Hedging Path-Dependent Options with Delta and Vanilla Options
Summary
The document asks how to maintain a hedge for path-dependent derivatives such as Asian options, where the payoff depends on the path of the underlying rather than only its terminal price. The answer sketches the standard delta-hedging setup: offset an option position with a quantity of the underlying, then use a pricing framework to describe the resulting portfolio and its risks. It notes that removing delta exposure does not remove volatility exposure, and identifies vanilla options as instruments that can be used to hedge that remaining risk in exotic positions.
The response also argues that derivative pricing and hedging are linked through risk-neutral valuation and the Black–Scholes partial differential equation, rather than requiring a directional forecast of the underlying. However, it does not derive an Asian option hedge, provide a formula for the required hedge quantities, or discuss practical rebalancing, averaging-state variables, model risk, or liquidity. Its broad claims about model construction are presented without derivation, so the note is a conceptual introduction rather than an implementation guide.
Key ideas
- Delta hedging an option uses the underlying to offset first-order price exposure.
- A delta-neutral position can still carry implied-volatility exposure.
- Vanilla options can be used to hedge volatility risk in exotic option positions.
- Risk-neutral pricing and hedging arguments connect to the Black–Scholes equation.
- Path-dependent products require hedge details beyond the general outline provided here.
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Full text
# Hedging exotic options
# Hedging exotic options
How can exotic and other path dependent, such as asian options be hedged? For example in the case of an asian option, what is the replicating portfolio: what instruments to keep in it and “how much”?
It is known from the standard Black-Scholes model that when we replicate a vanilla European we have to hold $\Delta_{t}$ (the partial derivative of the option PV corresponding to the variable of the stock price) underlying in every $t$, but how is the replication of an asian option (or any other exotic option) maintained in theory/practice?
In general, literature firstly always discuss what the price of an option is as calculating a tipically very tough expectation. It is always good to know what the price is, but the other important question is how to hedge these options, i.e. what strategy to use in order to construct a replicating portfolio. I think this second question is rarely discussed, even though it is probably more important then knowing the price. (Additionally, in my opinion determining the price is also part of the “strategy”, but it is just my opinion.)
## Answer by THATS MY QUANT MY QUANTITATIVE (score 0)
https://quant.stackexchange.com/a/77753
If you've hedged away delta using a replicating portfolio, you become exposed to implied volatility, hence vanilla options are used to hedge exotics.
In regards to your 2nd question, you are thinking of pricing derivatives backwards. Risk-neutral pricing is just an accounting formula and the stock dynamics is the conclusion of that formula.
We construct a portfolio, $\Pi$ that is short an option, $V$ and long $a$ shares.
$$\Pi = - V + aS$$ $$d\Pi = ...$$ Then using the risk-neutral accounting formula and the Feynmann-kac formula, we get the Black-Scholes PDE. The black-scholes SDE is the probabilistic representation of the accounting PDE.
We don't require any assumptions about the dynamics of the stock so there is no "prediction". The Black-Scholes model is a consequence of the accounting formula. That's why the model was awarded a Nobel prize. Previous models assumed some type of asset dynamics to price options, whilst in Black and Scholes' paper, they didn't need to make any assumptions of the underlying’s dynamics.
It's the same with the Heston model. We don't assume the stock's dynamics is like the Heston model - it's a consequence of hedging volatility.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.