# Discrete Hedging of Options

Assume that a stock $S_t$ follows simple geometric Brownian motion. Let's say we sold option whose payoff is $f(S_T)$. Now, we are only allowed to trade 2 times in the interval [0,T]. What kind of strategy should we use to minimize the variance of (gains through trading - payoff of option)? This accounts to figuring out 2 Stopping times $T_1 \leq T_2 \leq T$ and the amount to invest in the stock at these times. I tried to solve it by starting with given time $t_1$ and $t_2$ but I did not get far from here. Any help is highly appreciated. Thanks

• How can one have any hope of answering this question without knowing anything about the asset price process $\{ S_t \}$? – user32416 Sep 13 '15 at 1:30
• Assume that $S_t$ follows simple geometric Brownian motion. Thanks – Chandrasekhar Sep 13 '15 at 1:39
• I don't have a full answer but to help you get started you could notice that the "Hedging Error" (Difference from actual portfolio value vs theoretical BS) is a random variable coming from the Chi-Squared distribution. The total hedging error at expiration should have a standard deviation proportional to the square root of the time between re-hedges. This leads me to believe that optimal re-balancing would be at T/3 and 2T/3, but I can provide no proof. However, with discrete hedging your PNL will be highly path-dependent, so in terms of practicality you would hedge based on that. – meh Sep 14 '15 at 17:54