Exact Formulas for Expected Hedging Error and Transactions Costs in Option Replication

In this paper, I derive exact formulas for expected hedging error and transactions costs in option replication for the Black-Scholes economy with exogenously fixed trading points. I derive the formulas using two different volatilities which allow the hedger to use a transactions costs adjusted volatility to determine the hedge portfolio. The expected hedging error is written in an easily recognized form. The four terms in the expectation can be interpreted as terms from Black and Scholes' (1973) formula with adjusted parameters. This interpretation holds for all future hedging periods even though the expectation is conditional on the stock price at the time of the hedging scheme's initiation. I also derive an approximation of the expected transactions costs. This approximation has a simple interpretation: for each of the future hedging periods, the approximate expected transactions costs incurred at the end of each hedging period are proportional to the option's gamma with adjusted parameters, multiplied by the squared expected value of the underlying asset. For the risk neutral economy with no volatility adjustment, I show that present values of the approximate expected transactions costs are identical for each of the future hedging intervals. Moreover, I illustrate that the approximation to the expected transactions costs is accurate except for hedging periods close to the maturity of the contingent claim. Here, the exact expectation tends to be larger than the approximation, even though the expectation is taken only with knowledge of the initial stock price. Finally, I derive an approximation of the variance of the hedging scheme's cash-flow (the hedging error minus the transactions costs) for each of the future hedging periods. This approximation facilitates evaluation of the tradeoff between cost and variance of the replication strategy.