Delta-Gamma Hedging with Options of Different Maturities

Summary

The document asks whether the usual delta-gamma hedging procedure changes when the options used for the hedge have different maturities. The answer says that, within the Black-Scholes framework, options on the same underlying can have different strikes, maturities, and call or put types while still contributing to one portfolio delta and one portfolio gamma.

To calculate those aggregate sensitivities, add the individual option deltas and gammas, accounting for each position’s size and direction. The resulting portfolio exposures can then be used in the familiar system of equations to determine hedge quantities, such as the underlying position and another option position. The response provides no worked example or discussion of hedge rebalancing. Its conclusion is limited to the stated Black-Scholes setting; it does not address model risk, transaction costs, volatility-surface effects, or sensitivities beyond delta and gamma.

Key ideas

• Options on the same underlying can be aggregated across different maturities in the Black-Scholes framework.
• Portfolio delta is the position-weighted sum of the individual option deltas.
• Portfolio gamma is the position-weighted sum of the individual option gammas.
• The usual delta-gamma hedge equations can use those aggregate sensitivities.
• The answer does not cover costs, rebalancing, or risks beyond delta and gamma.

Tags

# Effect of different maturity options in delta-gamma-hedging


# Effect of different maturity options in delta-gamma-hedging












I read about hedging with options and think i got it. However there is a case am not sure how to handle.

Is there any exception in the delta-gamma-hedging-(calculaton-)technique? - say: solve an set of equations to get the needed stock -and options-amount.

All examples i have seen so far were using options with the same maturity. Will the procedure change if i consider using options with different maturity dates?

## Answer by Alex C (score 1, accepted)

https://quant.stackexchange.com/a/24582

In a Black-Scholes world a portfolio of options (some calls, some puts) of different maturities and strikes on the same underlying still has one delta and one gamma, which can be calculated by summing over the deltas and gammas. So you still have the same setup as with a single option situation.

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