Then, foraself-financing strategy$\pi$wehave$\hat{W}_{t}^{\pi}=w_{0}+\Sigma_{k=1}^{t}B_{k}^{-1}\alpha_{k}S_{k-1}(\rho_{k}-$

$r_{k})$whichis amartingaleu.r.t.$p^{*}$. Denote by $\mathcal{J}_{t,T}$ thefinite set of stopping timestaking values in

$\{t, t+1, \cdots, T\}$. A callable contingent claim is acontractbetween

an

issuer A and an investor $B$

addressingthe assetwitha maturity$T$. The issuer canchoose astoppingtime $\sigma$ to call backthe

claim with the payoff function$Y_{\sigma}$ and the investorcan also choose astopping time $\tau$ to exercise

his/her right with the payoff function $X_{\tau}$ at any time before the maturity. Should neither of

them stop before the maturity, the payoffshould be $Z_{t}$

.

The payoff always goes from the issuer

to the investor. We

assume

$0\leq X_{t}\leq Z_{t}\leq Y_{t},$ $0\leq t<T$

and

$X_{T}=Z_{T}$ (2.5)

The investor wishes to exercise the right so

as

to maximize the expected payoff. On the other

hand, theissuer wants to call the contract so

as

to minimizethe payment to the investor. Then,

for any pair of the stopping times $(\sigma, \tau)$, define the payoff function by

A hedge against a callable CC with a maturity $T$ is a pair $(\sigma, \pi)$ of a stopping time $\sigma$ and

a

self-financing investment strategy $\pi$ such that

$W_{\sigma\wedge t}^{\pi}\geq R(\sigma, t),$ $t=0,1,$ $\cdots,$$T$.

The price $v^{*}$ ofacallable CC is the infinum of_{$v\geq 0$} such that there exists ahedge $(\sigma, \pi)$ against

this callable CC with $W_{o}^{\pi}=v$.

Theorem 1 (Kifer [1$1J)$ Let $P^{*}=\{p_{t}^{*}, 1-p_{t}^{*}\}^{T}$ be the probability on the space $\Omega$ with _{$p_{t}^{*}=$}

$\perp r-arrow ur_{t}-d_{t}’ t\leq T<\infty$, and $E^{*}$ be the expectation with respect to $P^{*}$

.

Then, the price $v^{*}$

of

the

callable $CC$ equals $v_{0,T}^{*}$ which

can

be obtained

from

the recursive equations

as

follows;

$v_{T,T}^{*}=\Pi_{t=1}^{T}(1+r_{t})^{-1}Z_{T}$

and

$v_{t,T}^{*}= \min\{\Pi_{k=1}^{t}(1+r_{k})^{-1}Y_{t}, \max[\Pi_{k=1}^{t}(1+r_{k})^{-1}X_{t}, E^{*}(v_{t+1,N}^{*})]\}$ (2.7)

Moreover,

_for

$t=0,1,$$\cdots,$$T$

$v_{t,T}^{*}$ $=$ $\min_{\sigma\in J_{t,T}}\max_{\tau\in J_{t,T}}E^{*}[\Pi_{k=1}^{-\sigma\wedge\tau}(1+r_{k})^{-1}R(\sigma, \tau)|\Im_{t}]$

$=$ $\max_{\tau\in}\min_{J_{t,T}\sigma\in J_{t,T}}E^{*}[\Pi_{k=1}^{-\sigma\wedge\tau}(1+r_{k})^{-1}R(\sigma, \tau)|\Im_{t}]$, (2.8)

for

each $t=0,1,$ $\cdots,$$T$, the stopping times

$\sigma_{t,T}^{*}=\min\{k\geq t|\Pi_{l=1}^{k}(1+r_{l})^{-1}Y_{k}=v_{k,T}^{*}\}$ (2.9)

and

$\tau_{t,T}^{*}=\min\{k\geq t|\Pi_{l=1}^{k}(1+r_{l})^{-1}X_{k}=v_{k,T}^{*}\}$ (2.10)

belong to $\mathcal{J}_{t},\tau$ and $v_{T,T}^{*}=\Pi_{t=1}^{T}(1+r_{t})^{-1}Z_{T}$

.

The inequalities

$E^{*}[\Pi_{k=1}^{\sigma_{t,T}^{*}\wedge\tau}(1+r_{k})^{-1}R(\sigma_{t,T}^{*}, \tau)|\Im_{t}]$ $\leq$ $v_{t,T}^{*}$

$\sigma\wedge\tau^{*}$

$\leq$ $E^{*}[\Pi_{k=1}^{t,T}(1+r_{k})^{-1}R(\sigma, \tau_{t,T}^{*})|\Im_{t}]$ (2.11)

hold

_for

any $\sigma,$$\tau\in \mathcal{J}$

Remark 1 The model can be extended to the

_infinite

case $Tarrow\infty$, provided that _{$r_{k}=r$}

for

all$k$

$\lim_{Tarrow\infty}(1+r)^{-T}Y_{T}=0$ with $v_{T,T}=Z_{T}$ (2.12)

with $p^{*}$-probability 1.

If

$Y_{t}=(K-S_{t})+\delta_{t}$, then equation (2.12) can be replaced by

$\lim_{tarrow\infty}(1+r)^{-t}\delta_{t}=0$ (2.13)

Remark 2 Defining $Z_{t}=\Pi_{k=1}^{t}(1+r_{k})^{-1}W_{t}^{\pi}$, then we obtain

$Z_{t}=w+\Sigma_{k=1}^{t}\Pi_{l=1}^{k}(1+r_{l})^{-1}\alpha_{k}S_{k-1}(\rho_{k}-r_{k})$ (2.14)

which is a martingale w.r.t. $P^{*}=\{p^{*}, 1-p^{*}\}^{T}$

Corollary 1 Assume that equation (2.12) holds. Then, the limit value

$v^{*}= \lim_{Tarrow\infty}v_{0,T}^{*}$ (2.15)

exists.

3

Optimal Policies and their Analytic Properties

In this section we propose a different approach from Kifer [11] and Dynkin [6]. Since the

asset price process follows a (non-stationary) binominal process, the payoffprocesses of $X_{t}$ and

$Y_{t}$ are both Markov processes. So we formulate this optimal stopping problem

as a

Markov

decision process. In this section, we

assume

$r_{k}=r$ for all $k$ and put _{$\beta=(1+r)^{-1}$}

.

Let

$X_{t}=\beta^{t}X(S_{t}),$ $Y_{t}=\beta^{t}Y(S_{t})$ and $Z_{t}=\beta^{T}Z(S_{T})$. It follows from these new notations that $\prod_{k=1}^{t}(1+r_{k})^{-k}X_{t}=X(S_{t}),$ $\prod_{k=1}^{t}(1+r_{k})^{-k}Y_{t}=Y(S_{t})$ and $\prod_{k=1}^{t}(1+r_{k})^{-k}Z_{T}=Z(S_{T})$

.

Put $V_{0}(s)=Z(s)$ and define for $n\geq 1$ $v^{n+1}(s)$ $\equiv$ $(\mathcal{U}v^{n})(s)$

$\equiv$ $\min(Y(s), \max(X(s), \beta E_{s}[v^{n}(\tilde{S}_{n+1})]))$ (3.1)

where $E_{s}$ is the expectation $w.r.S_{n}=s$

Let $V$ be the set of all bounded functions and its limit with the norm $\Vert v\Vert=\sup_{s\in\Omega}|v(s)|$

.

For

$u,$$v\in V$ we write $u\leq v$ if$u(s)\leq v(s)$ for all $s\in\Omega$. A mapping is called a contraction mapping if

$\Vert \mathcal{U}u-\mathcal{U}v\Vert\leq\beta\Vert u-v\Vert$

for some $\beta<1$ and for all $u,$$v\in V$.

Lemma 1 The mapping$\mathcal{U}$ as

defined

by equation(3.1) is a contmction mapping

Proof.

For any $u,$ $v\in V$ we have

$(\mathcal{U}u)(s)-(\mathcal{U}v)(s)$ $=$ $\min(Y(s), \max(X(s), \beta E_{s}[u(\tilde{S})]))$

- $\min(Y(s), \max(X(s), \beta E_{s}[v(\tilde{S})]))$

$(\mathcal{U}u)(s)-(\mathcal{U}v)(s)$ $\leq$ $\min(Y(S), \beta E_{s}[u_{s}(\tilde{S})])$

$\max(X(S), \beta E_{s}[v_{s}(\tilde{S})])$

$\leq$ $\beta E_{s}[u(\tilde{S})]-\beta E_{s}[v_{s}(\tilde{S})]$

$\leq$ $\beta E_{s}[\sup(u(\tilde{S})-v(\tilde{S}))]$

Hence, we obtain

$\sup_{s\in\Omega}(\mathcal{U}u)(s)-(\mathcal{U}v)(s)\leq\beta\Vert u-v\Vert$. (3.2)

By taking the roles of$u$ and $v$ reversely, we have

$\sup_{s\in\Omega}(\mathcal{U}v)(s)-(\mathcal{U}u)(s)\leq\beta\Vert v-u\Vert$ (3.3)

Putting_{equation(3.2) and (3.3) together we obtain}

$\Vert \mathcal{U}u-\mathcal{U}v\Vert\leq\beta\Vert u-v\Vert$

$\square$

Corollary 2 There exists a unique

_function

$v\in V$ such that

$(\mathcal{U}v)(s)=v(s)$

for

all $s$ (3.4)

Furthermore,

_for

all$u\in V$

$(\mathcal{U}^{T}u)(s)arrow v(s)$ as $Tarrow\infty$

where $v(s)$ is equal to _the

fixed

_point

_defined

_{by equation(3.4), that is, $v(s)$ is a unique solution}

to

$v(s)= \min\{Y(s), \max(X(s), \beta E_{s}[v(\overline{S})])\}$

Since $\mathcal{U}$ is

a

contraction mapping from corollary 1, the optimal value function $v$ for the

perpetual contingent claim

can

be obtain

as

the limit by successively applyingan operator$\mathcal{U}$ to

any initial value function $v$ for a finite lived contingent claim.

Remark 3 When wespecialize thepriceprocess into the binominalprocess, theprobability space

can be reduced to $\Omega=\{0,1,2\cdots\}$ with a $\sigma$

–field

$\Im_{t}=i$ which represents the number

of

up-jumps by time $t$ and $P=(p, 1-p)$

Assumption 2

_If

$v(s)$ is monotone _in $s$, then $E_{s}v(\tilde{S})$ is monotone in

$s$.

Lemma

_{2 Suppose that Assumption 2 holds. Then,}

i$)$ $(\mathcal{U}^{n}v)(s)$ is monotone in _$s$

for

_{$v\in V$}

.

ii) $v$ satisfying $v=\mathcal{U}v$ is monotone in _$s$

.

iii) there exists a pair $(s_{n}^{*}, s_{n}^{**})$

of

the optimal boundaries such that

$v^{n+1}(s)\equiv(\mathcal{U}v^{n-1})(s)=\{\begin{array}{ll}Y(s) if s\geq s_{n}^{*}\max(X(s), \beta E_{s}[v^{n-1}(\tilde{S})]) if s\leq s_{n}^{**}, n=1,2, \cdots, T\end{array}$

with $v_{0}=Z$.

i$)$ The proof follows by induction on $n$. For $n=1$, we have

$( \mathcal{U}v^{0})(s)=\min\{Y(s), \max[X(s), \beta E_{s}Z(\tilde{S})]\}$

Suppose that $X(s),$$Y(s)$ and $Z(s)$ is increasing in $s$.

which is monotone in $s$, provided that Assumption 2 holds. Suppose that $v_{n}$ is monotone

for $n>1$. Then,

$v^{n+1}(s) \equiv(\mathcal{U}v^{n})(s)=\min\{X(s), \max(Y(s), \beta E_{s}[v^{n}(\tilde{S})])\}$

which is again monotone in $s$ since the maximum of the monotone functions is monotone.

ii) Since $\lim_{narrow\infty}(\mathcal{U}^{n}v_{o})(s)$ point-wisely converges to the limit $v(s)$ from corollary 2, the limit

function $v(s)$ is also monotone in $s$

.

iii) Should $v^{n}=(\mathcal{U}^{n-1}v_{0})(s)$ be monotone in $s$, then there exists at least

one

pairof boundary

values $s^{*}$ and $s_{n}^{**}$ such that

$v^{n}=\{\begin{array}{ll}Y(s) if s\geq s^{*}\max[X(s), \beta E_{s}(v^{n-1}(\tilde{S})] otherwise\end{array}$

$\max(X(s), E_{s}[v^{n-1}(\tilde{S})])=\{\begin{array}{ll}X(s) for s\leq s^{**}E_{s}[v^{n-1}(\tilde{S})] otherwise\end{array}$

$\square$

From equation(2.11), $v^{n}$ is monotone increasing in $n$ since $X_{n}(s)\leq v^{n}(s)\leq Y_{n}(s)$

.

Define for

the issuer

$S_{n}^{I}=\{s|V^{n}(s)=Y(s)\}$ (3.5)

$s_{n}^{*}= \inf\{s|s\in S_{n}^{I}\}$ (3.6)

and for the investor

$S_{n}^{II}=\{s|V^{n}(s)=X(s)\}$ (3.7)

$s_{n}^{**}= \inf\{s|s\in S_{n}^{II}\}$ (3.8)

It is easy to show that

$s_{n}^{*}\geq s_{n}^{**}$

for

each $n$ (3.9)

Remark 4 In game put options (Kifer $[$1$1],Kypr’ianou[13J)$ it is assumed that $X_{n}(S_{n})\equiv$

$\beta^{n}X(S_{n})$ and $Y_{n}(S_{n})=\beta^{n}(X(S_{n})+\delta)$ with $\delta>0$ where $X(S_{n})=(K-S_{n})^{+}$

.

_It is easy

4

A Simple

Random

Walk

Case

Suppose that the process $\{S_{t}, t=1,2, \cdots\}$ is a random walk, that is,

$S_{t+1}=S_{t}\cdot\tilde{X}_{t+1}$

where $\overline{X}_{1},\overline{X}_{2}\cdots$

are

independently distributed random variables with the finite

mean.

i$)$ We consider the

case

ofa callable call option where $X(s)=(s-K)^{+}$

and $Y(s)=X(s)+\delta,$ $\delta>0$

$\beta E_{s}(\tilde{S})=\beta s(1+p^{*}u+(1-p^{*})d)=\beta(1+r)s=s$

which is

a

martingale. So $\beta^{n}X(S_{n})=\max(\beta^{n}S-\beta^{n}K, 0)$ is

a

submartingale. Applying

the Optimal Sampling Theorem, we obtain that

$v_{t}(s)$ $=$

$\min_{\sigma\in J_{t,T}}\max_{\tau\in J_{t},\tau}E_{s}^{*}[\beta^{\sigma\wedge\tau}R(\sigma, \tau)]$

$=$ $\min$ $\max E_{S}^{*}[\beta^{\sigma\wedge\tau}(Y(S_{\sigma\wedge\tau})1_{\{\sigma<\tau\}}+X(S_{\sigma\wedge\tau})1_{\{\tau<\sigma\}}+Z_{T}1_{\{\sigma\wedge\tau=T\}})]$ $\sigma\in J_{t,T}\tau\in J_{t,T}$

$=$ _{$\min_{\sigma\in J_{t,T}}E_{s}^{*}[\beta^{\sigma}X(S_{\sigma})1_{\{\sigma<T\}}+\beta^{T}Z_{T}1_{\{\sigma=T\}}]$ (4.1)}

which can be represented in the following corollary;

Corollary 3

Callable-Putable

contingent claims with the maturity $T<\infty$ can be

degen-emted into callable ones.

This corollary correspondsto thewellknown result thatAmericancall options

are

identical

to the corresponding European call options. In the

case

of callable-putable call claims it

follows that the investor should exercise his$/her$ putable right at the maturity. However,

the issuer shouldchoose

an

optimal call stopping time which minimizethe expected payoff

function given by equation(4.1). From equation(2.11)

we

know that

$X_{t}\leq v_{t}\leq Y_{t}$

for

_{$0\leq t\leq T$}.

and the optimal stopping times for each $t=0,1,$$\cdots,$$T$ are

$\sigma_{t}^{*}=\min\{n\geq t : v_{t,T}=\beta^{n}Y_{n}(s)\}$_A$T$

and

$\tau_{t}^{*}=\{n\geq t:v_{t,T}(s)=\beta^{n}X_{n}(s)\}$.

Lemma 3 $V_{t}(s)-s$ is decreasing in $s$

for

each $t$ and decreasing in $t$

for

each _$s$

.

$S_{t}^{I}$ _{$=$ $\{s|v_{t}(s)-s\geq-k+\delta\}$}

for

$t<T$

.

$S_{T}^{I}=\phi$

$S_{t}^{II}$ _{$=$ $\{s|v_{t}(s)-s\leq-k\}=\phi$}

for

$t<TifE(\tilde{X})\geq 1$_.

Lemma 4 Let $X(s)= \max\{K-s, 0\},$$Y(s)=X(s)+\delta$ and $E(\overline{S}_{t})>-1$. _{$V_{t}(s)+s$} is

increasing in $s$

for

each $t$.

Proof.

For $t=T$ we have

$V_{T}(s)+s$ $=$ $\max\{X(s), 0\}+s$

$=$ $\max\{K-s, 0\}+s=\max\{K, s\}$

which is increasing in $s$

.

Suppose the assertion for $t+1$.

Then, putting$\mu=E(\tilde{S}_{t})$

$V_{t}(s)+s$ $=$ $\min\{(K-s)^{+}+\delta, \max\{(K-s)^{+}, \beta E_{s}V_{t+1}(s\tilde{S})+s\tilde{S}\}+\beta s(1+\mu)\}$

$=$ $\min\{K+\delta, \max[K, \beta E(V_{t+1}(s\tilde{S})+s\tilde{S})]+\beta s(1+\mu)\}$

$V_{t+1}(s\tilde{S})+s\tilde{S}$ is increasingin $s$ for all $\tilde{S}>0$and $s(1+\mu)$ is also increasingin$s$ for$\mu>-1$

.

So is $V_{t}(s)+s$. $\square$

For each $t$, define

$s_{t}^{I}$ _$=$ $\inf\{s:V_{t+1}(s)+s\geq K+\delta\}$

$s_{t}^{II}$ _$=$ $\inf\{s:V_{t+1}(s)+s\geq K\}$

where $s_{t}^{I}$ and $s_{t}^{II}$ equal $\infty$ when these sets

are

empty.

Lemma 5 $s_{t}^{I}$ and $s_{t}^{II}$ are increasing in $t$

Proof.

$s_{t}^{I}$ _$=$ $\inf\{s|V_{t}(s)+s\geq K+\delta\}$

$\leq$ $\inf\{s|V_{t+1}(s)+s\geq K+\delta\}$

$=$ $s_{t+1}^{I}$

$s_{t}^{II}$ _$=$ $\inf\{s|V_{t}(s)+s\geq K\}$

$\geq$ $\inf\{s|V_{t+1}(s)+s\geq K\}$

$=$ $s_{t}^{II}$

Lemma 6

_If

$\frac{K}{S}F(\frac{K}{S})>1-\int_{\frac{K\infty}{s}}xdF(x)$, it is never optimal

for

the investor to exercise

Proof.

$S_{T}^{I}$ $\equiv$ $\{s|V_{T}(s)=K-s+\delta\}$

$=$ $\{s|\max(K-s, 0)+s=K+\delta\}$

$=$ $\{\phi\}$

$S_{T}^{II}$ $\equiv$ $\{s|V_{\tau}(s)=K-s\}$

$=$ $\{s|\max(K-s, 0)=K-s\}$

$=$ $\{K\}$

$V_{T-1}(s)$ $=$ $\min\{K-s+\delta, \max(K-s, \beta E[V_{T}(sX_{t})])\}$

$=$ $\min\{K-s+\delta, \max(K-s, \beta E\max(K-s, 0))\}$

$=$ $\min\{K-s+\delta,$$\max(K-s,$$\beta K\int_{0}^{\frac{K}{s}}dF(x)-\beta\int_{0}^{\frac{K}{s}}sxdF(x)\}$

$=$ $\min\{K-s+\delta,$$\max(K-s,$$\beta KF(\frac{K}{S})-\beta\int_{0}^{\frac{K}{s}}sxdF(x)\}$

$\beta KF(\frac{K}{S})-\beta s\int_{0}^{\frac{K}{s}}sxdF(s)$ _$=$

$\beta KF(\frac{K}{S})-\beta s(\int_{0}^{\infty}xdF(x)-\int_{\frac{K}{s}}^{\infty}xdF(x))$

$\leq$

$\beta KF(\frac{K}{S})-\beta s\mu+\beta s\int_{\frac{K}{s}}^{\infty}\frac{K}{s}dF(x))$

$\leq$ $\beta KF-\beta s\mu+\beta K(1-F)$

$=$ $\beta K-\beta s\mu\mu>-1$

$=$ $\beta(K-s\mu)$ _$-\mu>1$ $\leq$ $\beta(K-s)$ Hence, $V_{T-1}(s)$ $=$ $\min\{K-s+\delta, \beta(K-s)\}$ $<$ $K-s+\delta if(K-s+\delta)<\beta(K-s)$ $S_{T-1}^{I}$ $=$ $\{\phi\}$ $\square$ Theorem 3

i$)$ There exists an optimal call policy

for

the issuer asfollows;

If

the asset price is $s$ at time $t$ and $s>s_{t}^{I}$, then the issuer call the contingent claim.

ii) There exists

an

optimal exercise policy

_for

the investor

as

follows;

If

the asset price is $s$ at time $t$ and $s\leq s_{t}^{II}$, the investor exercises the contingent

Since $X\leq v_{t,T}\leq Y_{2}$

for

each $t\leq T$, the issuer should stop or call

if

$s\in D_{t}^{I}$ and the

investor should exercise

_if

$s\in D_{t}^{II}$

Lemma 7

$C_{t}^{I}\supset C_{t+1}^{I}$, $C_{t}^{II}\subset C_{t+1}^{II}$

$C_{t}^{I}\subset C_{t}^{I}$ and $C_{t}^{II}\supset C_{t+1}^{II}$

The pmof directly

_{follows from}

the result that $v_{t_{I}T}$ is increasing in $t$

.

Let

$( \mathcal{U}^{t}v)(s)=\min\{Y(s), \max(X(s), \beta^{t}E_{s}[v(s\tilde{S}_{t})])\}$

Lemma 8

_If

there exists $\theta>1$ and $\delta>0$

for

which $E[X_{t}^{\theta}|X_{0}, \cdots, X_{t-1}]\leq e^{\delta}$

for

_$t=$

$1,2,$$\cdots,$$T$, then we obtain

$V_{t}(s)\geq f(s)$

where $f(s)$ _{is given by}

$f(s)=\{\begin{array}{ll}\frac{s^{\theta}(\theta-1)^{\theta-1}}{\theta^{\theta}} if s\leq\frac{\theta}{\theta-1}K-S otherwise\end{array}$

5

Concluding

Remark

In this paper we consider the discrete time valuation model for callable contingent claims in

which the asset price follows a random walk including a binominal process

as

a special case. It

is shown that such valuation model canbe formulated

as

acoupled optimal stopping problem of

a two person gamebetween the issuer and the investor. We show under some assumptions that

these existsasimpleoptimal call policy for the issuer and optimal exercise policy for the investor

which can be described by the control limit values. Also, we investigate analytical properties of

such optimal stopping rules for the issuer and the investor, respectively, possessing amonotone

property.

It is of interest to extend it to the three person games among the issuer, investor and the

their party like stake holders. Furthermore,

we

might analyze

a

dynamic versionof the modelby

introducing the state ofthe economy which follows a Markov chain. In this extended dynamic

version the optimal stopping rules

as

well

as

their value functions should depend on the state of

the economy. We shall discuss such a dynamic valuation model somewhere in a near future.

Acknowledgment

This paper

was

supportedin part by the Grant-in-Aid for Scientific Research (No. 20241037)

of the Japan Society for the Promotion ofScience in 2008-2012.

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