Skip to content
All library documents

When Delta Hedging Replicates Exotic Option Payoffs

Article Quant Q&A · Author: hopflink
Read the original

Summary

The document asks whether the relationship between option value and accumulated dynamic delta hedging cost, familiar from vanilla calls, also applies to an up-and-out call. The answer is affirmative under a specific modeling condition: calculate deltas with the Black–Scholes model and evolve the underlying stock using that same model. In that internally consistent setting, delta hedging is said to replicate the payoff of any contract, including an exotic option.

The response is brief and states the result without a derivation or simulation evidence. Its conclusion depends on the assumed model for both pricing and stock dynamics; it does not discuss discrete rebalancing, transaction costs, market frictions, barrier monitoring, or model misspecification. Those practical effects can cause realized hedging costs to differ from the idealized replication result. The answer therefore describes a theoretical, model-consistent claim rather than a guarantee about hedging an exotic option in live markets.

Key ideas

  • The response says dynamic delta hedging can replicate exotic option payoffs under a consistent model.
  • The Black–Scholes model must be used both to calculate deltas and to evolve the stock price.
  • The claim is theoretical and does not address discrete hedging or market frictions.

Tags

OptionsDerivatives pricingRisk management
Full text
# Delta hedging cost of exotic options?


# Delta hedging cost of exotic options?












I'm simulating dynamic delta hedging for up-and-out call option. For plain vanilla call options, I heard that the option price is the expected value of the accumulated delta hedging cost. Does it also hold for exotic option, like up-and-out call options?

## Answer by Mark Joshi (score 1)

https://quant.stackexchange.com/a/19119

Yes.

If you use the BS Model for computing deltas and the same model for evolving the stock price then you should replicate the pay-off of any contract.

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.