Abstract
We introduce a new method to price American options based on Chebyshev interpolation. In each step of a dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. The key advantage of this approach is that it allows us to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalized (conditional) moments is computed by an appropriate numerical technique such as a Monte Carlo, PDE, or Fourier transform based method. Thanks to this methodological flexibility the approach applies to a large variety of models. Online, the backward induction is solved on a discrete Chebyshev grid, and no (conditional) expectations need to be computed. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. Moreover, the same family of (conditional) moments yield multiple outputs including the option prices for different strikes, maturities, and different payoff profiles. We provide a theoretical error analysis and find conditions that imply explicit error bounds for a variety of stock price models. Numerical experiments confirm the fast convergence of prices and sensitivities. An empirical investigation of accuracy and runtime also shows an efficiency gain compared with the least-squares Monte Carlo method introduced by Longstaff and Schwartz [Rev. Financ. Stud., 14 (2001), pp. 113--147]. Moreover, we show that the proposed algorithm is flexible enough to price barrier and multivariate barrier options.
Highlights
A challenging task for financial institutions is the computation of prices and sensitivities for large portfolios of derivatives such as equity options
We introduce a new method to price American options based on Chebyshev interpolation
We indicate the value function at each time step with subscript tu to directly refer to the time step tu
Summary
A challenging task for financial institutions is the computation of prices and sensitivities for large portfolios of derivatives such as equity options. In order to tackle this problem, we approximate the value function in each time step by Chebyshev polynomial interpolation. We express the value function Vt+1 as a finite sum of Chebyshev polynomials Tj(x) = cos(j acos(x)) times coefficients ctj+1 In this case, the conditional expectations become (1.1). We present the new approach to solve a dynamic programming problem (DPP) via backward induction using Chebyshev polynomial interpolation. VT (x) = g(T, x), Vtu (x) = f \bigl( g(tu, x), \BbbE [Vtu+1 (Xtu+1 )| Xtu = x]\bigr) , where t = t0 < \cdot \cdot \cdot < tn = T and f : \BbbR \times \BbbR \rightar \BbbR is Lipschitz continuous with constant Lf. At the initial time T = tn, we apply Chebyshev interpolation to the function g(T, x), i.e., for x \in \scrX ,.
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