Option Pricing with Orthogonal Polynomial Expansions

Abstract

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.

A Lognormal Type Stochastic Volatility Model With Quadratic Drift

“…, Z d ), then by plugging them into the above scheme we obtain the quadruplets (w (k) , m (k) , v (k) , y (k) ). As highlighted by Ackerer and Filipović (2019), raw Monte-Carlo simulation with w (k) ≡ 1/K requires far too many samples to produce an accurate approximation of the distribution. Instead, deterministic discretizations of the d-dimensional standard normal distribution, such as the quantization techniques of Pagès and Printems (2003) or Gaussian cubature rules, are preferred in order to keep K small.…”

Polynomial Jump-Diffusion Models

“…See for example the option pricing in Ackerer et al (2016); Ackerer and Filipović (2017). Otherwise, one has to numerically integrate (7.7) or, equivalently, (7.9) with respect to w(dx).…”

“…Ackerer et al (2016) price options using the polynomial expansion method of Section 7 for X = Y T and the auxiliary probability kernel w(dy) given by a Gaussian measure. The option pricing performance was improved by Ackerer and Filipović (2017) using as auxiliary probability kernel w(dy) a mixture of finitely many Gaussian measures.…”

“…Lipton and Sepp (2008) extended the diffusion model by adding jumps to X t of the form as in Example 9.1 with constant jump size. Ackerer and Filipović (2017) compare Fourier transform based option pricing to the polynomial expansion method from Section 7 for X = Y T and the auxiliary probability kernel w(dy) given by a mixture of finitely many Gaussian measures.…”

“…Sepp (2016) approximated the moment generating function of Y T by a power series. Ackerer and Filipović (2017) price options using the polynomial expansion method from Section 7 for X = Y T and the auxiliary probability kernel w(dy) given by a mixture of finitely many Gaussian measures. 12…”