数理解析研究所講究録
第 1818 巻 2012 年 1-16

Approximating the Early Exercise Boundary for
American-style Options*
Toshikazu Kimura
Department of Civil, Environmental & Applied System Engineering
Kansai University

1

Introduction

European-style options, which can only be exercised at its maturity, have closed-form
formulas for their values in the standard model pioneered by Black and Scholes [7] andMerton [25]. Although a vast majority of traded options are of American-style optimally
exercised before maturity, there are no closed-form formulas for their values even in thestandard model called vanilla. The original statements of the American options problem
are dating back to the work of Samuelson [28] and McKean [24]; see Barone-Adesi [2] fora concise review of the American options problem. The principal difficulty in analyzing
American options may be the absence of an explicit expression for the early exercise
boundary (EEB), which is an optimal level of critical asset value where early exercise
occurs; see Zhu [29, Equation (27)] for a complicated expression in an infinite-series form.
Kim [19], Jacka [16] and Carr, Jarrow and Myneni [11] provided the put value in
integral form as a function of the EEB. To implement their approach, we need to obtainan accurate EEB approximation possibly in closed form. Various approximations havebeen developed by many researchers; see, e.g., Barone-Adesi and Whaley [3], Bunch
and Johnson [8], Carr [9], Geske and Johnson [13], MacMillan [22], Zhu [30] and Zhu andHe [31]. Among them, however, there is no explicit approximation for the EEB. No doubt,
the simplest approximation is a flat boundary. Barone-Adesi and Whaley [3] proposeda flat approximation as an initial guess of their iterative procedure to find the optimal
EEB. With the aid of this approximation, Bjerksund and Stensland [6] analyzed American
options as barrier options with knockout feature. Huang, Subrahmanyam and Yu [15]assumed the EEB as a piecewise-constant function of time, and provided a recursivealgorithm for obtaining the optimal exercise levels; see also Bjerksund and Stensland [6].
Alternatively, Omberg [27] developed an exponential EEB, and Ju [17] approximated the
EEB as a piecewise-exponential function of time to maturity. In both approximations,however, there are no closed-form solutions for the bases and the exponents of those
exponential functions, which must be computed numerically in their approaches. The
multipiece EEB approximations in [15, 17] naturally have discontinuous points in theboundary, but the EEB should be smooth intrinsically [26]. Clearly, the discontinuity*This research was supported in part by the Grant-in-Aid for Scientific Research (No. 20241037) of
the Japan Society for the Promotion of Science (JSPS) in 2008-2012.

1

2
in the multipiece EEB approximations become an serious obstacle for accurate decision
making of the option holders. The purpose of this paper is to approximate the EEB by asingle exponential function with an explicit and asymptotically exact exponent.
This paper is organized as follows: To avoid prohxity, we primarily focus on the
problem of valuing the American put option, but we provide the corresponding results
for the associated American call case as well. In Section 2, we formulate the problem
by a free boundary problem in the classical Black-Scholes-Merton framework to obtaina basic partial differential equation for the American put value. In Section 3, following
Kimura [20], we adopt the Laplace-Carson transform approach to derive a functional
equation for the transformed EEB, from which we obtain two different exponential EEB
approximations in Section 4. In order to improve the accuracy of these approximations
near expiry, we develop a heuristic refinement in Section 5.

2

Black-Scholes-Merton Formulation

Assume that the capital market is well-defined and follows the efficient market hypothesis.
be the asset price govemed by the risk-neutralized diffusion process
Let$(S_{t})_{t\geq 0}$$\frac{dS_{t}}{S_{t}}=(r-\delta)dt+$ad$W_{t},$$t\geq 0$

(2.1)

,

is a continuous dividend rate, $\sigma>0$ is awhere $r>0$ is the risk-free interest rate,
is a standard Wiener process on a filteredvolatility of the asset returns. In (2.1),
is the natural filtration corresponding, whereprobability space
to $W$ and the probability measure is chosen risk-neutrally so that the asset has mean
rate of return . We consider an American put option written on the asset price process, which has maturity $T>0$ and strike price $K>0$ . Let$\delta\geq 0$$(W_{t})_{t\geq 0}$$(\mathcal{F}_{t})_{t\geq 0}$$(\Omega, (\mathcal{F}_{t})_{t\geq 0}, \mathcal{F}, \mathbb{P})$$\mathbb{P}$$r$$(S_{t})_{t\geq 0}$$P\equiv P(t, S_{t})=P(t, S_{t};K, r, \delta) , 0\leq t\leq T$

(2.2)

,

denote the value of the American put option at time . Similarly, let $C\equiv C(t, S_{t})=$$C(t, S_{t};K, r, \delta)(0\leq t\leq T)$ denote the value of the associated American call option withthe same parameters as those in the put option.
From the theory of arbitrage pricing, the fair value of the American put option at time
is given by solving an optimal stopping problem
$t$

$t$

$P(t, S_{t})= ess\sup_{T_{t}\in[t,T]}E[e^{-r(T_{t}-t)}(K-S_{T_{t}})^{+}|\mathcal{F}_{t}], 0\leq t\leq T$

,

(2.3)

and the conditional expectation isis a stopping time of the filtrationwhere
calculated under the risk-neutral probability measure . The random variable
is called an optimal stopping time if it gives the supremum value of the right-hand side of$T_{t}$$(\mathcal{F}_{t})_{t\geq 0}$$\mathbb{P}$$\tau_{t}*\in[t, T]$

3

$t$

Figure 1: Early exercise boundaries $B_{p}(t)(t\in[0, T])$ for American put options $(T=1,$$K=100,$ $r=0.05,$ $\delta=0.02,0.05,0.08,$ $\sigma=0.2)$(2.3). The relationship between the early exercise feature of American options and optimal
stopping problems was first analyzed by McKean [24] who studied the problem (2.3) underan actual probability measure rather than . Mathematically rigorous treatment of the
problem (2.3) was first established by Bensoussan [4] and Karatzas [18].
Solving the optimal stopping problem (2.3) is equivalent to find the pointsforwhich early exercise is optimal. Let and denote the stopping region and continuation
region, respectively. The stopping region is defined by$\mathbb{P}$$(t, S_{t})$

$S$

$C$

$S$

$S=\{(t, S)\in[0, T]\cross \mathbb{R}_{+}|P(t, S)=(K-S)^{+}\}$ .

(2.4)

Of course, the continuation region is the complement of in. The boundarythat separates
from is referred to as the early exercise boundary (EEB), which isdefined by
$C$

$S$

$S$

$[0, T]\cross \mathbb{R}_{+}$$C$$B_{p}(t)= \sup\{S\in \mathbb{R}_{+}|P(t, S)=(K-S)^{+}\}, 0\leq t\leq T$ .

(2.5)

Similarly, define the EEB for the American call option by.

$B_{c}(t)= \inf\{S\in \mathbb{R}_{+}|C(t, S)=(S-K)^{+}\}, 0\leq t\leq T$

Between these two boundaries
andney [10] derived a simple symmetric relation such that
$B_{p}(t)\equiv B_{p}(t;r, \delta)$$B_{c}(t)\equiv B_{c}(t;r, \delta)$

.

(2.6)

, Carr and Ches-(2.7)$B_{c}(t;r, \delta)B_{p}(t;\delta, r)=K^{2}, 0\leq t\leq T$

McKean [24] showed that the American put value and the early exercise boundary
can be obtained by jointly solving a free boundary problem, which is specified by theBlack-Scholes-Merton partial differential equation (PDE),$\frac{\partial P}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}P}{\partial S^{2}}+(r-\delta)S\frac{\partial P}{\partial S}-rP=0, S>B_{p}(t)$

(2.8)

4
together with the boundary conditions$\lim_{S\uparrow\infty}P(t, S)=0$(2.9)$\lim_{S\downarrow B_{p}(t)}P(t, S)=K-B_{p}(t)$$\lim_{S\downarrow B_{p}(t)}\frac{\partial P}{\partial S}=-1,$

and the terminal condition(2.10)$P(T, S)=(K-S)^{+}$ .

The second condition in (2.9) is often called the value-matching condition, while the third
condition is called the smooth-pasting or high-contact condition.
It is sometimes convenient to work with the equations where the current time isreplaced by the time to expiry $\tau\equiv T-t$ . For the sake of notational convenience, we
as theand $\tilde{B}_{p}(\tau)\equiv B_{p}(T-\tau)=B_{p}(t)$ , and we refer towrite. From $(2.8)-(2.10)$ , the put price for the backward
backward running process ofsatisfies the PDE
running process$t$$(\tilde{S}_{\tau})_{\tau\leq T}$$\tilde{S}_{\tau}\equiv S_{T-\tau}=S_{t}$$(S_{t})_{t\geq 0}$$\tilde{P}(\tau,\tilde{S}_{\tau})\equiv P(T-\tau, S_{T-\tau})=P(t, S_{t})$$- \frac{\partial\tilde{P}}{\partial\tau}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}\tilde{P}}{\partial S^{2}}+(r-\delta)S\frac{\partial\tilde{P}}{\partial S}-r\tilde{P}=0, S>\tilde{B}_{p}(\tau)$,(2.11)

with the boundary conditions$\lim_{s\uparrow\infty}\tilde{P}(\tau, S)=0$(2.12)$\lim_{s\downarrow\tilde{B}_{p}(\tau)}\tilde{P}(\tau, S)=K-\tilde{B}_{p}(\tau)$$s\iota_{p(\tau)}^{1_{\frac{i}{B}}m\frac{\partial\tilde{P}}{\partial S}=-1}$’

and the initial condition$\tilde{P}(0, S)=(K-S)^{+}$(2.13).

Similarly, we can show that the call price for the backward running process$C(T-\tau, S_{T-\tau})=C(t, S_{t})$ satisfies the PDE$- \frac{\partial\tilde{C}}{\partial\tau}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}\tilde{C}}{\partial S^{2}}+(r-\delta)S\frac{\partial\tilde{C}}{\partial S}-r\tilde{C}=0, S<\tilde{B}_{c}(\tau)$,$\tilde{C}(\tau,\tilde{S}_{\tau})\equiv$(2.14)

with the boundary conditions$\lim_{S\downarrow 0}\tilde{C}(\tau, S)=0$$\lim_{s\uparrow\tilde{B}_{c}(\tau)}\overline{C}(\tau, S)=\overline{B}_{c}(\tau)-K$$S \uparrow_{c}^{\frac{i}{B}}(\tau)1m\frac{\partial\tilde{C}}{\partial S}=1,$(2.15)

5and the initial condition(2.16)$\tilde{C}(0, S)=(S-K)^{+},$

where $\tilde{B}_{c}(\tau)\equiv B_{c}(T-\tau)=B_{c}(t)$ .

3

Valuation in the Laplace Domain

3.1

Laplace-Carson Transforms

For$\lambda>0$, define the Laplace-Carson transform (LCT) of the time-reversed quantities as$P^{*}( \lambda, S)=\mathcal{L}C[\overline{P}(\tau, S)]=\int_{0}^{\infty}\lambda e^{-\lambda\tau}\overline{P}(\tau, S)d\tau$

(3.1)

,

and
. No doubt, there is no essential difference between the LCT
and the Laplace transform ( $LT$ ) defined by$C^{*}(\lambda, S)=\mathcal{L}C[\tilde{C}(\tau, S)]$$\hat{P}(\lambda, S)=\int_{0}^{\infty}e^{-\lambda\tau}\tilde{P}(\tau, S)d\tau.$

Clearly, we havefor $\lambda>0$ . The principal reason why we prefer
LCT to $LT$ is that LCT generates relatively simpler formulas than $LT$ for option pricing
problems because constant values are invariant after taking transformation. In the context
of option pricing, LCTs were first used in the mndomization of Carr [9] for valuing an
American vanilla put option with an exponentially distributed random maturity . The
idea of randomization gives us another interpretation that the LCT
can beregarded as an exponentially weighted sum (integral) of the time-reversed valuefor (infinitely many) different values of the maturity, and hence forwhich makes LCTs well defined.
$P^{*}(\lambda, S)=\lambda\hat{P}(\lambda, S)$$T$$P^{*}(\lambda, S)$$\overline{P}(\tau, S)$$T\in \mathbb{R}_{+}$

3.2$\tau\in \mathbb{R}_{+},$

European Options

For American vanilla options, it is well known that the value of an American option canbe represented as the sum of the value of the corresponding European option and the early
exercise premium. Kim [19] proved that the option value has such a decomposition andthat the premium has an integral representation; see Kim [19, Equations (6) and (12)].
Here, as a preliminary for valuing American options, we derive closed-form LCTs of theEuropean values.
Consider a vanilla European put option written on the asset price processthat
has constant maturity and strike price $K$ . Let $p\equiv p(t, S_{t})$ denote the value of theEuropean put option at time $t\in[0, T]$ . Then, in much the same way as in Section 2,$(S_{t})_{t\geq 0}$$T$6
the put value for the backward running process$\tau=T-t$ can be obtained by solving the PDE$\tilde{p}(\tau,\tilde{S}_{\tau})\equiv p(T-\tau, S_{T-\tau})=p(t, S_{t})$$- \frac{\partial\tilde{p}}{\partial\tau}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}\tilde{p}}{\partial S^{2}}+(r-\delta)S\frac{\partial\tilde{p}}{\partial S}-r\tilde{p}=0, S>0$

,

for

(3.2)

with the boundary conditions$\lim_{s\downarrow 0}\tilde{p}(\tau, S)=Ke^{-r\tau}$(3.3)$\lim_{s\uparrow\infty}\tilde{p}(\tau, S)=0,$

and the same initial condition as in (2.13), i. e.,$\tilde{p}(0, S)=(K-S)^{+}$

.

(3.4)

For convenience, denote. We see from $(3.2)-(3.4)$ that
$ODE$
satisfies the ordinary differential equation ()$p^{*}(\lambda, S)=\mathcal{L}C\lceil\tilde{p}(\tau, S)]$$\frac{1}{2}\sigma^{2}S^{2}\frac{d^{2}p^{*}}{dS^{2}}+(r-\delta)S\frac{dp^{*}}{dS}-(\lambda+r)p^{*}+\lambda(K-S)^{+}=0, S>0$$p^{*}(\lambda, S)$

,

(3.5)

with the boundary conditions$hmp^{*}(\lambda, S)s\downarrow 0=\frac{\lambda K}{\lambda+r}$(3.6)$\lim_{S\uparrow\infty}p^{*}(\lambda, S)=0.$

Proposition 1 (Kimura [20]) The $LCTp^{*}(\lambda, S)$ for the European put value is given$by$$p^{*}(\lambda, S)=\{\begin{array}{ll}\xi(S)+\frac{\lambda K}{\lambda+r}-\frac{\lambda S}{\lambda+\delta}, S<K\eta(S) , S\geq K,\end{array}$where$\xi(S)=\frac{K}{\theta_{1}-\theta_{2}}\frac{\lambda}{\lambda+\delta}(1-\frac{r-\delta}{\lambda+r}\theta_{2})(\frac{S}{K})^{\theta_{1}} S<K$$\eta(S)=\frac{K}{\theta_{1}-\theta_{2}}\frac{\lambda}{\lambda+\delta}(1-\frac{r-\delta}{\lambda+r}\theta_{1})(\frac{S}{K})^{\theta_{2}} S\geq K,$

and the pammeters $\theta_{i}\equiv\theta_{i}(\lambda)(i=1,2, \theta_{1}>1, \theta_{2}<0)$ are two roots of the quadmtic
equation$\frac{1}{2}\sigma^{2}\theta^{2}+(r-\delta-\frac{1}{2}\sigma^{2})\theta-(\lambda+r)=0$

,

(3.7)

i. e.,$\theta_{i}=\frac{1}{\sigma^{2}}\{-(r-\delta-\frac{1}{2}\sigma^{2})-(-1)^{i}\sqrt{(r-\delta-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}(\lambda+r)}\},$$i=1,2.$

7
We can apply the same argument to the call case: Let
European call option at time $t\in[0, T],$and
. Solving the $ODE$$c\equiv c(t, S_{t})$be the value of thefor $\tau=T-t,$$\tilde{c}(\tau,\tilde{S}_{\tau})\equiv c(T-\tau, S_{T-\tau})=c(t, S_{t})$$c^{*}(\lambda, S)=\mathcal{L}C[\tilde{c}(\tau, S)]$$\frac{1}{2}\sigma^{2}S^{2}\frac{d^{2}c^{*}}{dS^{2}}+(r-\delta)S\frac{dc^{*}}{dS}-(\lambda+r)c^{*}+\lambda(S-K)^{+}=0, S>0$

,

(3.8)

together with the boundary conditions$\lim_{s\downarrow 0}c^{*}(\lambda, S)=0$(3.9)$\lim_{S\uparrow\infty}\frac{dc^{*}}{dS}<\infty,$

we have
Proposition 2 (Kimura [20]) The $LCTc^{*}(\lambda, S)$ for the European call value is given$by$$c^{*}(\lambda, S)=\{\begin{array}{ll}\xi(S) , S<K\eta(S)+\frac{\lambda S}{\lambda+\delta}-\frac{\lambda K}{\lambda+r}, S\geq K.\end{array}$

3.3

American Options

Now we apply the argument above to the American put option. From $(2.11)-(2.13)$ , the
LCTsatisfies the $ODE$$P^{*}(\lambda, S)$$\frac{1}{2}\sigma^{2}S^{2}\frac{d^{2}P^{*}}{dS^{2}}+(r-\delta)S\frac{dP^{*}}{dS}-(\lambda+r)P^{*}+\lambda(K-S)^{+}=0, S>B_{p}^{*}$,(3.10)

together with the boundary conditions$s\uparrow\infty hmP^{*}(\lambda, S)=0$$\lim_{s\downarrow B_{p}^{*}}P^{*}(\lambda, S)=K-B_{p}^{*}$(3.11)$\lim_{S\downarrow B_{\dot{p}}}\frac{dP^{*}}{dS}=-1,$

where, which is a constant in the Laplace world due to thememoryless property of the exponential distribution.
$B_{p}^{*}\equiv B_{p}^{*}(\lambda)=\mathcal{L}C[\tilde{B}_{p}(\tau)]$

Theorem 1 The $LCTP^{*}(\lambda, S)$ for the American put value is given by$P^{*}(\lambda, S)=\{\begin{array}{ll}K-S, S\leq B_{p}^{*}p^{*}(\lambda, S)+e_{p}^{*}(\lambda, S) , S>B_{p}^{*},\end{array}$where$e_{p}^{*}( \lambda, S)=-\frac{1}{\theta_{2}}\{\theta_{1}\xi(B_{p}^{*})+\frac{\delta}{\lambda+\delta}B_{p}^{*}\}(\frac{S}{B_{p}^{*}})^{\theta_{2}} S>B_{p}^{*},$

8
and$B_{p}^{*}(\leq K)$

is a unique positive solution of the functional equation$\lambda(\frac{B_{p}^{*}}{K})^{\theta_{1}}+\delta\theta_{1}\frac{B_{p}^{*}}{K}+r(1-\theta_{1})=0$.

(3.12)

Kim [19, Section 3] proved that the vanilla American put value has the decomposition$P(t, S_{t})=p(t, S_{t})+e_{p}(t, S_{t})$and that the premium$e_{p}(t, S_{t})$(3.13),

has the integral representation$e_{p}(t, S_{t})= \int^{T}\{rKe^{-r(u-t)}\Phi(-d_{-}(S_{t}, B_{p}(u), u-t))$$-\delta S_{t}e^{-\delta(u-t)}\Phi(-d_{+}(S_{t}, B_{p}(u), u-t))\}du$where$\Phi(\cdot)$(3.14),

is the cumulative standard normal distribution function, and for$d_{\pm}(x, y, \tau)=\frac{\log(x/y)+(r-\delta\pm\frac{1}{2}\sigma^{2})\tau}{\sigma\sqrt{\tau}}$$x,$ $y,$ $\tau>0$.

(3.15)

See also Jacka [16] and Carr et al. [11]. From these results, the function
can beinterpreted as the LCT of the time-reverse early exercise premium$S_{T-\tau})=e_{p}(t, S_{t})$ for
In much the same way, we can derive the LCT
for the American call value:
Solving the $ODE$$e_{p}^{*}(\lambda, S)$$\tilde{e}_{p}(\tau,\tilde{S}_{\tau})=e_{p}(T-$$S_{t}\equiv S.$$\tau,$$C^{*}(\lambda, S)$$\frac{1}{2}\sigma^{2}S^{2}\frac{d^{2}C^{*}}{dS^{2}}+(r-\delta)S\frac{dC^{*}}{dS}-(\lambda+r)C^{*}+\lambda(S-K)^{+}=0, S<B_{c}^{*}$,(3.16)

together with the boundary conditions$\lim_{S\downarrow 0}C^{*}(\lambda, S)=0$(3.17)$\lim_{s\uparrow B_{c}^{*}}C^{*}(\lambda, S)=B_{c}^{*}-K$$\lim_{S\uparrow B_{c}^{r}}\frac{dC^{*}}{dS}=1,$

where$B_{c}^{*}\equiv B_{c}^{*}(\lambda)=\mathcal{L}C[\tilde{B}_{c}(\tau)]$, we have

Theorem 2 The $LCTC^{*}(\lambda, S)$ for the American call value is given by$C^{*}(\lambda, S)=\{\begin{array}{ll}S-K, S\geq B_{c}^{*}c^{*}(\lambda, S)+e_{c}^{*}(\lambda, S) , S<B_{c}^{*},\end{array}$where$e_{c}^{*}( \lambda, S)=\frac{1}{\theta_{1}}\{\frac{\delta}{\lambda+\delta}B_{c}^{*}-\theta_{2}\eta(B_{c}^{*})\}(\frac{S}{B_{c}^{*}})^{\theta_{1}} S<B_{c}^{*},$

and$B_{c}^{*}(\leq K)$

is a unique positive solution of the functional equation$\lambda(\frac{B_{c}^{*}}{K})^{\theta_{2}}+\delta\theta_{2}\frac{B_{c}^{*}}{K}+r(1-\theta_{2})=0$.

(3.18)

9
The function
exercise call premium
integral representation$e_{c}^{*}(\lambda, S)$also can be interpreted as the LCT of the time-reverse early, which has thefor$\tilde{e}_{c}(\tau,\tilde{S}_{\tau})=e_{c}(T-\tau, S_{T-\tau})=e_{c}(t, S_{t})$$S_{t}\equiv S$$e_{c}(t, S_{t})= \int_{t}^{T}\{\delta S_{t}e^{-\delta(u-t)}\Phi(d_{+}(S_{t}, B_{c}(u), u-t))$$-rKe^{-r(u-t)}\Phi(d_{-}(S_{t}, B_{c}(u), u-t))\}du$;(3.19)

see Kwok [21, p. 277]

4

Asymptotic Approximations

4.1

Asymptotic Properties

The initial-value theorem in the theory of Laplace transforms,$\lim_{\lambdaarrow\infty}B_{p}^{*}(\lambda)=\lim_{\tauarrow 0}\tilde{B}_{p}(\tau)=B_{p}(T)$leads toProposition 3 (Kimura [20]) For the early exercise boundaries of the American put
and call options, we have$B_{p}(T)= \min(\frac{r}{\delta}, 1)K$and

$B_{c}(T)= \max(\frac{r}{\delta}, 1)K.$

See also Kwok [21, pp. 256-262] for another proof.
From the functional equations (3.12) and (3.18) for the LCTs
Theorems 1 and 2, we have
Lemma 1 For sufficiently small$B_{p}^{*}(\lambda)$and$B_{c}^{*}(\lambda)$

in$\lambda>0,$$B_{p}^{*}( \lambda)\sim\frac{r}{\delta}\frac{\theta_{1}-1}{\theta_{1}}K$$or$$B_{p}^{*}( \lambda)\sim\frac{\theta_{2}}{\theta_{2}-1}K,$$B_{c}^{*}( \lambda)\sim\frac{r}{\delta}\frac{\theta_{2}-1}{\theta_{2}}K$$or$$B_{c}^{*}( \lambda)\sim\frac{\theta_{1}}{\theta_{1}-1}K.$

Proof. From (3.12) and (3.18), we immediately obtain$B_{p}^{*}( \lambda)\sim\frac{r}{\delta}\frac{\theta_{1}-1}{\theta_{1}}K$

and$B_{c}^{*}( \lambda)\sim\frac{r}{\delta}\frac{\theta_{2}-1}{\theta_{2}}K,$by removing the first terms of the functional equations (3.12) and (3.18). Applying thebasic relations into (3. 12)$\{\begin{array}{l}\lambda+r=-\frac{1}{2}\sigma^{2}\theta_{1}\theta_{2}r-\delta=-\frac{1}{2}\sigma^{2}(\theta_{1}+\theta_{2}-1) ,\end{array}$

(4.1)

10
we have another expression of the equation (3.12) for$B_{p}^{*}$, which is$\lambda(1-\frac{r-\delta}{\lambda+r}\theta_{2})(\frac{B_{p}^{*}}{K})^{\theta_{1}}+\delta(1-\theta_{2})\frac{B_{p}^{*}}{K}+r\theta_{2}\frac{\lambda+\delta}{\lambda+r}=0$

Similarly, from (3.18) for$B_{c}^{*}$

.

(4.2)

.

(4.3)

, we have$\lambda(1-\frac{r-\delta}{\lambda+r}\theta_{1})(\frac{B_{c}^{*}}{K})^{\theta_{2}}+\delta(1-\theta_{1})\frac{B_{c}^{*}}{K}+r\theta_{1}\frac{\lambda+\delta}{\lambda+r}=0$

Deleting the first terms in (4.2) and (4.3) and using the approximation
for sufficiently small , we obtain the altemative approximations$(\lambda+\delta)/(\lambda+r)\approx\delta/r$$\lambda$$B_{p}^{*}( \lambda)\sim\frac{\theta_{2}}{\theta_{2}-1}K$

and$B_{c}^{*}( \lambda)\sim\frac{\theta_{1}}{\theta_{1}-1}K.$$\square$Proposition 4 (Kimura [20]) For the time-reverse early exercise boundaries of the American put and call options, we have$\underline{B}_{p}\equiv\lim_{\tauarrow\infty}\tilde{B}_{p}(\tau)=\frac{r}{\delta}\frac{\theta_{1}^{o}-1}{\theta_{1}^{o}}K=\frac{\theta_{2}^{o}}{\theta_{2}^{o}-1}K,$$\overline{B}_{c}\equiv\lim_{\tauarrow\infty}\tilde{B}_{c}(\tau)=\frac{r}{\delta}\frac{\theta_{2}^{o}-1}{\theta_{\mathring{2}}}K=\frac{\theta_{1}^{o}}{\theta_{1}^{o}-1}K,$

where$\theta_{i}^{o}=\lim_{\lambdaarrow 0}\theta_{i}(\lambda)$, i.e.,$\theta_{i}^{o}=\frac{1}{\sigma^{2}}\{-(r-\delta-\frac{1}{2}\sigma^{2})-(-1)^{i}\sqrt{(r-\delta-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}r}\}, i=1,2.$

4.2
Let

Put-Call Symmetry$v_{1}\equiv v_{1}(\lambda)>1$

and$\nu_{2}\equiv\nu_{2}(\lambda)<0$

be two real roots of the quadratic equation$\frac{1}{2}\sigma^{2}v^{2}+(\delta-r-\frac{1}{2}\sigma^{2})\nu-(\lambda+\delta)=0$

(4.4)

,

i.e.,
$\nu_{i}=\frac{1}{\sigma^{2}}\{-(\delta-r-\frac{1}{2}\sigma^{2})-(-1)^{i}\sqrt{(\delta-r-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}(\lambda+\delta)}\},$, denote. Clearly,
are symmetric with respect to and , namely,. For these quantities, we have

As in the case of$\theta_{i}(\lambda)(i=1,2)$

Lemma 2 For$v_{i}(\lambda)\equiv\nu_{i}(\lambda;r, \delta)$$\nu_{i}^{o}=\lim_{\lambdaarrow 0}\nu_{i}(\lambda)$$\theta_{i}(\lambda)\equiv\theta_{i}(\lambda;r, \delta)(i=1,2)$$\nu_{i}(\lambda;r, \delta)$$i=1,2.$$r$$\lambda\geq 0,$$\theta_{1}(\lambda)+\nu_{2}(\lambda)=1,$$\theta_{2}(\lambda)+\nu_{1}(\lambda)=1$$\delta$and$\theta_{i}(\lambda;\delta, r)=$11
Pmof. We only prove the first equation$\theta_{1}+v_{2}=1$. The second one follows similarly.

$v_{2}= \frac{1}{\sigma^{2}}\{-(\delta-r-\frac{1}{2}\sigma^{2})-\sqrt{(\delta-r-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}(\lambda+\delta)}\}$$=1- \frac{1}{\sigma^{2}}\{-(r-\delta-\frac{1}{2}\sigma^{2})+\sqrt{(\delta-r-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}(\lambda+r)+2\sigma^{2}(\delta-}r)\}$$=1- \frac{1}{\sigma^{2}}\{-(r-\delta-\frac{1}{2}\sigma^{2})+\sqrt{(r-\delta-\frac{1}{2}\sigma^{2})^{2}+2\sigma^{2}(\lambda+r)}\}$$=1-\theta_{1},$and hencefor$\theta_{1}(\lambda)+\nu_{2}(\lambda)=1$$\lambda\geq 0.$$\square$Proposition 5 Between two LCTs
andficiently small, there exists a symmetric relation, i. e.,$B_{p}^{*}(\lambda)\equiv B_{p}^{*}(\lambda;r, \delta)$$B_{c}^{*}(\lambda)\equiv B_{c}^{*}(\lambda;r, \delta)$

for suf-$\lambda>0$$B_{c}^{*}(\lambda;r, \delta)B_{p}^{*}(\lambda;\delta, r)\sim K^{2}.$In particular, letting$\lambdaarrow 0+$, we have$\overline{B}_{c}(r, \delta)\underline{B}_{p}(\delta, r)=K^{2}.$4.3

Exponential Approximations

Lemma 3 For sufficiently small$\lambda>0,$$\theta_{1}(\lambda)=\theta_{1}^{o}+\frac{2}{\sigma^{2}}\frac{\lambda}{\theta_{1}^{o}-\theta_{2}^{o}}+o(\lambda)$

2$\lambda$$\theta_{2}(\lambda)=\theta_{2}^{o}+-\sigma^{2}\overline{\theta_{2}^{o}-\theta_{1}^{o}}+o(\lambda)$

Proof. For simplicity, denote$\lambda>0$$\alpha\equiv r-\delta-\frac{1}{2}\sigma^{2}$,.

. Then, for $i=1,2$ and sufficiently small, we have$\theta_{i}(\lambda)=\frac{1}{\sigma^{2}}\{-\alpha-(-1)^{i}\sqrt{\alpha^{2}+2\sigma^{2}(\lambda+r)}\}$$= \frac{1}{\sigma^{2}}\{-\alpha-(-1)^{i}\sqrt{\alpha^{2}+2\sigma^{2}r}\sqrt{1+\frac{2\sigma^{2}\lambda}{\alpha^{2}+2\sigma^{2}r}}\}$$= \frac{1}{\sigma^{2}}\{-\alpha-(-1)^{i}\sqrt{\alpha^{2}+2\sigma^{2}r}(1+\frac{\sigma^{2}\lambda}{\alpha^{2}+2\sigma^{2}r})\}+o(\lambda)$$= \theta_{i}^{o}-(-1)^{i}\frac{\lambda}{\sqrt{\alpha^{2}+2\sigma^{2}r}}+o(\lambda)$$= \theta_{i}^{o}-(-1)^{i}\frac{2}{\sigma^{2}}\frac{\lambda}{\theta_{1}^{o}-\theta_{2}^{o}}+o(\lambda)$

where we have used the relation,$\theta_{1}^{O}-\theta_{2}^{o}=\frac{2}{\sigma^{2}}\sqrt{\alpha^{2}+2\sigma^{2}r}.$$\square$From Lemmas 1 and 3 and the consistency with the exact boundary values at maturity
shown in Proposition 3, we propose the following exponential approximations:12
Theorem 3 For sufficiently large $\tau>0$ , the time-reverse early exercise boundaries havethe asymptotically exponential expressions as follows:(i) For the American put option,$\frac{\tilde{B}_{p}(\tau)}{\underline{B}_{p}}\approx\beta_{p}(\tau)\equiv\{\begin{array}{ll}1+\frac{1}{\theta_{1}^{o}-1}\exp\{-\frac{1}{2}\sigma^{2}\theta_{1}^{o}(\theta_{1}^{o}-\theta_{2}^{o})\tau\}, r<\delta 1-\frac{1}{\theta_{\mathring{2}}}\exp\{-\frac{1}{2}\sigma^{2}(1-\theta_{2}^{o})(\theta_{1}^{o}-\theta_{2}^{o})\tau\}, r\geq\delta.\end{array}$(ii) For the American call option,$\frac{\tilde{B}_{c}(\tau)}{\overline{B}_{c}}\approx\beta_{c}(\tau)\equiv\{\begin{array}{ll}1+\frac{1}{\theta_{\mathring{2}}-1}\exp\{-\frac{1}{2}\sigma^{2}\theta_{2}^{o}(\theta_{2}^{o}-\theta_{1}^{o})\tau\}, r>\delta 1-\frac{1}{\theta_{\mathring{1}}}\exp\{-\frac{1}{2}\sigma^{2}(1-\theta_{1}^{o})(\theta_{2}^{o}-\theta_{1}^{o})\tau\}, r\leq\delta.\end{array}$

5

A Heuristic Refinement

Evans, Kuske and Keller [12] have derived explicit expressions valid near expiry for the
EEBs of American put and call options, which are, as $\tauarrow 0+,$$\frac{\tilde{B}_{p}(\tau)}{\underline{B}_{p}}\sim\{\begin{array}{ll}1-\sigma\sqrt{\tau\ln(\frac{\sigma^{2}}{8\pi(r-\delta)^{2_{\mathcal{T}}}})}, r>\delta 1-\sigma\sqrt{2\tau\ln(\frac{1}{4\sqrt{\pi}\delta\tau})}, r=\delta 1-\kappa\sigma\sqrt{2\tau}, r<\delta,\end{array}$

(5.1)

and

$\frac{\tilde{B}_{c}(\tau)}{\overline{B}_{c}}\sim\{\begin{array}{ll}1+\sigma\sqrt{\tau\ln(\frac{\sigma^{2}}{8\pi(r-\delta)^{2_{\mathcal{T}}}})}, r<\delta 1+\sigma\sqrt{2\tau\ln(\frac{1}{4\sqrt{\pi}\delta\tau})}, r=\delta 1+\kappa\sigma\sqrt{2\tau}, r>\delta,\end{array}$where the constant$\kappa\approx 0.4517$(5.2)

is the root of the transcendental equation$\int_{\kappa}^{\infty}e^{-(x^{2}-\kappa^{2})}dx=\frac{2\kappa^{2}-1}{4\kappa^{3}}.$

Clearly, the exponential approximations in Theorem 3 display different tangent behavior near expiry, e.g., for $r<\delta,$$\lim_{\tauarrow 0+}\frac{d}{d\tau}(\frac{\tilde{B}_{p}(\tau)}{\underline{B}_{p}})\approx\beta_{p}’(0)=-\frac{\sigma^{2}}{2}\frac{\theta_{1}^{o}(\theta_{1}^{o}-\theta_{2}^{o})}{\theta_{1}^{o}-1}<0,$13whereas the exact value $is-\infty$ ; see Figure 1. This may implies that our approximations
for put (call) tend to overestimate (underestimate) the true values for small $\tau>0$ . The
asymptotic properties near expiry seems to be helpful for refining our approximations.
However, the exact asymptotic expressions above cannot be directly applied to generatingrefined approximations for EEBs, because iffor the put (call) case, (i) theycannot be defined for all $\tau>0$ ; and (ii) for the region of where they can be defined,
they are not monotone functions of , being inconsistent with the exact results.
In order to eliminate this defect, we use a simple but rough approximation presented
earlier by Barone-Adesi and Whaley [3, Equations (33) and (A10)]. The idea of theirapproximation was also based on an asymptotic behavior near expiry. With a minor
modification of Bjerksund and Stensland [5] on the boundary value at maturity, it isgiven by, for put$r\geq\delta(r\leq\delta)$$\tau$$\tau$(5.3)$\overline{B}_{p}(\tau)\approx e^{-h_{p}(\tau)}B_{p}(T)+(1-e^{-h_{p}(\tau)})\underline{B}_{p}$where$h_{p}( \tau)=\frac{B_{p}(T)}{B_{p}(T)-\underline{B}_{p}}\{-(r-\delta)\tau+2\sigma\sqrt{\tau}\},$

and for call
(5.4)$\tilde{B}_{c}(\tau)\approx e^{-h_{c}(\tau)}B_{c}(T)+(1-e^{-h_{c}(\tau)})\overline{B}_{c}$where$h_{c}(\tau)=\overline{\overline{B}_{c}-B_{c}(T)}B_{c}(T)\{(r-\delta)\tau+2\sigma\sqrt{\tau}\}.$As shown in Barone-Adesi and Whaley [3, p. 310], their approximations also have thesame defect on the monotonicity as in (5.1) and (5.2), depending on the values of and
It is, however, relatively easy to eliminate this defect from (5.3) and (5.4). For sufficiently
small $\tau>0$ , we have$r$$h_{p}( \tau)\approx\frac{2B_{p}(T)}{B_{p}(T)-\underline{B}_{p}}\sigma\sqrt{\tau}$

and$\delta.$$h_{c}(\tau)\approx^{2B_{c}(T)}\sigma\sqrt{\tau}\overline{\overline{B}_{c}-B_{c}(T)},$

which are positive for all $\tau>0$ , and hence they keep the monotonous properties of EEBs.
These approximations and the results in Propositions 3 and 4 yields refined approximations of the time-reverse early exercise boundaries for the American put and call options,which are$\tilde{B}_{p}(\tau)\approx e^{-\gamma_{p}(\tau)}B_{p}(T)+(1-e^{-\gamma_{p}(\mathcal{T})})\underline{B}_{p}\beta_{p}(\tau)$

,

(5.5)

for put, where$\gamma_{p}(\tau)=\{\begin{array}{ll}2\theta_{1}^{o}\sigma\sqrt{\tau}, r<\delta 2(1-\theta_{2}^{o})\sigma\sqrt{\tau}, r\geq\delta,\end{array}$and for call$\tilde{B}_{c}(\tau)\approx e^{-\gamma_{c}(\tau)}B_{c}(T)+(1-e^{-\gamma_{\mathcal{C}}(\tau)})\overline{B}_{c}\beta_{c}(\tau)$

,

(5.6)

14
where
$\gamma_{c}(\tau)=\{\begin{array}{ll}-2\theta_{2}^{o}\sigma\sqrt{\tau}, r>\delta-2(1-\theta_{1}^{o})\sigma\sqrt{\tau}, r\leq\delta.\end{array}$

Note that both exponents$\gamma_{p}(\tau)$and$\gamma_{c}(\tau)$

are nonnegative and increasing functions of$\tau\geq 0.$

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Department of Civil, Environmental & Applied System Engineering
Faculty of Environmental & Urban Engineering
Kansai University, Suita 564-8680, Japan-mail address: t.kimura@kansai-u.ac.jp$E$$\ovalbox{\tt\small REJECT}\Phi\star\not\cong$. paxijxfflffi エ$\not\cong\Re_{rJ}\pi\Pi|ae-$