Understanding early exercise of options - The implicit put in an American call

Question

I am self-studying for an actuarial exam on models for financial economics. I am having a hard time grasping the concept highlighted in red:

I was wondering if someone could further elaborate on why there is an implicit put option that is lost when one early exercises an American call.

I tried making a concrete example for myself to demonstrate this using a binomial tree model constructed on forward prices, but using different interest rates, volatility, dividend yields, and times to expiration, I could never create a scenario where:

1. Early exercise of the American call is rational at a node
2. The stock price could decrease below the strike price at a subsequent node from the node that the stock is exercised early

But, let's just suppose we have an American call on a stock with strike $K$ expiring at time $T$, $C(S, K, T)$, and that it is rational to early exercise at $t < T$. Suppose that at time $T$, $S < K$.

Then at time $t$, the call holder exchanges $K$ for $S$, for a payoff of $S - K > 0$. But at time $T$, $S < K$, and so he now has $S - K < 0$. If he had not exercised early, he could have not exercised at expiration and in this case he would have $K > 0$.

A put option's payoff at time $T$ would be $\max{(K - S, 0)} > 0$, since $ S < K$.

I'm not seeing an implicit put here, since the payoffs are different. Could someone explain?

Answer 1

Let’s forget about dividends (actually assume there are no dividends). By Put Call parity $C^E(K)= P^E(K) + S - Ke^{-rt}$. Suppose that $S>K$ [otherwise you don’t even think about exercising!], if you exercise the American Call now you get $S - K$ that for sure is less than the intrinsic value of the European call, i.e. when the American Call is still alive its value is at least the value of the European meaning the following chain of inequalities: $$C^{AM}(K)\geq C^E(K)= P^E(K) + S - Ke^{-rt} > S-K$$ In particular you can see clearly that are losing the value of the implicit put $P^E(K)$.

By the way, with the American Put the same chain of inequalities doesn’t hold because $P^E(K)= C^E(K) - S + Ke^{-rt}$ hence the effect is ambiguous. To convince yourself, set S=0 and you’ll see that how much you get from exercising is higher than what you get by keeping the put alive.