Formulating Discrete Option Hedging as a Variance-Minimization Problem

Summary

The document frames a discrete hedging problem for a sold option when the underlying stock is assumed to follow geometric Brownian motion. The hedger may trade the stock only twice before expiry and wants to choose both the trading times and stock positions to minimize the variance of trading gains minus the option payoff. The decision times are stopping times, so the strategy can potentially respond to information revealed along the path rather than being fixed in advance.

The author says that beginning with two predetermined times did not lead to a solution and asks for guidance. The text provides no proposed strategy, derivation, numerical experiment, or evidence that identifies an optimal hedge. It is therefore useful as a research problem statement: solving it would require specifying admissible trading strategies and the precise objective, then handling the joint optimization over stopping times and positions under the assumed price process. Transaction costs and other market constraints are not discussed.

Key ideas

• The problem assumes the underlying stock follows geometric Brownian motion.
• The hedger has two opportunities to trade before the option expires.
• The objective is to minimize the variance of trading gains less the option payoff.
• Both trade times and stock positions are decision variables, with trade times modeled as stopping times.
• The document poses the optimization problem but does not provide a solution.

Tags

# Discrete Hedging of Options


# 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

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