数理解析研究所講究録
1488 巻 2006 年 123-129

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Optimal Stopping Problems for a Processwith a Jump
YOSHIFUMI MUROI*Institute for Monetary and Economic Studies, Bank of Japan
2-1-1 Nihonbashi-Hongokucho Chuo-Ku, Tokyo 103-8660, JAPAN.

Abstract
In the last two decades, the market of credit derivatives has expanded rapidly, and theimportance of pricing problems for credit derivatives has been recognized especially in thelast decade. Among these securities, the pricing problems of credit derivatives with an
early exercise feature, such as American put options, have not received enough attentions,
In order to evaluate American options on defaultable bonds, it is necessarily to consider
the optimal stopping problems for a process with a jump. The existence of the optimal
stopping time for a process with a jump is considered in this article.

Key words: pricing American put on defaultable bonds, optimal stopping problem.
JEL$\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{l}\mathrm{c}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:\mathrm{C}61,\mathrm{G}\mathrm{l}2$1Introduction

The market of credit derivatives has been grown rapidly in thedeacade and thepricing problems of credit derivatives have attracted great interest both from academicdand practioners. Two main methologies have been used in the analysis of the pricing
problems of credit derivatives -the structural and reduced modeling approaches. The
pricing problems of contingent claims with credit risk with the reduced modeling approachhas been considered in Duffie and Singleton (1999), Jarrow and Turnbull (1995) and Muroi$(2002,2005)$ , for example. Although there have been many investigations on the theory ofcredit derivatives, little is known about the mathematical theories relating to American
put options on defaultable bonds. It is necessary to consider the optimal stopping problem
for a process with a jump to evaluate the American put options on defaultable bonds
within the reduced modeling approach, since the defaultable bond price process has a$1\mathrm{a}s\mathrm{t}$’Vicws expresscd in this paper arc those of the author and do not necesgarily reflect thc opinion of
Institutc for Monetary and Economic Studies nor Bank of Japan.

124jump in the default time. See Modecki (1999) and Zhang (1995) for discussions on related
issues. The existence of the optimal stopping time for a process with a jump is shown inthis article. In general, the optimal stopping time is not necessarily achieved if the original
process has jumps, and it is not easy to prove the existence of an optimal stopping time.
This is possible, however, when the default time is represented by a totally inaccessible
stopping time, as will be shown in this paper. This paper is based on the Section 5 ofMuroi (2002).

2Optima stopping problems for a process with aJump

In the pricing problem of American put options on defaultable bonds within the reduced
modeling approach, it is necessary to prove the existence of an optimal stopping time fora continuous time stochastic process which has ajump at a totally inaccessible stopping
time. The assumption of the default time of the issuer of a bond as a totally inaccessible
stopping time is necessary also to derive the fair value of American put options on defaultable bonds. Karatzas and Shreve (1999) (in their Appendix D) proved many resultsof the optimal stopping problem without a regularity condition on the left continuity ofthe sample path, this assumption still plays an important role in proving the existence ofan optimal stopping time. In this article, however, the existence of an optimal stopping
time will be proved without the left continuity of the sample paths, by assuming that thejump time is a totally inaccessible stopping time.
Two non-negative stochastic processes, $Y(t)$ and $Y’(t)$ , are defined on the probability space $(\Omega, F, F, P)$ , with the assumptions that the two stochastic processes are$F=\{F_{t};t\leq T\}$ adapted and that they have continuous sample paths. The filtration,, satisfies the usual condition of stochastic analysis, and a random variable, ,is defined as a totally inaccessible stopping time. A totally inaccessible stopping time$\{\mathcal{F}_{t}\}_{0\leq t<T}$$\tau$means$P(\{(v :\tau=\xi<\infty\})=0$

for any predictable stopping time, . In other words, a totally inaccessible stopping
time means that, for any sequence of stopping times, the probability that the sequence
approaches from below is . This permits the definition of a new random variable,,$\xi$$\tau$$0$$X(\cdot)$by$X(t)=Y(t)1_{\{t<\tau\}}+Y’(t)1_{\{\tau\leq t\}}$

.

(2.1)

, has a possible jump, but only at a totally inaccessible stopThis stochastic process,ping time (already defined as ). A further assumption, termed the American Regularity$X(\cdot)$$\tau$

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Condition by Duffie (2001, p. 182),(2.2)$E[ \sup_{0\leq\iota\leq T}X(t)]<\infty$

is required to guarantee the existence of an optimal stopping time.
is introduced,, termed the Snell envelope of
Next, a new stochastic process,. Also,the Snell envelope being the smallest RCLL supermartingale which dominates, is defined which satisfies
a new stochastic process,$X(\cdot)$$Z^{0}(\cdot)$$X(\cdot)$$Z(\cdot)$$Z(t)=ess \sup_{t\leq\rho\leq T}E[X(\rho)|F_{t}]$,where $X^{*}=ess \sup\chi$ is the essential supremum satisfying conditions of (i) $X\leq X^{*}a.s$ .
for any $X\in\chi$ and (ii) $X^{*}\leq Ya.s$ . for $X\leq Y$ (all $X\in\chi$ ), and is the stopping time.
Applying Theorem D.7 of Karatzas and Shreve (1999, p354) leads to$\rho$$Z^{0}(t)=Z(t)(a.s.)$ .

To show the existence of the stopping time, , such that the expected reward is maximized,
is an important aim in the optimal stopping problem. Theorem 2.1, a basic result of thispaper, will play an important role in the valuation problem of an American put option
on a defaultable bond.
$/j$

Theorem 2.1 There exists an optimal stopping time under the assumptions of (2.2) for$X(t)$ of (2.1). Moreover, the explicit representation of the optimal stopping time is given$by$$\eta=\inf\{t\in[v, T]|Z^{0}(t)=X(t)\}$

.

(2.3)

[Proof] Random variables are defined by$D^{\lambda}(v)= \inf_{t\in[v,T]}\{\mathrm{A}Z^{0}(t)\leq X(t)\}\wedge T$

A $\in[0,1)(a.s.)$

and$D_{*}(v)= \lim_{\lambda\uparrow 1}D^{\lambda}(v)=n\uparrow\infty 1_{1}’\mathrm{m}D^{(1-\frac{1}{n})}(v)(a.s.)$, are the stopping times”just before the optimal stopping time”
The stopping times,
and $D_{*}(v)$ is the limit of these stopping times. In a continuous time process, it is nottrivial whether $D_{*}(v)$ is an optimal stopping time or not unless there is the assumptionof continuity of sample paths (see Karatzas and Shreve (1999, p. 358) Theorem D.12).
is bounded and monotone
Under the assumptions of the present paper,increasing in the wide sense. Hence, the random variable, $D_{*}(v)$ , is well-defined. It is$D^{\lambda}(v)$$\{D^{(1-\frac{1}{n})}(v)\}_{n=1}^{\infty}$

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possible to derive the following relationship$Z^{0}(v)$$=E[Z^{0}(D^{(1-\frac{1}{n})}(v))|F_{v}]$$\leq$$\frac{n}{n-1}E[X(D^{(1-\frac{1}{t\mathrm{t}})}(v))|\mathcal{F}_{v}]$$=$$\frac{n}{n-1}\{E[1_{\{\tau-<D^{(1}n^{)}(v)\}}-\perp X(D^{(1-\frac{1}{n})}(v))|F_{v}]$$+ \sum_{j=n}^{\infty}E[1X(D^{(1-_{\dot{n}}^{\mathrm{L}})}(v))|F_{v}]\{D^{(1-\mathrm{A})}f(v)\leq\tau-<D^{(1-;\Leftrightarrow)}(v)\}$$+E[1-*)1_{\{\tau\leq D.(v)\}}X1((D^{(1-_{n})}(v))| \mathcal{F}_{v}]\bigcap_{j=n}^{\infty}\{DJ(v)\leq\tau-\}\perp$$+$$E[1_{\{D.(v)<\tau\}}X(D^{(1-\frac{1}{n})}(v))|F_{v}]\}(a.s.)$

.

(2.4)

The first equality of (2.4) is a direct application of Proposition D.10 of Karatzas andShreve (1999, p. 356) which holds in the general case of the RCLL process.
Next, it is necessary to show that the third term of (2.4) equals . The proof of thisfact is quite lengthy and this fact will be shown later as Lemma 2.1. Once, Lemma 2.1,
has been proved, then, as n–$0$$+\infty$$Z^{0}(v)$$=$$\frac{n}{n-1}\{E[1\iota_{)}X(D^{(1-\frac{1}{n})}(v))|\mathcal{F}_{v}]\{\tau-<D^{(1-}n(v)\}$$+$$E[1_{\{D^{(1}n^{)}(v)\leq\tau-\leq D.(v)\}}-\perp X(D^{(1-\frac{1}{n})}(v))|F_{v}]$$+$$E[1_{\{D.(v)<\tau\}}X(D^{(1-\frac{1}{n})}(v))|F_{v}]\}$$arrow$$E[1_{\{\tau-<D.(v)\}}X(D_{*}(v))|\mathcal{F}_{v}]+E[1_{\{D_{*}(v)<\tau\}}X(D_{*}(v))|F_{v}]$$=$$E[1_{\{\tau\leq D_{*}(v)\}}X(D_{*}(v))|\mathcal{F}_{v}]+E[1_{\{D_{*}(v)<\tau\}}X(D_{*}(v))|F_{v}]$$E[X(D_{*}(v))|\mathcal{F}_{v}]$$=$$(a.s.)$(2.5).

Under the conditions$0 \leq X(D^{(11_{)}}-_{\mathfrak{n}}(v))\leq\sup_{0\leq t\leq T}X(t)$and$E[ \sup_{0\leq t\leq T}X(t)]<\infty$

for any integer , the theorem may be used on taking limits under the expectation sign,
as shown by Shiryaev (1989) (Theorem 2, p. 218), at (2.5).
The rest of the proof is essentially the same as in the second half of the proof for
Theorem D.12 of Karatzas and Shreve (1999, p. 358). By using the fact that$n$$Z^{0}(v)\leq E[X(D_{*}(v))|\mathcal{F}_{v}]\leq E[Z^{0}(D_{*}(v))|F_{v}]\leq Z^{0}(v)(a.s.)$,

it can be proved that $E[X(D_{*}(v))|F_{v}]=Z^{0}(v)=ess \sup_{v\leq\rho\leq T}E[X(\rho)|\mathcal{F}_{v}]$ .
To prove the second half of Theorem 2.1, it must be shown that
$D_{*}(v)= \inf\{t\in[v, T]|Z^{0}(t)=X(t)\}(a.s.)$.

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Using the supermartingale property of the stochastic process,$Z^{0}(\cdot)$, leads to[ (D. $(v))$ ] $=E[Z^{0}(v)]\geq E[Z^{0}(D_{*}(v))]$ .

$E$ $X$On the other hand,dominates $X($ . . These two facts can be unified to conclude
that $X(D_{*}(v))=Z^{0}(D_{*}(v))(a.s.)$ , leading to the consequence that$Z^{0}(\cdot)$$)$$D_{*}(v) \geq\inf\{t\in[v, T]|Z^{0}(t)=X(t)\}$.

To show the reverse inequality, it is first noticed that
$X(t)<\lambda Z^{0}(t)\leq Z^{0}(i)(a.s.)$for all$t\in(v, D^{\lambda}(v))$

and $\lambda\in(0,1)$ , leading to$D_{*}(v)= \lim_{\lambda\uparrow 1}D^{\lambda}(v)\leq\inf\{t\in(v, T]|Z^{0}(t)=X(t)\}(a.s.)$.

Finally, it must be shown that $D_{*}(v)=v(a.s.)$ on the set $\{Z^{0}(v)=X(v)\}$ . On theset, $\{Z^{0}(v)=X(v)>0\}$ , the right continuity ofand $X$ imply that $D^{\lambda}(v)=v$ for
all A $\in(0,1)(a.s.)$ leading to $D_{*}(v)=v(a.s.)$ . Using the optimal stopping theorem of
Elliott (1982) (Theorem 4.12, p.36) leads to$Z^{0}$[$E$ $1_{\{Z^{0}(v)=0\}}Z^{0}$( $(v+\theta)$ A $T)$ ] $\leq E[1_{\{Z^{0}(v)=0\}}Z^{0}(v)]=0$

for any$\theta\geq 0$.

with the conclusion that $D_{*}(v)=v(a.s.)$ for $v\leq t\leq T(a.s.)$ on the set, $\{Z^{0}(v)=$$X(v)=0\}$ . On this set, $D_{*}(v)=v(a.s.)$ . (Q.E.D.)

The remaining task is to show the following Lemma 2.1.
Lemma 2.1 The third term of (2.4) equals . $i.e$ .
$0$$E[1_{\bigcap_{j=n}^{\infty}\{D^{(1-1)}\mathrm{J}(v)\leq\tau-\}}1_{\{\tau\leq D.(v)\}}X(D^{(1-\frac{1}{\tau\iota})}(v))|F_{v}]=0$.

[Proof] The first indicator function equals 1 on the region satisfying
for an arbitrary integer such that $n\leq j$ . Hereafter, conditioning on the events which
satisfy $D^{(1-1_{)}}j(v)\leq\tau$ -for an arbitrary such that $n\leq j$ , leads to$D^{(1-\iota_{)}}j(v)\leq\tau-$

$j$

$j$

$D^{(1-\frac{\iota}{\mathrm{j}})}(v)\leq\tau-\Leftrightarrow D^{(1-1_{)}}j(v)<\tau$for any$n\leq j$,

and hence$D_{*}(v)= \sup_{n}D^{(1-_{n}}(v)\leq\tau\iota_{)}$.

On the other hand, the assumption thatsatisfies $P[D_{*}(v)=\tau|F_{v}]>0$ leads to thecontradiction as follows. By virtue of the continuity property of the stochastic process,$D_{*}(v)$128, which is the same stochastic process as $X(t)$ until the jump time, in (2.2),, eachsatisfies the relation $D^{(1-\frac{1}{n})}(v)<D_{*}(v)\leq\tau$ . Therefore, there exists a positive, such that
integer-valued, random variable,$Y(t)$$\tau$$D^{(1-\frac{1}{\tau})}‘(v)$$l(\omega)$$v=D^{(1-\frac{1}{1})}(v)=D^{\langle 1-\frac{1}{2})}(v)=\cdots=D^{(1-\frac{1}{l(\omega)})}(v)<D^{(1-\frac{1}{l(\omega)+1})}(v)<\cdots(a.s.)$

To show the relation (2.6), it is evident that for eachvariable,, such that$\omega$.

(2.6)

there is an integer values random$l(\omega)$$(1- \frac{1}{l(\omega)+1})Z^{0}(v)<X(v)\leq(1-\frac{1}{l(\omega)})Z^{0}(v)(a.s.)$,

is a strictly monotone increasing sequence of stopping times
andutilizing the continuity property of the stochastic process, $Y(t)$ , which must satisfy$\mathrm{a}_{n}(\mathrm{w})$$=D^{(1-\frac{1}{n+l(\cdot)})}(v)$.

$\alpha_{n}arrow D_{*}(v)(narrow\infty)(a.s.)$

On the other hand, there is an assumption of $P[D_{*}(v)=\tau|F_{v}]>0$ , leading to$P[\alpha_{n}\uparrow\tau(narrow\infty)|F_{v}]>0$.

However, given , the stopping time, , is a totally inaccessible stopping time, leading
to a contradiction. This will lead to the relation, $D_{*}(v)<\tau(a.s.)$ , as a consequence. In
contrast, the second indicator in the third term of (2.4) equals in the region satisfying$\tau>D_{*}(v)$ . Therefore, the lemma is shown. (Q.E.D.)$v$$\tau$$0$

If the jump time of the process is a predictable stopping time, generally it is notpossible tonor is it possible to show the existence of an optimal stoppingtime. Amin and Jarrow (1993) have provided an example where there is no optimal
stopping time owing to the predictability of the jump time.
$\mathrm{p}\mathrm{r}o\mathrm{v}\mathrm{e}(2.5)$

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Duffie, D. and Singleton, K. (1999) Modeling term structures of defaultable bonds. The
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Elliott, R.J. (1982) Stochastic Calculus and Applications, Springer.

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