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Early Exercise and Bull Call Spread Payoffs in American Options

Article Quant Q&A · Author: FawaMop
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Summary

The note compares a bull call spread’s value before expiration with its payoff at expiration, then asks whether early exercise of both American calls could realize the larger terminal spread payoff. It explains that an American call’s value is bounded below by its immediate exercise value: when the underlying is above the long call’s strike, exercising that call should not produce less than the intrinsic value captured by selling or exercising it.

The short call is unlikely to be exercised when it has no intrinsic value, so simultaneous exercise is not the expected outcome in the example. The note also cautions that the displayed current-value curve comes from a Black–Scholes valuation, which does not generally describe American options. The discussion is conceptual and does not provide a full valuation method; actual early-exercise incentives depend on contract terms and market conditions.

Key ideas

  • An American call’s value cannot fall below its immediate exercise value.
  • The short call is unlikely to be exercised when it has no intrinsic value.
  • A spread’s current option value can differ from its payoff at expiration.
  • A Black–Scholes curve for European options does not generally represent American option values.

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# Understanding American option payoff at T+0


# Understanding American option payoff at T+0












The above picture shows the payoff at expiry(in gold) and at current time T+0(in blue) for a bull call spread.

I am trying to understand American options and to know if it has any significant advantage over the European options in terms of payoff.

Looking at the payoff, assuming volatility and other factors are constant, if price rises to $120 then we would be earning close to 5 at T+0, but if that was at expiration, we would be earning approximately the max profit potential of the spread which is somewhat above 12.5. This is the case for European options.

Now my question is, assuming both options that make up the spread are American options, and I exercise the long call and my short call owner(the buyer) exercises at the exact same time(at T+0), is my payoff still going to be a profit of 5(blue payoff) or above 12.5(gold payoff) at T+0?

## Answer by MrLCh (score 1, accepted)

https://quant.stackexchange.com/a/77758

In general American options behave similar to European options but face another arbitrage boundary because of early exercise.

For example in the case where the underlying price rises to $S = \\\$120$ the value of your long call $C(K)$ with strike $K = 100$ has a lower limit. $C(K) \geq S - K = 120 - 100 = 20$. Otherwise you could buy the option and immediately realize the profit.

The problem is that your short call will never be exercised as it has no intrinsic value at the strike. But if the counterparty exercises you will realize the profit of about $ \\\$ 12.50$.

Note that the blue line does not apply in general to your case as this is the price of the option(s) under Black-Scholes model which does not apply to American options.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.