The role of Gamma in replicating a put

Question

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 1

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.)