An American binary (or digital) call (resp. put) option with maturity \(T>0\) on an underlying asset process \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}=\left(\mathrm{e}^{r t+\sigma B_{t}-\sigma^{2} t / 2}\right)_{t \in \mathbb{R}_{+}}\)can be exercised at any time \(t \in[0, T]\), at the choice of the option holder.

The call (resp. put) option exercised at time \(t\) yields the payoff \(\mathbb{1}_{[K, \infty)}\left(S_{t}\right)\) (resp. \(\left.\mathbb{1}_{[0, K]}\left(S_{t}\right)\right)\), and the option holder wants to find an exercise strategy that will maximize his payoff.

a) Consider the following possible situations at time \(t\) :

i) \(S_{t} \geqslant K\), ii) \(S_{t}

In each case \((i)\) and (ii), tell whether you would choose to exercise the call option immediately, or to wait.

b) Consider the following possible situations at time \(t\) :
i) \(S_{t}>K\), ii) \(S_{t} \leqslant K\).
In each case \((i)\) and (ii), tell whether you would choose to exercise the put option immediately, or to wait.

c) The price \(C_{d}^{\mathrm{Am}}\left(t, T, S_{t}\right)\) of an American binary call option is known to satisfy the BlackScholes PDE \[r C_{d}^{\mathrm{Am}}(t, T, x)=\frac{\partial C_{d}^{\mathrm{Am}}}{\partial t}(t, T, x)+r x \frac{\partial C_{d}^{\mathrm{Am}}}{\partial x}(t, T, x)+\frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} C_{d}^{\mathrm{Am}}}{\partial x^{2}}(t, T, x) .\]
Based on your answers to Question (a), how would you set the boundary conditions \(C_{d}^{\mathrm{Am}}(t, T, K), 0 \leqslant t

d) The price \(P_{d}^{\mathrm{Am}}\left(t, T, S_{t}\right)\) of an American binary put option is known to satisfy the same Black-Scholes PDE \[\begin{equation*}
r P_{d}^{\mathrm{Am}}(t, T, x)=\frac{\partial P_{d}^{\mathrm{Am}}}{\partial t}(t, T, x)+r x \frac{\partial P_{d}^{\mathrm{Am}}}{\partial x}(t, T, x)+\frac{1}{2} \sigma^{2} x^{2} \frac{\partial^{2} P_{d}^{\mathrm{Am}}}{\partial x^{2}}(t, T, x) . \tag{15.39}
\end{equation*}\]
Based on your answers to Question (b), how would you set the boundary conditions \(P_{d}^{\mathrm{Am}}(t, T, K), 0 \leqslant tK ?\)

e) Show that the optimal exercise strategy for the American binary call option with strike price \(K\) is to exercise as soon as the price of the underlying asset reaches the level \(K\), i.e. at time \[\tau_{K}:=\inf \left\{u \geqslant t: S_{u}=K\right\}\]
starting from any level \(S_{t} \leqslant K\), and that the price \(C_{d}^{\operatorname{Am}}\left(t, T, S_{t}\right)\) of the American binary call option is given by \[C_{d}^{\mathrm{Am}}(t, x)=\mathbb{E}\left[\mathrm{e}^{-\left(\tau_{K}-t\right) r} \mathbb{1}_{\left\{\tau_{K}

f) Show that the price \(C_{d}^{\mathrm{Am}}\left(t, T, S_{t}\right)\) of the American binary call option is equal to \[\begin{aligned}
& C_{d}^{\mathrm{Am}}(t, T, x)=\frac{x}{K} \Phi\left(\frac{\left(r+\sigma^{2} / 2\right)(T-t)+\log (x / K)}{\sigma \sqrt{T-t}}\right) \\
& \quad+\left(\frac{x}{K}\right)^{-2 r / \sigma^{2}} \Phi\left(\frac{-\left(r+\sigma^{2} / 2\right)(T-t)+\log (x / K)}{\sigma \sqrt{T-t}}\right), \quad 0 \leqslant x \leqslant K, \end{aligned}\]
that this formula is consistent with the answer to Question (c), and that it recovers the answer to Question (a) of Exercise 15.10 as \(T\) tends to infinity.

g) Show that the optimal exercise strategy for the American binary put option with strike price \(K\) is to exercise as soon as the price of the underlying asset reaches the level \(K\), i.e. at time

\[\tau_{K}:=\inf \left\{u \geqslant t: S_{u}=K\right\},\]

starting from any level \(S_{t} \geqslant K\), and that the price \(P_{d}^{\mathrm{Am}}\left(t, T, S_{t}\right)\) of the American binary put option is

\[P_{d}^{\mathrm{Am}}(t, T, x)=\mathbb{E}\left[\mathrm{e}^{-\left(\tau_{K}-t\right) r_{1}} \mathbb{1}_{\left\{\tau_{K}

h) Show that the price \(P_{d}^{\mathrm{Am}}\left(t, T, S_{t}\right)\) of the American binary put option is equal to \[\begin{aligned}
& P_{d}^{\mathrm{Am}}(t, T, x)=\frac{x}{K} \Phi\left(\frac{-\left(r+\sigma^{2} / 2\right)(T-t)-\log (x / K)}{\sigma \sqrt{T-t}}\right) \\
& \quad+\left(\frac{x}{K}\right)^{-2 r / \sigma^{2}} \Phi\left(\frac{\left(r+\sigma^{2} / 2\right)(T-t)-\log (x / K)}{\sigma \sqrt{T-t}}\right), \quad x \geqslant K, \end{aligned}\]
that this formula is consistent with the answer to Question (d), and that it recovers the answer to Question (b) of Exercise 15.10 as \(T\) tends to infinity.

i) Does the standard call-put parity relation hold for American binary options?