Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 4

Take $r>0$, $\sigma >0$ and proportional continuous dividend yield $\delta \ge 0$, the usual perpetual-option setting. Write $L$ for the prescribed downward exercise trigger, with $0<L<X$. Above the trigger there is no dependence on calendar time, so the Black-Scholes equation with continuous stock dividends reduces toSubstitute $V=S^{\beta }$. The characteristic equation and its roots areSince the root product is $-2r/{\sigma }^2<0$, ${\beta }_{-}<0<{\beta }_{+}$. The boundary condition $V(S)\to 0$ as $S\to \infty$ excludes the positive-power solution. Value matching at the trigger requires $V(L)=X-L$, giving the fixed-rule priceThe lower branch means exercise immediately if the starting price is already below the trigger. Above it, the formula also equals ${\mathbb{E}}_Q[e^{-r{\tau }_L}(X-L)1_{\{{\tau }_L<\infty \}}]$, where ${\tau }_L$ is the first passage to $L$. To verify this expectation, apply Itô formula to the bounded stopped discounted solution on $S\ge L$; the generator equation makes it a martingale before the hit, and the residual term on surviving paths vanishes as the time horizon tends to infinity because $r>0$. A fixed trigger is a specified exercise rule, not yet the optimized American price.

For a starting price above the trigger, maximizing $V_L(S)$ amounts to maximizing $g(L)=(X-L)L^{-{\beta }_{-}}$ on $0<L<X$. Its derivative isBecause ${\beta }_{-}<0$, this is positive below ${\beta }_{-}X/({\beta }_{-}-1)$ and negative above it. Both endpoint limits of $g$ are zero. Thus the unique maximizing trigger isEquivalently smooth fit requires ${\beta }_{-}(X-S^{\ast })/S^{\ast }=-1$, matching the derivative of $X-S$ below the boundary. The optimized perpetual put option value is $X-S$ for $S\le S^{\ast }$ and $(X-S^{\ast })(S/S^{\ast }{)}^{{\beta }_{-}}$ above it.

We can verify optimality among all stopping rules, not only constant triggers. The continuation branch is convex, with value and slope matching $X-S$ at $S^{\ast }$, so it dominates $X-S$ above the boundary; it is also positive, and hence dominates $(X-S{)}^{+}$ everywhere. Its discounted generator vanishes in continuation. In the exercise region, the generator of $X-S$ is $\delta S-rX$. The root equation givesso for $\delta \ge 0$ this generator is nonpositive throughout $S\le S^{\ast }$. Smooth fit eliminates a local-time contribution at the boundary. Itô formula and localization therefore make $e^{-rt}V(S_t)$ a nonnegative supermartingale, providing an upper bound for every discounted exercise payoff. The threshold strategy attains that bound by the martingale calculation in part (a). This proves the boundary is globally optimal.

The positive-interest assumption is substantive. If $r=0$ and $\delta \ge 0$, the stock tends to zero almost surely, so the supremum of perpetual put payoffs has value $X$, approached by triggers $L\downarrow 0$. No positive finite trigger attains that supremum for positive spot. The displayed finite-boundary formula belongs to the $r>0$ case. If $r<0$ and $\delta \ge 0$, the stock still tends to zero, while the discount factor grows without bound. Even exercise at deterministic times tending to infinity makes the expected discounted put payoff unbounded, so there is no finite perpetual price in that extension.

Differentiate the negative-root equation $F({\beta }_{-},\delta )=0$, keeping $r$ and $\sigma$ fixed. Its derivatives are $F_{\delta }=-{\beta }_{-}$ andThus $d{\beta }_{-}/d\delta ={\beta }_{-}/F_{\beta }=-{\beta }_{-}/\Delta >0$. Differentiating the trigger ratio then givesThis is the dividend-yield sensitivity of the perpetual put trigger. A larger dividend yield lowers the risk-neutral drift of the ex-dividend stock; waiting for further price reductions becomes more attractive to the put holder, so exercise occurs at a lower price. As checks, at $\delta =0$ the negative root is $-2r/{\sigma }^2$, giving $S^{\ast }=2rX/(2r+{\sigma }^2)$. As $\delta \to \infty$, ${\beta }_{-}\to 0-$ and $S^{\ast }\to 0$.