Abstract
This paper explains how to calibrate a stochastic collocation polynomial against market option prices directly. The method is first applied to the interpolation of short-maturity equity option prices in a fully arbitrage-free manner and then to the joint calibration of the constant maturity swap convexity adjustments with the interest rate swaptions smile. To conclude, we explore some limitations of the stochastic collocation technique.
Highlights
The market provides option prices for a discrete set of strikes and maturities
A Lagrange polynomial gN cannot always be used to interpolate directly on the collocation points implied by the market option strikesi=0,...,N, because on one side N might be too large for the method to be practical, and on the other side, there is no guarantee that the Lagrange polynomial will be monotonic, even for a small number of strikes
Let us recall shortly some of the different approaches to build an arbitrage-free implied volatility interpolation, or equivalently, to extract the risk-neutral probability density
Summary
The market provides option prices for a discrete set of strikes and maturities. In order to price over-the-counter vanilla options with different strikes, or to hedge more complex derivatives with vanilla options, it is useful to have a continuous arbitrage-free representation of the option prices, or equivalently of their implied volatilities. We will explore how to calibrate the stochastic collocation polynomial directly to market prices, without going through an intermediate model This is of particular interest to the richer collocated local volatility (CLV) model, which allows to price exotic options through Monte Carlo or finite difference methods (Grzelak 2016). A collocation polynomial calibrated to the vanilla options market is key for the application of this model in practice Another application of our model-free stochastic collocation is to price constant maturity swaps (CMS) consistently with the swaption implied volatility smile. In the standard approximation of Hagan (2003), the CMS convexity adjustment consists in evaluating the second moment of the distribution of the forward swap rate It can be computed in closed form with the stochastic collocation.
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