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Why Rolling a Long Call Up Can Lower a Call Spread’s Cost

Article Quant Q&A · Author: Thomas
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Summary

The note considers why a trader might roll a long call to a higher strike after the underlying rises, especially when a quick reversal is expected. It frames the issue through a proposed inequality for Black–Scholes call prices, but the response does not prove or discuss that inequality.

Instead, the answer describes the cash-flow effect in a call spread: selling the existing lower-strike long call and buying a higher-strike call can produce a net credit. That credit reduces the trade’s cumulative net debit, and the answer says this lowers the new breakeven and raises return on investment and probability of profit. The short call is assumed to remain at its original strike. These are qualitative claims without numerical examples or a payoff analysis; the result depends on the option prices and spread structure, and the note does not establish that the roll improves expected value in general.

Key ideas

  • Rolling the long call up closes the existing call and opens a higher-strike call.
  • The answer describes a net credit when the sold call is worth more than the replacement call.
  • A net credit reduces the trade’s cumulative net debit.
  • The explanation assumes the short call strike remains unchanged and does not prove the proposed pricing inequality.

Tags

OptionsDerivatives pricingRisk management
Full text
# rolling a call up when the underlying goes up and then reverse course


# rolling a call up when the underlying goes up and then reverse course












Is it always true that for $S<S'$ and $K<K'$, $$ C(S,K') + C(S',K) - C(S',K') - C(S,K) > 0. $$ Here $C(S,K)=C(S,K; r,q,\sigma,T-t)$ is the BSM call price, assuming the other 4 parameters are set in stone.

If it is true, then it gives a rough explanation (by ignoring time-decay, for instance) of why, when longing a call (in certain options strategies) and the underlying price goes up, then it is beneficial to roll the call up to a higher strike, assuming that the chance is high that the underlying price will reverse course quickly. (Such a rolling is a recommendation by traders, but it is never clear to me what the rationale is.)

## Answer by Dr. Michael J. Stefano (score 1)

https://quant.stackexchange.com/a/85328

Here is an example with my favorite call spread:

If price rises, then when you roll the long call up to a new, higher strike, you are essentially closing the old position and opening a new one, receiving a credit for the old, lower-strike option and paying a debit for the new, higher-strike one, resulting in a NET CREDIT.

The total net debit paid for the entire trade (original debit minus the net credit received from the roll) decreases. Because the total cost of the trade is now lower, the new roi is higher, and the new breakeven point shifts downward, increasing your probability of profit and your expected value, making it easier to be profitable.

In this scenario, only the long call is rolled up and the short call strike remains the same.

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.