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Why Delta Hedging Works and Extending It to Multiple Risks

Article Quant Q&A · Author: Jaood
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Summary

The document explains the basic logic behind replicating an option in the Black–Scholes setting: when the underlying asset is the model’s sole source of uncertainty, a position in that asset and a risk-free account can hedge the option’s exposure. It presents the hedge as a local Taylor approximation of the option value, with delta describing sensitivity to the underlying price.

The response extends the intuition to payoffs affected by additional risk factors. A hedge must then account for those sources of uncertainty as well; for a foreign stock, for example, the exchange rate may matter alongside the stock price. This is a conceptual answer rather than a mathematical derivation of replication or a worked foreign exchange hedge. Its completeness depends on the model and available instruments: additional risks may require additional hedge assets, and the response points readers toward continuous-time arbitrage theory for further study.

Key ideas

  • In Black–Scholes, the underlying and a risk-free account can replicate an option when the underlying is the only risk source.
  • Delta represents the option’s local sensitivity to the underlying price.
  • A Taylor approximation provides intuition for why the hedge tracks changes in option value.
  • Payoffs exposed to multiple risk factors require hedges that address those factors.
  • A foreign equity payoff may depend on both the share price and the exchange rate.

Tags

OptionsDerivatives pricingRisk management
Full text
# Why does this delta hedge work, and what to do in more general circumstances?


# Why does this delta hedge work, and what to do in more general circumstances?












In the simple Black-Scholes model, we can replicate an option by investing its $\Delta$ in the underlying, and keeping that portfolio self-financing via the bank account.

I have two questions. I don't necessarily expect answers served on a platter, some references that I can read on my own would be okay too:

- Why does that strategy work? I understand it on an intuitive basis, but I don't know how to prove it via the math.

- What would you do in more general situations where the payoff of the option depends on something other than the domestic stock asset? Say it was a foreign stock asset. If the exchange rate is X (stochastic), what is the corresponding replicating portfolio?

## Answer by LetTheHuntBegin (score 2)

https://quant.stackexchange.com/a/32744

In the Black-Scholes world, one assumes that there is only one source of uncertainty in the model (namely the value of the underlying asset). So, to hedge, you only need to invest in a risk free asset and the underlying. In general, if your payoff dépends on other sources of risk, your hedging portfolio would also have to depend on those sources of risk.

One can think of a hedging portfolio as a Taylor approximation of the value of the asset. So, if the value of your asset is dépendent on several variables, your Taylor expansion (i.e. your hedge portfolio) will have to also depend on those variables.

As far as litterature on the subject goes.......I can't think of one in particular. Bjork's book "Arbitrage theory in continuous time" might be a good place to look.

Hope this helps!

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