American Options

Charles Edouard Delort (1841–1895) was a French painter.

The topics are

In contrast to the holder of a European option, the holder of an American option is allowed to exercise his or her right to buy (sell) the underlying asset at any time before or at the expiry date. This special feature of American-style option, and more generally of American claims, makes the arbitrage pricing of American options much more involved than the valuation of standard European claims. We know already that arbitrage valuation of American claims is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose his or her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, under risk-neutral probability). Therefore, the uniqueness of the arbitrage price of an American claim holds. For an exhaustive survey of results and techniques related to the arbitrage pricing of American options, we refer to Myneni \cite{myn}.

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

Valuation of American Claims.

We place ourselves within the classic Black-Scholes setup. Hence, the prices of primary securities; that is, the stock price \(S\), and the
savings account \(B\), are modelled by means of the following SDEs
\[dS_{t}=\mu\cdot S_{t}dt+\sigma\cdot S_{t}dW_{t}\mbox{ for }S_{0}>0,\]
where \(\mu\in \mathbb{R}\) and \(\sigma >0\) are real numbers, and
\[dB_{t}=r\cdot B_{t}dt\mbox{ for }B_{0}=1,\]
with \(r\in \mathbb{R}\), respectively. As usual, we denote by \(W\) the standard Brownian motion defined on a filtered probability space \((\Omega ,{\bf F},\mathbb{P})\), where \({\bf F}={\bf F}^{W}\). For the sake of notational convenience, we assume here that the underlying Brownian motion \(W\) is one-dimensional.

In the context of arbitrage valuation of American contingent claims, it is convenient (although not necessary) to assume that an individual may withdraw funds to finace his or her comsumption needs. For any fixed \(t\), we denote by \(A_{t}\) the cumulative amount of funds that are withdrawn and consumed (The term “consumed” refers to the fact that the wealth is dynamically diminished according to the process \(A\).) by an investor up to time \(t\). The process \(A\) is assumed to be progressively measurable (We assume also that \(A\) is an RCLL process; that is, almost all sample paths of \(A\) are right-continuous functions with finite left-hand limits.) with nondecreasing sample paths; also, by convention, \(A_{0}=A_{0-}=0\). We say that \(A\) represents the consumption strategy, as opposed to the trading strategy \(\boldsymbol{\phi}\). It is thus natural to call a pair \((\boldsymbol{\phi},A)\) a trading and consumption strategy in \((S,B)\).

Definition. A trading and consumption strategy \((\boldsymbol{\phi},A)\) in \((S,B)\) is {\bf self-financing} on \([0,T]\) if its wealth process \(V(\boldsymbol{\phi},A)\), which equals
\begin{equation}{\label{museq2064}}\tag{1}
V_{t}(\boldsymbol{\phi},A)=\phi_{t}^{1}\cdot S_{t}+\phi_{t}^{2}\cdot B_{t}\mbox{ for all }t\in [0,T],
\end{equation}
satisfies for every \(t\in [0,T]\)
\begin{equation}{\label{musrq81}}\tag{2}
V_{t}(\boldsymbol{\phi},A)=V_{0}(\boldsymbol{\phi},A)+
\int_{0}^{t}\phi_{s}^{1}dS_{s}+\int_{0}^{t}\phi_{s}^{2}dB_{s}-A_{t}. \sharp
\end{equation}

In view of (\ref{museq81}), it is clear that \(A\) models the flow of funds that are not reinvested in primary securities, but rather are put aside forever (Let us stress that, in contrast to the trading-consumption optimization problems, the role of the consumption process is not essential in the present context. Indeed, one can alternatively assume that these funds are invested in risk-free bonds.). By convention, we say that the amount of funds represented by \(A_{t}\) is consumed by the holder of the dynamic portfolio \((\boldsymbol{\phi},A)\) up to time \(t\). From (\ref{museq81}), we have
\begin{equation}{\label{museq2065}}\tag{3}
dV_{t}=\phi_{t}^{1}\cdot dS_{t}+\phi_{t}^{2}dB_{t}-dA_{t}=
\phi_{t}^{1}\cdot (\mu\cdot S_{t}dt+\sigma\cdot S_{t}dW_{t})+\phi_{t}^{2}\cdot r\cdot B_{t}dt-dA_{t}.
\end{equation}
From (\ref{museq2064}), we get \(r\cdot\phi_{t}^{2}\cdot B_{t}=r\cdot V_{t}-r\cdot\phi_{t}^{1}\cdot S_{t}\). Therefore, equality (\ref{museq2065}) implies
\[dV_{t}=r\cdot V_{t}dt+\phi_{t}^{1}\cdot S_{t}\cdot\left ((\mu -r)dt+\sigma dW_{t}\right )-dA_{t}.\]
Equivalently,
\begin{equation}{\label{museq82}}\tag{4}
dV_{t}=r\cdot V_{t}dt+\xi_{t}\cdot (\mu -r)dt+\sigma\cdot\xi_{t}dW_{t}-dA_{t},
\end{equation}
where \(\xi_{t}=\phi_{t}^{1}\cdot S_{t}\) represents the amount of cash invested in shares at time \(t\). The unique solution of the linear SDE (\ref{museq82}) is given by an explicit formula
\[V_{t}=B_{t}\left (V_{0}+\int_{0}^{t}(\nu -r)\cdot\xi_{s}\cdot B_{s}^{-1}
ds-\int_{0}^{t}B_{s}^{-1}dA_{s}+\int_{0}^{t}\sigma\cdot\xi_{s}\cdot B_{s}^{-1}dW_{s}\right ),\]
which holds for every \(t\in [0,T]\). We conclude that the wealth process of any self-financing trading and consumption strategy is uniquely determined by the following quantities: the initial endowment \(V_{0}\), the consumption process \(A\), and the process \(\xi\) representing the amount of cash invested in shares. In other words, given an initial endowment \(V_{0}\), there is one-to-one correspondence between self-financing trading and consumption strategies \((\boldsymbol{\phi},A)\) and two-dimensional process \((\xi ,A)\). Recall that the unique martingale measure \(\mathbb{P}^{*}\) for the Black-Scholes spot market satisfies
\[\frac{d\mathbb{P}^{*}}{d\mathbb{P}}=\exp\left (\frac{r-\mu}{\sigma}\cdot W_{T}-\frac{1}{2}
\cdot\frac{(r-\mu )^{2}}{\sigma^{2}}\cdot T\right )\mbox{ \(\mathbb{P}\)-a.s.}\]
It is easily seen that the dynamics of the wealth process \(V\) under the martingale measure \(\mathbb{P}^{*}\) are given by the following expression
\[dV_{t}=r\cdot V_{t}dt+\sigma\cdot\xi_{t}dW_{t}^{*}-dA_{t},\]
where \(W^{*}\) follows the standard Brownian motion under \(\mathbb{P}^{*}\). This yields immediately
\[V_{t}=B_{t}\left (V_{0}-\int_{0}^{t}B_{s}^{-1}dA_{s}+\int_{0}^{t}\sigma\cdot\xi_{s}\cdot B_{s}^{-1}dW_{s}^{*}\right ).\]
Therefore, an auxiliary process \(Z\), which is given by the formula
\[Z_{t}\equiv V_{t}^{*}+\int_{0}^{t}B_{s}^{-1}dA_{s}=V_{0}+\int_{0}^{t}\sigma\cdot\xi_{s}\cdot B_{s}^{-1}dW_{s}^{*},\]
where \(V_{t}^{*}=V_{t}/B_{t}\), follows a local martingale under \(\mathbb{P}^{*}\). We say that a self-financing trading and consumption strategy \((\boldsymbol{\phi},A)\) is admissible when the condition
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\int_{0}^{T}\xi_{s}^{2}ds\right ]=\mathbb{E}_{\mathbb{P}^{*}}\left [
\int_{0}^{T}(\phi_{s}^{1}\cdot S_{s})^{2}ds\right ]<\infty\]
is satisfied so that \(Z\) is a \(\mathbb{P}^{*}\)-martingale. This assumption is imposed in order to exclude pathological examples of arbitrage opportunities from the market model. We are now in a position to formally introduce the concept of a contingent claim of American style. To this end, we take an arbitrary continuous {\bf reward function} \(g:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) satisfying the linear growth condition
\[|g(s,t)|\leq k_{1}+k_{2}s\]
for some constants \(k_{1},k_{2}\). An {\bf American claim} with the reward function \(g\) and expiry date \(T\) is a financial security which pays to its holder the amount \(g(S_{t},t)\) when exercised at time \(t\). We deliberately restrict the attention to path-independent American contingent claims; that is, to American claims whose payoff at exercise depends on the value of the underlying asset at the exercise date only.

The writer of an American claim with the reward function \(g\) accepts the obligation to pay the amount \(g(S_{t},t)\) at any time \(t\). It should be emphasized that the choice of the exercise time is at discretion of the holder of an American claim (that is, of a party assuming a long position). In order to formalize the concept of an American claim, we need to introduce a suitable class of admissible exercise times. Since we exclude clairvoyance, the admissible exercise time \(\tau\) is assumed to be a stopping time of filtration \({\bf F}\) (A random variable \(\tau :(\Omega ,{\cal F}_{T},\mathbb{P})\rightarrow [0,T]\) is a stopping time of filtration \({\bf F}\) if, for every \(t\in [0,T]\), the event \(\{\tau\leq t\}\) belongs to the \(\sigma\)-field \({\cal F}_{t}\).). Since in the Black-Shcoles model we have \({\bf F}= {\bf F}^{W}={\bf F}^{W^{*}}={\bf F}^{S}\), any stopping time of the filtration \({\bf F}\) is also a stopping time of the filtration \({\bf F}^{S}\) generated by the stock price process \(S\). In intuitive terms, it is assumed throughout that the decision to exercise an American claim at time \(t\) is based on the observations of stock price fluctuations up to time \(t\), but not after this date. This interpretation is consistent with the general assumption that the \(\sigma\)-field \({\cal F}_{t}\) represents the information available to all investors at time \(t\). Let us denote by \({\cal T}_{[t,T]}\) the set of all stopping times of the filtration \({\bf F}\) which satisfy \(t\leq\tau\leq T\) (with probability \(1\)).

Definition. An American contingent claim \(X^{a}\) with the reward function \(g:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\) is a financial instrument consisting of:

  • an expiry date \(T\);
  • he selection of a stopping time \(\tau\in {\cal T}_{[0,T]}\);
  • a payoff \(X_{\tau}^{a}=g(S_{\tau},\tau )\) on exercise. \(\sharp\)

Typical examples of American claims are American options with constant strike price \(K\) and expiry date \(T\). The payoff of American call and put options, when exercised at the random time \(\tau\), are equal to \(X_{\tau}=(S_{\tau}-K)^{+}\) and \(Y_{\tau}=(K-S_{\tau})^{+}\), respectively. In a slightly more general case, when the strike price is allowed to vary in time, the corresponding reward functions \(g^{c}(s,t)=(s-K_{t})^{+}\) and \(g^{p}(s,t)=(K_{t}-s)^{+}\) for American call and put options, respectively. Here \(K:[0,T]\rightarrow \mathbb{R}_{+}\) is a deterministic function which represents the variable level of the strike price. The aim is to derive the “rational” price and to determine the “rational” exercise time of an American contingent claim by means of purely arbitrage arguments. To this end, we shall first introduce a specific class of trading strategies. For expositional simplicity, we concentrate on the price of an American claim \(X^{a}\) at time \(0\); the general case can be treated along the same lines, but is more cumbersome from the notational viewpoint. It will be sufficient to consider a very special class of trading strategies associated with the American contingent claim \(X^{a}\), namely the {\bf buy-and-hold} strategies. By a buy-and-hold strategy associated with an American claim \(X^{a}\), we mean a pair \((c,\tau )\), where \(\tau\in {\cal T}_{[0,T]}\) and \(c\) is a real number. In financial interpretation, a buy-and-hold strategy \((c,\tau )\) assumes that \(c>0\) units of the American security \(X^{a}\) are acquired (or shorted, if \(c<0\)) at time \(0\), and then held in the portfolio (Observe that such a strategy excludes trading in the American claim after the initial date. In other words, dynamic trading in the American claim is not considered at this stage.) up to the exercise time \(\tau\). Let us assume that there exists a “market” price, say \(U_{0}\), at which the American claim \(X^{a}\) trades in the market at time \(0\). The first task is to find the right value of \(U_{0}\) by means of no-arbitrage arguments (as mentioned above, the arguments which lead to the arbitrage valuation of the claim \(X^{a}\) at time \(t>0\) are much the same as in the case of \(t=0\).).

Definition. By a self-financing trading strategy in \((S,B,X^{a})\), we mean a collection \((\boldsymbol{\phi},A,c,\tau )\), where \((\boldsymbol{\phi},A)\) is a trading and consumption strategy in \((S,B)\) and \((c,\tau )\) is a buy-and-hold strategy associated with \(X^{a}\). In addition, we assume that on the random interval \((\tau ,T]\) we have
\begin{equation}{\label{museq83}}\tag{5}
\phi_{t}^{1}=0\mbox{ and }\phi_{t}^{2}=\phi_{\tau}^{1}\cdot S_{\tau}
\cdot B_{\tau}^{-1}+\phi_{\tau}^{2}+c\dot g(S_{\tau},\tau )\cdot B_{\tau}^{-1}. \sharp
\end{equation}

It will soon become apparent that it is enough to consider the cases of \(c=1\) and \(c=-1\); that is, the long and short positions in the American claim \(X^{a}\). An analysis of condition (\ref{museq83}) shows that the definition of a self-financing strategy \((\boldsymbol{\phi},A,c,\tau )\) implicitly assumes that the American claim is exercised at a random time \(\tau\), existing positions in shares are closed at time \(\tau\), and all the proceeds are invested in risk-free bonds. For brevity, we shall sometimes write \(\tilde{\boldsymbol{\psi}}\) to denote the dynamic portfolio \((\boldsymbol{\phi},A,c,\tau )\) in what follows. Note that the wealth process \(V(\tilde{\boldsymbol{\psi}})\) of any self-financing strategy in \((S,B,X^{a})\) satisfies the following initial and terminal conditions
\[V_{0}(\tilde{\boldsymbol{\psi}})=\phi_{0}^{1}\cdot S_{0}+\phi_{0}^{2}
+c\dot U_{0}\mbox{ and }V_{T}(\tilde{\boldsymbol{\psi}})=
e^{r(T-t)}\cdot\left (\phi_{\tau}^{1}\cdot S_{\tau}+c\cdot g(S_{\tau},\tau )\right )+e^{rT}\cdot\phi_{\tau}^{2}.\]
In what follows, we restrict the attention to the class of admissible trading strategies \(\tilde{\boldsymbol{\psi}}= (\boldsymbol{\phi},A,c,\tau )\) in \((S,B,X^{a})\), which are defined in the following way

Definition. A self-financing trading strategy \((\boldsymbol{\phi},A,c,\tau )\) in \((S,B,X^{a})\) is said to be {\bf admissible} if a trading and consumption strategy \((\boldsymbol{\phi},A)\) is admissible and \(A_{T}=A_{\tau}\). The class of all admissible strategies \((\boldsymbol{\phi},A,c,\tau )\) is denoted by \(\tilde{\boldsymbol{\psi}}\). \(\sharp\)

Let us introduce the class \(\tilde{\boldsymbol{\psi}}^{0}\) of those admissible trading strategies \(\tilde{\boldsymbol{\psi}}\) for which the initial wealth satisfies \(V_{0}(\boldsymbol{\psi})<0\), and the terminal wealth has the nonnegative value; that is, \(V_{T}(\tilde{\boldsymbol{\psi}})=\phi_{T}^{2}\cdot B_{t}\geq 0\) (Since the existence of a strictly positive savings account is assumed, one can alternatively define the class \(\boldsymbol{\psi}^{0}\) as the set of those strategies \(\tilde{\boldsymbol{\psi}}\) from \(\tilde{\boldsymbol{\psi}}\) for which \(V_{0}(\tilde{\boldsymbol{\psi}})=0\) and \(V_{T}(\tilde{\boldsymbol{\psi}})=\phi_{T}^{2}\cdot B_{T}\geq 0\), and the latter inequality is strict with positive probability.). In order to precisely define an arbitrage opportunity, we have to take into account the early exercise feature of American claims. It is intuitively clear that it is enough to consider two cases; a long and a short position in one unit of an American claim. This is due to the fact that we need to exclude the existence of arbitrage opportunities for both the seller and the buyer of an American claim. Indeed, the position of both parties involved in a contract of American style is no longer symmetric, as it was in the case of European claims. The holder of an American claim can actively choose his or her exercise policy. The seller of an American claim, on the contrary, should be ready to meet his or her obligations at any (random) time. We therefore set down the following definition of arbitrage and an arbitrage-free market model.

\begin{equation}{\label{musd815}}\tag{6}\mbox{}\end{equation}

Definition \ref{musd815}. There is arbitrage in the market model with trading in the American claim \(X^{a}\) with initial price \(U_{0}\).

  • Long arbitrage. There exists a stopping time \(\tau\) such that for some trading and consumption strategy \((\boldsymbol{\phi},A)\) the strategy \(((\boldsymbol{\phi},A,1,\tau )\) belongs to the class \(\tilde{\boldsymbol{\psi}}^{0}\).
  • Short arbitrage. There exists a trading and consumption strategy \((\boldsymbol{\phi},A)\) such that for any stopping time \(\tau\) the strategy \((\boldsymbol{\phi},A,-1,\tau )\) belongs to the class \(\tilde{\boldsymbol{\psi}}^{0}\).

In the absence of arbitrage in the market model, we say that the model is arbitrage-free. \(\sharp\)

Definition \ref{musd815} can be reformulated in the following way: there is absence of arbitrage in the market if the following conditions are satisfied:

  • the strategy \((\boldsymbol{\phi},A,1,\tau )\) is not in \(\tilde{\boldsymbol{\psi}}^{0}\);
  • for any trading and consumption strategy \((\boldsymbol{\phi},A)\),there exists a stopping time \(\tau\) such that the strategy \((\boldsymbol{\phi},A,-1,\tau )\) is not in \(\tilde{\boldsymbol{\psi}}^{0}\).

Intuitively, under the absence of arbitrage in the market, the holder of an American claim is unable to find an exercise policy \(\tau\) and a trading and consumption strategy \((\boldsymbol{\phi},A)\) that would yield a risk-free profit. Also, under the absence of arbitrage, it is not possible to make risk-free profit by selling the American claim at time \(0\), provided that the buyer makes a clever choice of the exercise date. More precisely, there exists an exercise policy for the long party which prevents the short party from locking in a risk-free profit.

By definition, the arbitrage price at time \(0\) of the American claim \(X^{a}\), denoted by \(\Pi_{0}(X^{a})\), is that level of the price \(U_{0}\) which makes the model arbitrage-free. The aim is now to show that the assumed absence of arbitrage in the sense of Definition \ref{musd815} leads to a unique value for the arbitrage price \(\Pi_{0}(X^{a})\) of \(X^{a}\) (as already mentioned, it is not hard to extend this reasoning in order to determine the arbitrage price \(\Pi_{t}(X^{a})\) of the American claim \(X^{a}\) at any date \(t\in [0,T]\)). Also, we shall find the {\bf rational} exercise policy of the holder; that is, the stopping time that excludes the possibility of short arbitrage.

The following auxiliary result relates the value process associated with the specific optimal stopping problem to the wealth process of a certain admissible trading strategy. For any reward function \(g\), we define an adapted process \(V\) by setting
\begin{equation}{\label{museq84}}\tag{7}
V_{t}=\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [\left .
e^{-r(\tau -t)}\cdot g(S_{\tau},\tau )\right |{\cal F}_{t}\right ]
\end{equation}
for every \(t\in [0,T]\), provided that the right-hand side in (\ref{museq84}) is well-defined.

Proposition. Let \(V\) be an adapted process defined by formula (\ref{museq84}) for some reward function \(g\). Then there exists an admissible trading and consumption strategy \((\boldsymbol{\phi},A)\) such that \(V_{t}=V_{t} (\boldsymbol{\phi},A)\) for every \(t\in [0,T]\). \(\sharp\)

Definition. An admissible trading and consumption strategy \((\boldsymbol{\phi},A)\) is said to be a {\bf perfect hedging} against the American contingent claim \(X^{a}\) with reward function \(g\) if, with probability \(1\),
\[V_{t}(\boldsymbol{\phi})\geq g(S_{t},t)\mbox{ for all }t\in [0,T].\]
We write \(\boldsymbol{\phi}(X^{a})\) to denote the class of all perfect hedging strategies against the American contingent claim \(X^{a}\). \(\sharp\)

The goal is now to explicitly determine \(\Pi_{0}(X^{a})\) by assuming that trading in the American claim \(X^{a}\) would not destroy the arbitrage-free features of the Black-Scholes model.

Theorem. There is absence of arbitrage (in the sense of Definition \ref{musd815}) in the market model with trading in an American claim if and only if the price \(\Pi_{0}(X^{a})\) is given by the formula
\[\Pi_{0}(X^{a})=\sup_{\tau\in {\cal T}_{[0,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [e^{-r\tau}\cdot g(S_{\tau},\tau )\right ].\]
More generally, the arbitrage price at time \(t\) of an American claim with reward function \(g\) equals
\[\Pi_{t}(X^{a})=\mbox{ess}\sup_{\tau\in {\cal T}_{[0,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [
\left .e^{-r(\tau -t)}\cdot g(S_{\tau},\tau )\right |{cal F}_{t}\right ].\]

Proof. Sketch of the proof. Let us assume that the “market” price of the option is \(U_{0}>V_{0}\). We shall show that in this case, the American claim is overpriced; that is, a short srbitrage is possible. On the other hand, suppose that \(U_{0}<V_{0}\) so that the American claim is under priced. We shall construct an arbitrage opportunity for the buyer of this claim. We then conclude that the arbitrage price \(\Pi_{0}(X^{a})\) necessarily coincides with \(V_{0}\) since otherwise arbitrage opportunities would exist in the market model. \(\sharp\)

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

American Call and Put Options.

From now on we restrict the attention to the class of American call and put options. We allow the strike price to vary in time; the strike price is represented by a deterministic function \(K:[0,T]\rightarrow \mathbb{R}_{+}\) which satisfies
\[K_{t}=K_{0}+\int_{0}^{t}k_{s}ds\mbox{ for all }t\in [0,T],\]
for a bounded function \(k:[0,T]\rightarrow \mathbb{R}_{+}\). The reward functions we shall study in what follows are \(g^{c}(s,t)=(s-K_{t})^{+}\) and \(g^{p}(s,t)=(K_{t}-s)^{+}\), where the rewards \(g^{c}\) and \(g^{p}\) correspond to the call and put options, respectively. It will be convenient to introduce the discounted rewards
\[X_{t}^{*}=\frac{S_{t}-K_{t})^{+}}{B_{t}}\mbox{ and } Y_{t}^{*}=\frac{K_{t}-S_{t})^{+}}{B_{t}}.\]
For a continuous semimartingale \(Z\), and a fixed \(a\in \mathbb{R}\), we denote by \(L_{t}^{a}(Z)\) the (right) semimartingale {\bf local time} of \(Z\), given explicitly by the formula
\[L_{t}^{a}(Z)\equiv |Z_{t}-a|-|Z_{0}-a|-\int_{0}^{t}sgn(Z_{s}-a)dZ_{s}\]
for every \(t\in [0,T]\) (by convention we set \(sgn(0)=-1\)). It is well known that the local time \(L^{a}(Z)\) of a continuous semimartingale \(Z\) is an adapted process whose sample paths are almost all continuous and nondecreasing functions. Moreover, for an arbitrary convex function \(f:\mathbb{R}\rightarrow \mathbb{R}\), the following decomposition, referred to as the Ito-Tanaka-Meyer formula, is valid
\[f(Z_{t})=f(Z_{0})+\int_{0}^{t}f'(Z_{s})dZ_{s}+\frac{1}{2}\int_{\mathbb{R}}L_{t}^{a}(Z)d\mu (a),\]
where \(f’\) is the left-hand side derivative of \(f\), and \(\mu\) denotes the second derivative of \(f\), in the sense of distributions. An applications of the Ito-Tanaka-Meyer formula yields
\[X_{t}^{*}=X_{0}^{*}+\int_{0}^{t}I_{\{S_{s}>K_{s}\}}\cdot B_{s}^{-1}\cdot
(\eta_{s}ds+\sigma\cdot S_{s}dW_{s}^{*})+\frac{1}{2}\int_{0}^{t}B_{s}^{-1}dL_{s}^{0}(S-K),\]
where \(\eta_{s}=r\cdot K_{s}-k_{s}\). Similarly, for the process \(Y^{*}\) we get
\[Y_{t}^{*}=Y_{0}^{*}-\int_{0}^{t}I_{\{S_{s}<K_{s}\}}\cdot B_{s}^{-1}\cdot
(\eta_{s}ds-\sigma\cdot S_{s}dW_{s}^{*})+\frac{1}{2}\int_{0}^{t}B_{s}^{-1}dL_{s}^{0}(K-S).\]
If the strike price \(K_{t}=K>0\) is a constant, then
\[L_{t}^{0}(S-K)=L_{t}^{K}(S)\mbox{ and }L_{t}^{0}(K-S)=L_{t}^{-K}(-S).\]
We can also show that \(L_{t}^{-K}(-S)=L_{t}^{K}(S)\). Therefore, the discounted rewards \(X^{*}\) and \(Y^{*}\) satisfy
\[X_{t}^{*}=X_{0}^{*}+\int_{0}^{t}I_{\{S_{s}>K\}}\cdot B_{s}^{-1}\cdot
(r\cdot Kds+\sigma\cdot S_{s}dW_{s}^{*})+\frac{1}{2}\int_{0}^{t}B_{s}^{-1}dL_{s}^{K}(S)\]
and
\[Y_{t}^{*}=Y_{0}^{*}-\int_{0}^{t}I_{\{S_{s}<K\}}\cdot B_{s}^{-1}\cdot
(r\cdot Kds+\sigma\cdot S_{s}dW_{s}^{*})+\frac{1}{2}\int_{0}^{t}B_{s}^{-1}dL_{s}^{0}(K-S).\]
Since the local time is known to be an increasing process, it is evident that if the strike price is a positive constant and the interest rate is nonnegative, then the discounted reward process \(X^{*}\) follows a submartingale under the martingale measure \(\mathbb{P}^{*}\), that is,
\[\mathbb{E}_{\mathbb{P}^{*}}[X_{t}^{*}|{\cal F}_{s}]\geq X_{s}^{*}\mbox{ for all }s\leq t\leq T.\]
The above property follows directly from Jensen’s conditional inequality. On the other hand, the discounted reward \(Y^{*}\) of the American put option with constant exercise price is a submartingale under \(\mathbb{P}^{*}\), provided that \(r\leq 0\). Summarizing, we have the following useful result.

Proposition. The discounted reward \(X^{*}\) (resp. \(Y^{*}\)) of the American call option (resp. put option) with constant strike price follows a submartingale under \(\mathbb{P}^{*}\) if \(r\geq 0\) (resp. if \(r\leq 0\)). \(\sharp\)

Let us now examine the rational exercise policy of the holder of an option of American style. We shall use throughout the superscripts \(c\) and \(p\) to denote the quantities associated with the reward functions \(g^{c}\) and \(g^{p}\), respectively. In particular,
\begin{align*}
V_{t}^{c} & =\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [
\left .e^{-r(\tau -t)}\cdot (S_{\tau}-K_{\tau})^{+}\right |{\cal F}_{t}\right ]\\
& =e^{rt}\cdot\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}
[X_{\tau}^{*}|{\cal F}_{t}]\end{align*}
and
\begin{align*}
V_{t}^{p} & =\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [
\left .e^{-r(\tau -t)}\cdot (K_{\tau}-S_{\tau})^{+}\right |{\cal F}_{t}\right ]\\
& =e^{rt}\cdot\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}
[Y_{\tau}^{*}|{\cal F}_{t}]
\end{align*}
The Snell envelope \(J\) of the process \(X\) is defined to be the smallest supermartingale majorant to the process \(X\). From the general theory of optimal stopping, we know that
\[J_{t}=\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}
[X_{\tau}|{\cal F}_{t}]\mbox{ for all }t\in [0,T].\]
We know already that in the case of a constant strike price and under a nonnegative interest rate \(r\), the discounted reward \(X^{*}\) of a call option follows a \(\mathbb{P}^{*}\)-submartingale. Consequently, the Snell envelope \(J^{c}\) of \(X^{*}\) equals
\[J_{t}^{c}=\mbox{ess}\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}
[X_{\tau}^{*}|{\cal F}_{t}]=\mathbb{E}_{\mathbb{P}^{*}}[X_{T}^{*}|{\cal F}_{t}].\]
It says that
\[V_{t}^{c}=e^{rt}\cdot \mathbb{E}_{\mathbb{P}^{*}}[X_{T}^{*}|{\cal F}_{t}].\]
This in turn implies that for every date \(t\), the rational exercise time after time \(t\) of the American call option with a constant strike price \(K\) is the option expiry date \(T\) (the same property holds for the American put option, provided that \(r\leq 0\)). In other words, under a nonnegative interest rate, an American call option with constant strike price should never be exercised before its expiry date, and thus its arbitrage price coincides with the Black-Scholes price of a European call. Put another way, in usual circumstances, the American call option written on a non-dividend-paying stock is always worth more alive than dead, hence it is equivalent to the European call option with the same contractual features. (Let us qualify here that this remark does not apply to American currency call options.) This shows that in the case of constant strike price and nonnegative interest rate \(r\), only an American put option written on a non-dividend-paying stock requires further examination.

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

Early Exercise Representation of an American Put.

Proposition. The Snell envelope \(J^{p}\) admits the following decomposition
\[J_{t}^{p}=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .e^{-rT}\cdot (K_{T}-S_{T})^{+}\right |
{\cal F}_{t}\right ]+\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\int_{t}^{T}e^{-rs}\cdot
I_{\{\tau_{s}=s\}}\cdot (r\cdot K_{s}-k_{s})ds\right |{\cal F}_{t}\right ].\sharp\]

Recall that \(V_{t}^{p}=e^{rt}\cdot J_{t}^{p}\) for every \(t\). Consequently, the price \(P_{t}^{a}\) of an American put option satisfies
\[P_{t}^{a}=V_{t}^{p}=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .e^{-r(T-t)}\cdot
(K_{T}-S_{T})^{+}\right |{\cal F}_{t}\right ]+\mathbb{E}_{\mathbb{P}^{*}}\left [\left .
\int_{t}^{T}e^{-r(s-t)}\cdot I_{\{\tau_{s}=s\}}\cdot (r\cdot K_{s}-k_{s})ds\right |{\cal F}_{t}\right ].\]
Furthermore, in view of the Markov property of the stock price process \(S\), it is clear that for any stopping time \(\tau\in {\cal T}_{[t,T]}\), we have
\[\mathbb{E}_{\mathbb{P}^{*}}\left [\left .e^{-r(\tau -t)}\cdot (K_{\tau}-S_{\tau})^{+}
\right |{\cal F}_{t}\right ]=\mathbb{E}_{\mathbb{P}^{*}}\left [\left .e^{-r(\tau -t)}\cdot
(K_{\tau}-S_{\tau})^{+}\right |S_{t}\right ]\]
We conclude that the price of an American put equals \(P_{t}^{a}=\mathbb{P}^{a}(S_{t},T-t)\) for a certain function \(\mathbb{P}^{a}:\mathbb{R}_{+}\times [0,T]\rightarrow \mathbb{R}\). To be a bit more explicit, we define the function \(\mathbb{P}^{a}(s,t)\) by setting
\[\mathbb{P}^{a}(s,T-t)=\sup_{\tau\in {\cal T}_{[t,T]}}\mathbb{E}_{\mathbb{P}^{*}}\left [\left .
e^{-r(\tau -t)}\cdot (K_{\tau}-S_{\tau})^{+}\right |S_{t}=s\right ].\]

We assume from now on that the strike price \(K\) is constant; that is, \(K_{t}=K\) for every \(t\in [0,T]\). It is possible to show that the function \(\mathbb{P}^{a}(s,t)\) is decreasing and convex in \(s\), and increasing in \(t\). Let us denote by \({\cal C}\) and \({\cal D}\) the continuation region and stopping region, respectively. The stopping region \({\cal D}\) is defined as that subset of \(\mathbb{R}_{+}\times [0,T]\) for which the stopping time \(\tau_{t}\) satisfies
\[\tau_{t}=\int\{s\in [t,T]:(S_{s},s)\in {\cal D}\}\]
for every \(t\in [0,T]\). The continuation region \({\cal C}\) is the complement of \({\cal D}\) in \(\mathbb{R}_{+}\times [0,T]\). Note that in terms of the function \(\mathbb{P}^{a}\), we have
\[{\cal D}=\{(s,t)\in \mathbb{R}_{+}\times [0,T]:\mathbb{P}^{a}(s,T-t)=(K-s)^{+}\}\]
and
\[{\cal C}=\{(s,t)\in \mathbb{R}_{+}\times [0,T]:\mathbb{P}^{a}(s,T-t)>(K-s)^{+}\}.\]
Let us define the function \(b^{*}:[0,T]\rightarrow \mathbb{R}_{+}\) by setting
\[b^{*}(T-t)=\sup\{s\in \mathbb{R}_{+}:\mathbb{P}^{a}(s,T-t)=(K-s)^{+}\}.\]
It can be shown that the graph of \(b^{*}\) is contained in the stopping region \({\cal D}\). This means that it is only rational to exercise the put option at time \(t\) if the current stock price \(S_{t}\) is at or below the level \(b^{*}(T-t)\). For this reason, the value \(b^{*}(T-t)\) is commonly referred to as the critical stock price at time \(t\). It is sometimes convenient to consider a function \(c^{*}:[0,T]\rightarrow \mathbb{R}_{+}\) which is given by the equality \(c^{*}(t)=b^{*}(T-t)\). For any \(t\in [0,T]\), the optimal exercise time \(\tau_{t}\) after time \(t\) satisfies
\[\tau_{t}=\inf\{s\in [t,T]:(K-S_{s})^{+}=\mathbb{P}^{a}(S_{s},T-s)\},\]
or equivalently
\[\tau_{t}=\inf\{s\in [t,T]:S_{s}\leq b^{*}(T-s)\}=\inf\{s\in [t,T]:S_{s}\leq c^{*}(s)\}.\]

Proposition. The function \(c^{*}\) is nondecreasing, infinitely smooth over \((0,T)\) and \(\lim_{t\uparrow T}c^{*}(t)=\lim_{(T-t)\downarrow 0}b^{*}(T-t)=K\). \(\blacksquare\)

By virtue of the next result, the price of an American put option may be represented as the sum of the arbitrage price of the corresponding European call option and the so-called early exercise premium.

\begin{equation}{\label{musc831}}\tag{8}\mbox{}\end{equation}

Proposition \ref{musc831}. The following decomposition of the price of an American put option is valid
\[P_{t}^{a}=P_{t}+\mathbb{E}_{\mathbb{P}^{*}}\left [\left .\int_{t}^{T}e^{-r(s-t)}\cdot
I_{\{S_{s}<b^{*}(T-s)\}}\cdot t\cdot Kds\right |{\cal F}_{t}\right ],\]
where \(P_{t}^{a}=\mathbb{P}^{a}(S_{t},T-t)\) is the price of the American put option, and \(P_{t}=P(S_{t},T-t)\) is the Black-Scholes price of a European put with strike \(K\). \(\sharp\)

Decomposition of the price provided by Proposition \ref{musc831} is commonly referred to as the early exercise premium representation of an American put. Using Proposition \ref{musc831}, we get for \(t=0\)
\begin{equation}{\label{museq88}}\tag{9}
P_{0}^{a}=P_{0}+\mathbb{E}_{\mathbb{P}^{*}}\left [\int_{0}^{T}e^{-rs}\cdot I_{\{S_{s}<b^{*}(T-s)\}}\cdot r\cdot Kds\right ],
\end{equation}
where \(P_{0}\) is the price at time \(0\) of the European put. Observe that if \(r=0\), then the early exercise premium vanishes, which means that the American put is equivalent to the European put and thus should not be exercised before the epiry date. Taking into account the dynamics of the stock price under \(\mathbb{P}^{*}\), we can make representation (\ref{museq88}) more explicitly, namely
\[\mathbb{P}^{a}(S_{0},T)=P(S_{0},T)+r\cdot K\cdot\int_{0}^{T} e^{-rs}\cdot
N\left (\frac{\ln b^{*}(T-s)-\ln S_{0}-\rho\cdot s}{\sigma\cdot\sqrt{s}}\right )ds,\]
where \(\rho =r-\sigma^{2}/2\), and \(N\) denotes the standard Gaussian cumulative distribution function. A similar decomposition is valid for any instant \(t\in [0,T]\), provided that the current stock price \(S_{t}\) belongs to the continuation region \({\cal C}\); that is, the option should not be exercised immediately. We have
\[\mathbb{P}^{a}(S_{t},T-t)=P(S_{t},T-t)+r\cdot K\cdot\int_{t}^{T} e^{-r(s-t)}
\cdot N\left (\frac{\ln b^{*}(T-s)-\ln S_{t}-\rho\cdot (s-t)}{\sigma\cdot\sqrt{s-t}}\right )ds,\]
where \(P(S_{t},T-t)\) stands for the price of a European put option of maturity \(T-t\), and \(S_{t}\) is the current level of the stock price. A
change of variables leads to the following equivalent expression
\[\mathbb{P}^{a}(s,t)=P(s,t)+r\cdot K\cdot\int_{t}^{T} e^{-rs}
\cdot N\left (\frac{\ln b^{*}(t-u)-\ln s-\rho\cdot u}{\sigma\cdot\sqrt{u}}\right )du,\]
which is valid for every \((s,t)\in {\cal C}\). If \(S_{t}=b^{*}(T-t)\), then we have necessarily \(\mathbb{P}^{a}(S_{t},T-t)=K-b^{*}(T-t)\) so that clearly \(\mathbb{P}^{a}(b^{*}(t),t)=K-b^{*}(t)\) for every \(t\in [0,T]\). This simple observation leads to the following integral equation, which is satisfied by the optimal boundary function \(b^{*}\)
\[K-b^{*}(t)=P(b^{*}(t),t)+r\cdot K\cdot\int_{t}^{T} e^{-rs}
\cdot N\left (\frac{\ln b^{*}(t-s)-\ln b^{*}(t)-\rho\cdot s}{\sigma\cdot\sqrt{s}}\right )ds.\]
Unfortunately, a solution to this integral equation is not explicitly known, and thus it needs to be solved numerically. On the other hand, the following bounds for the price \(\mathbb{P}^{a}(s,t)\) are easy to derive
\[\mathbb{P}^{a}(s,t)-P(s,t)\leq r\cdot K\cdot\int_{0}^{t} e^{-ru}\cdot N\left (\frac{\ln K-\ln s-\rho\cdot u}
{\sigma\cdot\sqrt{u}}\right )du\]
and
\[\mathbb{P}^{a}(s,t)-P(s,t)\geq r\cdot K\cdot\int_{0}^{t} e^{-ru}
\cdot N\left (\frac{\ln b_{\infty}^{*}-\ln s-\rho\cdot u}{\sigma\cdot\sqrt{u}}\right )du,\]
where \(b_{\infty}^{*}\) stands for the optimal exercise boundary of a perpetual put; that is, an American put option with expiry date \(T=\infty\). To this end, it is enough to show that for any maturity \(T\), the values of the optimal stopping boundary \(b^{*}\) lie between the strike price \(K\) and the level \(b_{\infty}^{*}\), i.e., \(K\leq b^{*}(t)\leq b_{\infty}^{*}\) for every \(t\in [0,T]\). The value of \(b_{\infty}^{*}\) is known to be
\begin{equation}{\label{museq89}}\tag{10}
b_{\infty}^{*}=\frac{2r\cdot K}{2r+\sigma^{2}}.
\end{equation}

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

Analytical Approach.

As usual, we shall focus on the valuation within the Black-Scholes framework of an American put written on a stock which pays no dividends during the option’s lifetime. For a fixed expiry date \(T\) and constant strike price \(K\), we denote by \(\mathbb{P}^{a}(S_{t},T-t)\) the price of an American put at time \(t\in [0,T]\); in particular, \(\mathbb{P}^{a}(S_{T},0)=(K-S_{T})^{+}\). It will be convenient to denote by \({\cal L}\) the following differential operator
\[{\cal L}v=\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot\frac{\partial^{2}v}
{\partial s^{2}}+r\cdot s\cdot\frac{\partial v}{\partial s}={\cal A}s-r\cdot v,\]
where \({\cal A}\) stands for the infinitesimal generator of the one-dimensional diffusion process \(S\) (considered under the martingale measure \(\mathbb{P}^{*}\)). Also, let \({\cal L}_{t}\) stand for the following differential operator
\[{\cal L}_{t}v=\frac{\partial v}{\partial t}+{\cal L}v=v_{t}+{\cal A}v-r\cdot v.\]

Proposition. The American put value function \(\mathbb{P}^{a}(s,t)\) is smooth on the continuation region \({\cal C}\) with \(-1\leq P_{s}^{a}(s,T)\leq 0\) for all \((s,t)\in {\cal C}\). The optimal stopping boundary \(b^{*}\) is a continuous and nonincreasing function on \((0,t]\), hence \(c^{*}\) is a nondecreasing function on \([0,T)\), and
\[\lim_{(T-t)\downarrow 0}b^{*}(T-t)=\lim_{t\uparrow T}c^{*}(t)=K.\]
On \({\cal C}\), the function \(\mathbb{P}^{a}\) satisfies
\[P_{t}^{a}(s,t)={\cal L}\mathbb{P}^{a}(s,t);\]
that is,
\[P_{t}^{a}(s,t)=\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot P_{ss}^{a}
(s,t)+r\cdot s\cdot P_{s}^{a}(s,t)-r\cdot \mathbb{P}^{a}(s,t)\mbox{ for all }(s,t)\in {\cal C}.\]
Furthermore, we have
\begin{align*}
\lim_{s\downarrow b^{*}(t)}\mathbb{P}^{a}(s,t) & =K-b^{*}(t),\mbox{ for all }t\in (0,T],\\
\lim_{t\rightarrow 0}\mathbb{P}^{a}(s,t) & =(K-s)^{+},\mbox{ for all }s\in \mathbb{R}_{+},\\
\lim_{s\rightarrow\infty} \mathbb{P}^{a}(s,t) & =0,\mbox{ for all }t\in (0,T],\\
\mathbb{P}^{a}(s,t) & \geq (K-s)^{+},\mbox{ for all }(s,t)\in \mathbb{R}_{+}\times (0,T]. \sharp
\end{align*}

In order to determine the optimal stopping boundary, one needs to impose an additional condition, known as the {\bf smooth fit principle}, which reads as follows.

Proposition. The partial derivative \(\mathbb{P}^{a}_{s}(s,t)\) is continuous a.e. across the stopping boundary \(c^{*}\); that is,
\[\lim_{s\downarrow c^{*}(t)} \mathbb{P}^{a}_{s}(s,t)=-1\]
for almost every \(t\in [0,T]\). \(\sharp\)

We are now in a position to state the result, which characterizes the price of an American put option as the solution to the free boundary problem.

Theorem. Let \({\cal G}\) be an open domain in \(\mathbb{R}_{+}\times [0,T)\) with continuously differentiable boundary \(c\). Assume that \(v:\mathbb{R}_{+}\times [0,T]\) is a continuous function such that \(u\in C^{3,1}({\cal G})\), the function \(g(s,t)=v(e^{s},t)\) has Tychonov growth (A function \(g\in \mathbb{R}_{+}\times (0,T]\) has Tychonov growth if \(g\) and partial derivatives \(g_{s},g_{ss},g_{st}\), and \(g_{sss}\) have growth at most \(\exp (o(s^{2}))\), uniformly on a compact sets, as \(s\) tends to infinity.), and \(v\) satisfies \({\cal L}v=0\) on \({\cal G}\); that is,
\begin{equation}{\label{museq810}}\tag{11}
v_{t}(s,t)+\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot v_{ss}(s,t)+r\cdot s\cdot v_{s}(s,t)-r\cdot v(s,t)=0
\end{equation}
for every \((s,t)\in {\cal G}\), and
\begin{align*}
v(s,t) & >(K-s)^{+}\mbox{ for all }(s.t)\in {\cal G},\\
v(s,t) & =(K-s)^{+}\mbox{ for all }(s.t)\in {\cal G}^{c},\\
v(s,T) & =(K-s)^{+}\mbox{ for all }s\in \mathbb{R}_{+},\\
\lim_{s\downarrow c(t)} v_{s}(s,t) & =-1\mbox{ for all }t\in [0,T].
\end{align*}
Then the function \(\mathbb{P}^{a}(s,t)=v(s,T-t)\) for every \((s,t)\in \mathbb{R}_{+}\times [0,T]\) is the value function of the American put option with strike price \(K\) and maturity \(T\). Moreover, the set \({\cal C}={\cal G}\) is the option’s continuation region, and the function \(b^{*}(t)=c(T-t)\) for \(t\in [0,T]\), represents the critical stock price. \(\sharp\)

We shall now apply the above theorem to a perpetual put; that is, an American put option which has no expiry date (i.e. with maturity \(T=\infty\)). Since the time to expiry of a perpetual put is always infinite, the critical stock price becomes a real number \(b_{\infty}^{*}\leq K\), and the PDE (\ref{museq810}) becomes the following ordinary differential equation for
the function \(v_{\infty}(s)=v(s,\infty )\)
\begin{equation}{\label{museq811}}\tag{12}
\frac{1}{2}\cdot\sigma^{2}\cdot s^{2}\cdot\frac{d^{2}v_{\infty}(s)}
{ds^{2}}+r\cdot s\cdot\frac{dv_{\infty}(s)}{ds}-r\cdot v_{\infty}(s)=0\mbox{ for all }s\in (b_{\infty}^{*},\infty )
\end{equation}
with \(v_{\infty}(s)=K-s\) for every \(s\in [0,b_{\infty}^{*}]\). The aim is to show that equality (\ref{museq89}) is valid. For this purpose, observe that ODE (\ref{museq811}) admits a general solution of the form
\[v_{\infty}(s)=c_{1}\cdot s^{d_{1}}+c_{2}\cdot s^{d_{2}}\mbox{ for all }s\in (b_{\infty}^{*},\infty ),\]
where \(c_{1},c_{2},d_{1}\) and \(d_{2}\) are constants. Using the boundary condition
\[v_{\infty}(b_{\infty})=K-b_{\infty},\]
and the smooth fit condition
\[\lim_{s\downarrow b_{\infty}}\frac{dv_{\infty}(s)}{ds}=-1,\]
we find that
\[v_{\infty}(s)=(K-b_{\infty})\cdot\left (\frac{b_{\infty}}{s}
\right )^{2r/\sigma^{2}}\mbox{ for all }s\in (b_{\infty}^{*},\infty ),\]
and thus, in particular, equality (\ref{museq89}) is valid.

In contrast to the free boundary approach, the formulation of the optimal stopping problem in terms of variational inequalities allows us to treat the domain of the option as an entire region. In other words, we do not need to introduce explicitly here teh stopping boundary \(b^{*}\). Jaillet et al. \cite{jai} have exploited the general theory of variational inequalities to study the optimal stopping problem associated with American claims, They show that the techniques of variational inequalities provide an adequate framework for the study of numerical methods related to American options.

Theorem. Assume that \(v:\mathbb{R}_{+}\times [0,T]\) is a continuous function such that the function \(g(s,t)=v(e^{s},t)\) satisfies certain growth conditions. Suppose that for every \((s,t)\in \mathbb{R}_{+}\times [0,T]\) we have
\begin{align*}
{\cal L}_{t}v(s,t) & \leq 0,\\
v(s,t) & \geq (K-s)^{+},\\
v(s,T) & = (K-s)^{+},\\
\left ((K-s)^{+}-v(s,t)\right )\cdot {\cal L}_{t}v(s,t) & = 0.
\end{align*}
Equivalently, we have
\[v(s,T)=(K-s)^{+}\mbox{ and }\max\left\{(K-s)^{+}-v(s,t),{\cal L}_{t}v(s,t)\right\}=0.\]
Then \(v\) is unique and \(v(s,t)=\mathbb{P}^{a}(s,T-t)\) for every \((s,t)\in \mathbb{R}_{+}\times [0,T]\). \(\sharp\)

As one might expect, a solution to the problem above is not known explicitly. Numerical methods of solving variational inequalities associated with American options were developed by Jaillet et al. \cite{jai}.

\begin{equation}{\label{e}}\tag{E}\mbox{}\end{equation}

Option on a Dividend-Paying Stock.

Since most traded options on stocks are unprotected American call options written on dividend-paying stocks, it is worthwhile to comment briefly on the valuation of these contracts. A call option is said to be unprotected if it has no contarcted “protection” against the stock price decline that occurs when a dividend is paid. It is intuitively clear that an unprotected American call written on a dividend-paying stock is not equivalent to the corresponding option of European style in general. Suppose that a known dividend \(D\) will be paid to each shareholder with certainty at a prespecified date \(T_{D}\) during the option’s lifetime. Furthermore, assume that the ex-dividend stock price decline equals \(\delta D\) for a given constant \(\delta\in [0,1]\). Let us denote by \(S_{T_{D}}\) and \(P_{T_{D}}=S_{T_{D}}-\delta D\) respectively the cum-dividend and ex-dividend stock prices at time \(T_{D}\). It is clear that the option should eventually be exercised just before the dividend is paid; that is, an instant before \(T_{D}\). Consequently, as first noted by Black \cite{bla}, the lower bound for the price of such an option is the price of the European call option with expiry date \(T_{D}\) and strike price \(K\). This lower bound is a good estimate of the exact value of the price of the American option whenever the probability of early exercise is large; that is, when the probability \(P\{C_{T_{D}}<S_{T_{D}}-K\}\) is large, where \(C_{T_{D}}=C(P_{T_{D}},T-T_{D},K)\) is the Black-Scholes price of the European call option with maturity \(T-T_{D}\) and exercise price \(K\), and the shorter the time-period \(T-T_{D}\) between expiry and dividend payment dates. An analytic valuation formula for unprotected American call options on stocks with known dividends was established by Roll \cite{rol}. However, it seems to us that Roll’s original reasoning, which refers to options that expire an instant before the ex-dividend date. To avoid this discrepancy, we prefer instead to consider European options which expire on the ex-dividend date, i.e., after the ex-dividend stock price decline.

Let us denote by \(b^{*}\) the cum-dividend stock price level above which the original American option will be exercised at time \(T_{D}\) so that
\begin{equation}{\label{museq812}}\tag{13}
C(b^{*}-\delta D,T-T_{D},K)=b^{*}-K.
\end{equation}
It is worthwhile to observe that \(C(s-\delta D,T-T_{D},K)<s-K\) when \(s\in (b^{*},\infty )\), and \(C(s-\delta D,T-T_{D},K)>s-K\) for every \(s\in (0,b^{*})\). Note that the first two terms on the right-hand side of equality (\ref{museq813}) below represent the values of European options, written on a stock \(S\), which expire at time \(T\) and on the ex-dividend date \(T_{D}\), respectively. \({\bf CO}_{t}(T_{D},b^{*}-K)\) represents the price of a so-called {\bf compound option}. To be more specific, we deal here with a European call option with strike price \(b^{*}-K\) which expires on the ex-dividend date \(T_{D}\), and whose underlying asset is the European call option, written on \(S\), with maturity \(T\) and strike price \(K\). The compound option will be exercised by its holder at the ex-dividend date \(T_{D}\) if and onyl if he or she is prepared to pay \(b^{*}-K\) for the underlying European option. Since the value of the underlying option after the ex-dividend stock price decline equals \(C(P_{T_{D}},T-T_{D},K)\), the compound option is exercised whenever
\[C(P_{T_{D}},T-T_{D},K)=C(S_{T_{D}}-\delta D,T-T_{D},K)>b^{*}-K,\]
that is, when the cum-dividend stock price exceeds \(b^{*}\) (this follows from the fact thet the price of a standard European call option is an increasing function of the stock price, combined with equality (\ref{museq812})).

Proposition. The arbitrage price \(\tilde{C}_{t}^{a}(T,K)\) of an unprotected American call option with expiry date \(T>T_{D}\) and strike price \(K\), written on a stock which pays a known dividend \(D\) t time \(T_{D}\), equals
\begin{equation}{\label{museq813}}\tag{14}
\tilde{C}_{t}^{a}(T,K)=\tilde{C}_{t}(T,K)+C_{t}(T_{D},b^{*})-{\bf CO}_{t}(T_{D},b^{*}-K)
\end{equation}
for \(t\in [0,T_{D}]\), where \(b^{*}\) is the solution of \((\ref{museq812})\).

Proof. Note that the first term in (\ref{museq813}) represents the price of an option written on a dividend-paying stock, hence it is not given by the standard Black-Scholes formula. On the other hand, on the ex-dividend date \(T_{D}\) we have
\[\tilde{C}_{T_{D}}(T,K)=C(S_{T_{D}}-\delta D,T-T_{D},K)=C(P_{T_{D}},T-T_{D},K).\]
From the reasoning above, it is clear that in order to check the validity of (\ref{museq813}), it is enough to conisder the value of the portfolio of options on the ex-dividend date \(T_{D}\). Let us assume first that the cum-dividend stock price \(S_{T_{D}}\) is above the early exercise level \(b^{*}\). The value of the portfolio is
\[C(P_{T_{D}},T-T_{D},K)+(S_{T_{D}}-b^{*})-\left (C(P_{T_{D}},T-T_{D},K)-(b^{*}-K)\right )=S_{T_{D}}-K,\]
as expected. Assume now that the stock is below the level \(b^{*}\). In this case, the right-hand side of (\ref{museq813}) equals simply \(C(P_{T_{D}},T-T_{D},K)\), as the remaining options are worthless. This completes the derivation of (\ref{museq813}). \(\blacksquare\)

 

 

標籤
Hsien-Chung Wu
Hsien-Chung Wu
文章: 183

相關文章

發佈留言

發佈留言必須填寫的電子郵件地址不會公開。 必填欄位標示為 *