A PDE approach to risk measures of derivatives

This paper proposes a partial differential equation (PDE) approach to calculate coherent risk measures for portfolios of
derivatives under the Black–Scholes economy. It enables us to deŽ ne the risk measures in a dynamic way and to deal
with American options in a relatively effective way. Our risk measure is based on the representation form of coherent
risk measures. Through the use of some earlier results the PDE satisŽ ed by the risk measures are derived. The PDE
resembles the standard Black–Scholes type PDE which can be solved using standard techniques from the mathematical Ž nance literature. Indeed, these results reveal that the PDE approach can provide practitioners with a more applicable and � exible way to implement coherent risk measures for derivatives in the context of the Black– Scholes model.