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.
Highlights
The arbitrage-free option price function defined across strike price and estimated from the available quotes has been studied extensively by researchers and practitioners
Using the properties of the Bernstein polynomial basis, the inequality constraints on BN (x; f ) and its the derivatives can be transformed to the linear inequality constraints involving βN only
The estimated state price density functions are positive in the entire domain, which shows that our proposed estimator satisfies the inequality arising from the convexity constraint
Summary
The arbitrage-free option price function defined across strike price and estimated from the available quotes has been studied extensively by researchers and practitioners (see, e.g., [ – ]). Aït-Sahalia and Duarte [ ] proposed a two-step method to estimate the arbitrage-free call price function They use a constrained least-squares procedure which incorporate no-arbitrage shape constraints of monotonicity and convexity and employ smoothing using local polynomials. We use a Bernstein polynomial basis [ ] to estimate the arbitrage-free option price function The choice of this basis is straightforward as the constraints of monotonicity, convexity, and other bounds on the function and its first derivative on the entire domain leads to a set of linear inequality constraints on unknown parameters. Section describes our proposed methodology to approximate the call price function using a Bernstein polynomial basis under the various inequality constraints arising from no-arbitrage conditions and derive the quadratic programming formulation of the estimation problem. An arbitrage-free call price function must satisfy the set of inequality constraints given by ( ), ( ), and ( )
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