P&L Analysis of Exotic Derivatives

Abstract

In this paper, we introduce a theoretical framework to formalize the analysis of the drivers of an exotic trader's P&L (profit and loss). The framework is divided into five steps: the definition of the trading context, the choice of the hedging instruments and the choice of the valuation/hedging model are the first three steps. Once these elements are defined, the fourth step consists in breaking down the dynamic of the global P&L, including both the exotic derivative exposure and the related hedged portfolio, into specific pieces. This allows to achieve the fifth and final step whose purpose is an analysis of the residual terms of the P&L and in particular, the impact of modeling assumptions on these terms. Two illustrations are proposed to detail the approach: the first one with a generic exotic equity derivative managed with a Local Volatility model, delta-vega hedged and the second one with a generic exotic interest rate derivative managed with a Hull-White model, also delta-vega hedged. These two applications led us to propose a conjecture on the writing of the P&L for models with a single state variable and raise questions regarding the calibration of model parameters.