Call option price function in Bernstein polynomial basis with no-arbitrage inequality constraints

Abstract

We propose an efficient method for the construction of an arbitrage-free call option price function from observed call price quotes. The no-arbitrage theory of option pricing places various shape constraints on the option price function. For each available maturity on a given trading day, the proposed method estimates an option price function of strike price using a Bernstein polynomial basis. Using the properties of this basis, we transform the constrained functional regression problem to the least-squares problem of finite dimension and derive the sufficiency conditions of no-arbitrage pricing to a set of linear constraints. The resultant linearly constrained least square minimization problem can easily be solved using an efficient quadratic programming algorithm. The proposed method is easy to use and constructs a smooth call price function which is arbitrage-free in the entire domain of the strike price with any finite number of observed call price quotes. We empirically test the proposed method on S&P 500 option price data and compare the results with the cubic spline smoothing method to see the applicability.

Universal Arbitrage-Free Estimation of State Price Density

“…-Various kernel functions have been applied to approximate the call option price surface in previous studies, such as low order polynomial (Kundu et al (2016)), radial basis function (Lai (2011)) and cubic spline kernel (Fengler (2009)). In this paper, since the smallest relative distance is obtained by using a cubic spline kernel, we compare our method with Fengler (2009).…”