Local Time for the SABR Model: Connection with the 'Complex' Black Scholes and Application to CMS and Spread Options

It is well known that the cost of a call and put option is equal to its intrinsic value plus the cost of a stop loss strategy. This stop loss strategy can be re-expressed in terms of the local time. It provides easily closed forms solution for model like Black Scholes [8] or [3]. This paper examines the theory of local time for stochastic volatility models and in particular the SABR model [5]. It gives an approximated formula for the local time in SABR and shows that this model can be valued using a Black Scholes formula but where all the terms are complex number. This formula turns out to be more robust for low and high strikes. This solves in particular the problem of valuing the whole smile in SABR as required in the replication method for CMS and the copula integration for CMS spread options.