Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 / 3

Use the usual frictionless option model with nonnegative deterministic interest rates and no stock dividends. Let $B(t,T)\le 1$ be the price of a unit zero-coupon bond, $C_E$ the European call price, and $C_A$ the American call price. A call payoff dominates $S_T-X$, so no-arbitrage pricing gives $C_E\ge S_t-XB(t,T)$, and $C_A\ge C_E$. If $S_t>X$, thenThe last term is the immediate exercise payoff. Waiting retains downside optionality and defers payment of the strike. For positive interest and $X>0$ the inequality is strict while exercise has positive payoff. If $S_t\le X$, exercising for a zero payoff provides no advantage over keeping the right. Thus early exercise of a non-dividend-paying call is never strictly beneficial under these assumptions, and $C_A=C_E$. For completeness, discounted European call value is a true martingale in the standard pricing model. For any exercise stopping time $\tau \le T$, its pointwise dominance of immediate payoff gives ${\mathbb{E}}_Q[B(t,\tau )(S_{\tau }-X{)}^{+}\mid {\mathcal{F}}_t]\le {\mathbb{E}}_Q[B(t,\tau )C_E(\tau )\mid {\mathcal{F}}_t]=C_E(t)$. Taking the supremum over exercise rules proves $C_A\le C_E$, and waiting to maturity gives the reverse inequality. This proves no early exercise of a call without dividends.

The contrast for a put is the interest earned on a strike received sooner. The European put has payoff at most $X$, so $P_E\le XB(t,T)$. When interest is positive and $S_t<X(1-B(t,T))$, immediate exercise gives $X-S_t>XB(t,T)\ge P_E$. Hence committing never to exercise before maturity is worse than an available early-exercise strategy in such a deep-in-the-money state. This does not say that every put should be exercised early, only that a general prohibition is not rational.

The interest-rate hypothesis is essential to the printed claim. If $r<0$ and the stock is deterministic, $S_u=S_t{e}^{r(u-t)}$ with no dividends, choose spot sufficiently large that it stays in the money through $T$. The European call price is $S_t-X{e}^{-r(T-t)}<S_t-X$, so immediate exercise is better. At zero interest, early exercise can be indifferent in degenerate cases; it is the usual positive-interest setting that makes the non-exercise comparison strict.

Assume the stated cash dividend is the only dividend before maturity and interest rates are nonnegative. Before its ex-dividend date, compare exercise now with retaining the option until any later deterministic date $u<t_1$. There is no dividend between these dates, and retaining the option has value at least that of a European call expiring at $u$, namely at least $S_t-XB(t,u)$. This is at least the immediate in-the-money payoff, and is strictly larger with positive interest. Thus exercise earlier in the open pre-dividend interval is not optimal; the only possible dividend-induced early-exercise opportunity is the last cum-dividend instant, conventionally denoted $t_1-$.

At that instant write $S_{-}=S_{t_1-}$ and $S_{+}=S_{-}-D$, the ex-dividend price. Immediate in-the-money exercise pays $S_{-}-X$. Continuing across the dividend and retaining the call to maturity gives at least $S_{+}-XB(t_1,T)$, since no further dividend remains. Therefore continuation minus immediate exercise is bounded below byConsequently the dividend-date call-exercise criterion isIf the pre-dividend call is out of the money, exercising gives no benefit in the first place. After the dividend, part (a) excludes later early exercise. The comparison is a sufficient condition: failing it does not prove exercise is optimal, because continuation also includes put/time value not present in this lower bound.

The date convention matters. A dividend can make exercise just before the ex-dividend date optimal. Thus a literal assertion excluding $t_1-$ under all dividends would be false; the pre-dividend non-exercise argument excludes earlier dates, leaving this last cum-dividend opportunity to be tested. Further dividends would require separate tests at their own dates.

For European options with the same strike and maturity, the terminal payoff identity isThus long the call and short the put has the same terminal payoff as one stock less $X$ units of the maturity-$T$ zero-coupon bond. With no dividends, the stock strategy has no intervening payments. The law of one price gives put-call parity:Equivalently the two nonnegative terminal-payoff portfolios, call plus strike bond and put plus stock, agree in every state. The parity proof does not require lognormal stock prices or positive interest.

For American options the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+{\Pi }_P$ with early-exercise premium ${\Pi }_P\ge 0$. Hencewhich generally differs from European parity. For a concrete limiting state, if the stock is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. European parity is not a general American-option equality. It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.

The two binary terminal payoffs are complementary:in every state, including $S_T=X$. Long one of each is therefore exactly a unit zero-coupon bond. The law of one price gives digital put-call parity:With deterministic interest, the two separate values are $B(t,T)Q(S_T\ge X\mid {\mathcal{F}}_t)$ and $B(t,T)Q(S_T<X\mid {\mathcal{F}}_t)$. There is no need to assume a continuous terminal price distribution: the strict inequality for the put and weak inequality for the call allocate any atom at the strike exactly once.