Derivatives Performance Attribution

This paper shows how to decompose the dollar profit earned from an option into two basic components: ¶ mispricing of the option relative to the asset at the time of purchase, and · profit from subsequent fortuitous changes or mispricing of the underlying asset. This separation hinges on measuring the “true relative value” of the option from its realized payoff. The payoff from any one option has a huge standard error about this value which can be reduced by averaging the payoff from several independent option positions. It appears from simulations that 95% reductions in standard errors can be further achieved by using the payoff of a dynamic replicating portfolio as a Monte Carlo control variate. In addition, it is shown that these low standard errors are robust to discrete rather than continuous dynamic replication and to the likely degree of misspecification of the benchmark formula used to implement the replication. The first basic component, the option mispricing profit, can be further decomposed into profit due to superior estimation of the volatility (volatility profit) and profit from using a superior option valuation formula (formula profit). In order to make this decomposition reliably, the benchmark formula used for the attribution needs to be similar to the formula implicitly used by the market to price options. If so, then simulation indicates that this further decomposition can be achieved with low standard errors. The second basic component can be further decomposed into profit from a forward contract on the underlying asset (asset profit) and what I term pure option profit. The asset profit indicates whether or not the investor was skillful by buying or selling options on mispriced underlying assets. However, asset profit could also simply be just compensation for bearing risk -a distinction beyond the scope of this paper. Although simulation indicates that the attribution procedure gives an unbiased allocation of the option profit to this source, its standard error is large -a feature common with attempts by others to measure performance of assets. May 3, 1998 Mark Rubinstein is the Paul Stephens Professor of Applied Investment Analysis at the University of California at Berkeley. The author would like to thank Barra, Inc. for encouraging this research. 2 Derivatives Performance Attribution 2