On the Profit and Loss Distribution of Dynamic Hedging Strategies

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Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black-Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P&L conditioned on the present value of the underlying. We consider time dynamics of this probability distribution for an arbitrary hedging strategy. We assume log-normal process for the value of the underlying asset and use convolution formula to relate conditional probability distribution of P&L at any two successive time moments. It leads to a simple PDE on the probability measure parameterized by a hedging strategy. For risk-neutral replication the P&L probability distribution collapses to a delta-function at the Black-Scholes price of the contingent claim. Therefore, our approach is consistent with the Black-Scholes one and can be viewed as its generalization. We further analyze the PDE and derive formulae for hedging strategies targeting various objectives, such as minimizing variance or optimizing distribution quantiles. The developed method of computing the profit and loss distribution for a given hedging scheme is applied to the classical example of hedging a European call option using the stop-loss strategy. This strategy refers to holding 1 or 0 shares of the underlying security depending on the market value of such security. It is shown that the stop-loss strategy can lead to a loss even for an infinite frequency of re-balancing. The analytical method allows one to compute profit and loss distributions without relying on simulations. To demonstrate the strength of the method we reproduce the Monte Carlo results on stop-loss strategy given in Hull's book, and improve the precision beyond the limits of regular Monte-Carlo simulations.