Abstract
In this article, we show how the scaling symmetry of the SABR model can be utilized to efficiently price European options. For special kinds of payoffs, the complexity of the problem is reduced by one dimension. For more generic payoffs, instead of solving the 1 + 2 dimensional SABR PDE, it is sufficient to solve Nv uncoupled 1 + 1 dimensional PDE's, where Nv is the number of points used to discretize one dimension. Furthermore, the symmetry argument enables us to obtain prices of multiple options, whose payoffs are related to each other by convolutions, by valuing one of them. The results of the method are compared with the Monte Carlo simulation. Our implementation can price 10 year swaptions with all strikes in as little as 25 milliseconds.
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