Volatility estimation from observed option prices

Abstract

It is well established that the standard Black-Scholes model does a very poor job in matching the prices of vanilla European options. The implied volatility varies by both time to maturity and by the moneyness of the option. One approach to this problem is to use the market option prices to back out a local volatility function that reproduces the market prices. Since option price observations are only available for a limited set of maturities and strike prices, most algorithms require a smoothing technique to implement this approach. In this paper we modify the implementation of Andersen and Brotherton-Ratcliffe to provide another way of dealing with this issue. Numerical examples indicate that our approach is reasonably successful in reproducing the input prices. Mathematics Subject Classification (2000): 91B28Journal of Economic Literature Classification: C61, C63

A New Representation of the Local Volatility Surface

“…Instead of numerically implementing Dupire's equation followed by a regularisation (e.g., [2,6] amongst others), we derived a semi-explicit solution of this equation. This solution involved functions that satisfy a first-order nonlinear system of PDEs.…”

Volatility Smile by Multilevel Least Square

“…Let us just mention Andersen et al [1,2] and Boyle et al [5] where an efficient procedure is suggested based on the use of the PDE itself to extract the volatility; there is too a pioneering study by Avellaneda et al [3] based on the maximization of an entropy function and also Lagnado et al [15,16], Jackson et al [13] and Coleman et al [7] who used, like us here, a least square fit to the financial data.…”

Numerical Procedure for Calibration of Volatility with American Options

“…Calibration with European option is made easier by the linear character of the equations, as seen by Dupire (1997). In Andersen and Brotherton-Ratcliffe (1998), Boyle and Thangaraj (2000); an efficient procedure is suggested based on the use of the PDE itself to extract the volatility; Lagnado and Osher (1997a, b), Jackson et al (1998), Coleman et al (1999), Achdou and Pironneau (2002) used, like us here, a least square fit to the financial data. By and large, least square methods are easier to stabilize than a direct use of the PDE with parameter reduction of the volatility surface by splines or other.…”