Chebyshev interpolation for parametric option pricing

Abstract

Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce runtimes while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.

Linear Credit Risk Models

“…This straightforward interpolation has the advantage to prevent the Runge's phenomenon. We refer to (Gaß, Glau, Mahlstedt, and Mair 2015) for more details on the multidimensional Chebyshev interpolation, and for an interesting financial application of multivariate function interpolation in the context of fast model estimation or calibration. The tables report the minimal, maximal, median, and root mean squared errors in basis point by maturity over the entire time period for the three different specifications.…”

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

“…As far as the author knows, this is the first proposal on the quantum method for Bermudan option pricing. Chebyshev interpolation is a widely used method for function approximation, 5 and has already been used in some (classical) methods for Bermudan option pricing [44][45][46][47][48][49][50]. In the proposed method, given the access to the quantum circuit (or, the oracle) for time evolution of underlying asset prices, we calculate the continuation values at the interpolation nodes by the quantum algorithm, and find Chebyshev interpolation using these values.…”

Improved error bound for multivariate Chebyshev polynomial interpolation

“…Here, computationally expensive integrals have to be evaluated for a large set of different parameters. At this point interpolation in the parameter space promises to be highly beneficial as shown in Gaß et al (2016).…”