An Option to Reduce Transaction Costs

Under the assumptions of the market of Black and Scholes, options are redundant since, through the classic Black-Scholes delta hedging argument, they can be replaced by an equivalent combination the risky asset underlying the option and a risk free asset. We show that options are not redundant when small proportional transaction costs of size ε are added to the model, which provides mathematical evidence for the common belief of traders that delta-gamma hedging is superior to just delta hedging. When options are not present, the minimal loss in the value function for the expected final utility due to the presence of transaction costs is O ( ε2/3 ) . When options are present, we show the minimal loss can be reduced to O ( ε6/7 ) . Further, we compute the explicit leading order optimal strategy and the coefficient for the O ( ε6/7 ) term.