A New Approach for American Option Pricing: The Dynamic Chebyshev Method

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 to shift the model-dependent computations into an offline phase prior to the time-stepping. In the offline part a family of generalised (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-square Monte-Carlo method introduced by Longstaff and Schwartz (2001).

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

“…Therefore, it seems that the error on the option price exponentially grows as K increase, and so does the complexity sufficient to achieve a given error tolerance. Similar situations arose in the error analyses for LSM [53,55,57,[59][60][61] and classical Chebyshev interpolation-based methods [48]. However, we should note that ( 58) is an upper bound on the error, and that the actual error might not necessarily grows exponentially against K .…”

“…In fact, the idea that we approximate the continuation value by Chebyshev interpolation is not novel. There are some classical methods for Bermudan option pricing based on Chebyshev interpolation [44][45][46][47][48][49][50]. However, in addition to whether we use QAE or other classical methods for calculating the nodal continuation values, there are the following differences between the above proposed method and the existing methods.…”

“…Let us also mention the differences from [48]. The major difference is that, in the method in [48], the continuation value is not the target of either Monte Carlo integration or Chebyshev interpolation.…”

“…Let us also mention the differences from [48]. The major difference is that, in the method in [48], the continuation value is not the target of either Monte Carlo integration or Chebyshev interpolation. Instead, the method calculates the conditional expectations of Chebyshev polynomials by Monte Carlo or other methods, and find the Chebyshev interpolation of not the continuation value but the option price at each exercise date.…”

“…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.…”