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Delta-Hedged Call Spread P&L Under Black–Scholes Assumptions

Article Quant Q&A · Author: roz
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Summary

The document considers buying a call and selling a higher-strike call below the spread’s theoretical value, then delta-hedging through expiry. One response says that under Black–Scholes dynamics, with the hedge using the true volatility, the initial pricing difference is preserved as a deterministic profit independent of the price path.

A second response emphasizes that the theoretical value depends on the model and volatility assumption used to compare it with the market price. Under idealized continuous trading and no execution costs, delta hedging leads to a payoff tied to the spread’s intrinsic value at expiry; the hedge itself depends on the chosen volatility. These conclusions rely on restrictive assumptions. The discussion does not account for discrete rebalancing, transaction costs, volatility mismatch, or departures from Black–Scholes, so it does not establish a typical realized or average profit in practice.

Key ideas

  • The claimed fixed profit depends on Black–Scholes dynamics and hedging with the actual volatility.
  • The apparent discount to theoretical value depends on the pricing model and volatility input.
  • In the idealized continuous-hedging setting, the terminal outcome is linked to the spread’s intrinsic value.
  • The hedge ratio changes with the volatility assumption used to calculate delta.
  • Real-world trading frictions and model errors can invalidate the theoretical result.

Tags

OptionsDerivatives pricingRisk management
Full text
# Delta Hedged PnL on Call Spread


# Delta Hedged PnL on Call Spread












Suppose I buy a call and then sell a call one dollar in strike higher. Suppose I get into this position for 10 cents lower than it is theoretically worth. (I.e if this spread is worth 0.50 I just bought it for 0.40). Then I delta hedge the spread to expiry. What will be my PnL? What will be my average PnL?

## Answer by P. Carr (score 2, accepted)

https://quant.stackexchange.com/a/51235

If the actual dynamics are those of Black Scholes and if the vol used in the delta hedge is the actual vol, then the P&L will be 10 cents i.e. not random and not dependent on the path.

## Answer by David Duarte (score 0)

https://quant.stackexchange.com/a/51223

When you compare the market price to the theoretical price, your difference is model dependent.

Let's say you are using Black Scholes and you're using some other vol that you think is more appropriate (you're not using the IV because that would give you the market price)

Under the Black Scholes assumptions of continuous trading and no execution costs your P/L would be the intrinsic value of the position when you delta hedged. Since you are long, if you delta hedge when the options are OTM, your P/L will probably be close to zero because the optionaliy will eventually expire and its as if the settlement price is the initial hedge price. If the options are ITM you would guarantee the intrinsic value.

This is obviously very theoretical, since these BS assumption are not observed in the real world. Also, don't forget from the start that your delta depends on the vol you considered.

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