How Gamma Affects Discrete Delta Hedging of a Put
Summary
The document explains the role of gamma when replicating a protective put by adjusting positions in the underlying asset. Under idealized continuous hedging with no transaction costs, delta hedging can replicate the option payoff, so gamma does not create a separate replication problem. In practice, hedging occurs at discrete intervals, leaving slippage between adjustments.
That slippage has a mean of zero in the stated constant-volatility geometric Brownian motion setting, but its variance grows with the option’s gamma. More frequent rebalancing reduces that variance, although it entails more trading. Because a linear position in the underlying has no gamma, it cannot by itself offset the gamma of a nonlinear option payoff; hedging gamma requires another nonlinear instrument. The explanation is deliberately narrow: it frames the result under idealized assumptions and does not quantify transaction costs, jumps, changing volatility, or the optimal rebalancing frequency in real markets.
Key ideas
- Continuous delta hedging without transaction costs can replicate the option in the stated idealized setting.
- Discrete rebalancing creates random slippage around the replication result.
- Higher gamma increases the variance of slippage from discrete hedging.
- More frequent adjustments reduce slippage variance but require more trading.
- The underlying asset alone cannot hedge option gamma because its payoff is linear.
Tags
Full text
# The role of Gamma in replicating a put # The role of Gamma in replicating a put I am analyzing portfolio protection by replication of a put. Having my portfolio with value $V$ I could buy put giving me the payoff $P$ resulting in a call like pay-off scenario $C=V+P$. Say, I don't want to buy the put but replicate it by taking positions according to the Delta. I know there are problems involved: - Black-Scholes is wrong, we have jumps, changing volatility and other things - however if we do it nevertheless then we have to trade frequently (reestimate volatility, take positions with the new Delta, ...) If I do this often and correctly. What about the Gamma of the put. I am a bit confused. Do I have to address Gamma? Gamma punishes me if I do not trade frequent enough - I know. But how does Gamma influence the success of my procedure. Say vol is constant and the stock price follows GBM and the only decision is how often I trade. How does Gamma harm me? Can I do something else besides buying other options to hedge Gamma risk or can I do something using the underlying (I assume not)? ## Answer by Mark Joshi (score 5) https://quant.stackexchange.com/a/16908 If you could hedge continuously with zero transaction costs, the gamma would be irrelevant: you would perfectly replicate with delta hedging and be done. In practice, hedging is discrete and there is a certain amount of slippage giving a random outcome with mean zero. The larger the gamma, the bigger the variance of slippage. Trading more frequently reduces the variance. You need a non-linear pay-off to get a non-zero gamma so the underlying will not help with hedging gamma risk. (see eg my book "more mathematical finance" for further discussion.)
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.