Nonparametric estimation of an implied volatility surface

Given standard di usion-based option pricing assumptions and a set of traded European option quotes and their pay-o s at maturity, we identify a unique and stable set of di usion coe cients or volatilities. E ectively, we invert a set of option prices into a stateand time-dependent volatility function. Our problem di ers from the standard direct problem in which volatilities and maturity pay-o s are known and the associated option values are calculated. Speci cally, our approach, which is based on a small parameter expansion of the option value function, is a nite di erence-based procedure. This approach builds on previous work which has followed Tikhonov's treatment of integral equations of the Fredholm or convolution type. An implementation of our approach with CBOE S&P 500 option data is also discussed. In this paper, we address the general problem of inverting option prices into a stateand time-dependent volatility function. Speci cally, we build on the foundation of research by Rubinstein[24], Shimko[25], Derman and Kani[11] and DuPire [13]. Like Rubinstein, we use an optimization-based method with a nite set of option prices; like Shimko, Derman-Kani and Dupire we work with the continuous-time speci cation. Taken together, these works leave open two important questions regarding their volatility surface estimates: uniqueness and stability. These two characteristics are important in estimating timeand state-dependent option volatility, and the volatility surface estimation problem belongs to the mathematical class of \ill-posed" problems. The ill-posedness problem is seen in nance in the context of multicollinearity and regression, estimation of nonnormal interest rate process parameters, extraction of a term structure from a set of bond prices and asset pricing factor identi cation. Ill-posedness implies that small changes in data inputs can generate large changes in parameter estimates and estimates of parameter signi cance. In the option valuation and hedging context, the e ect of ill-posedness on derivative estimates that are used as hedging parameters is especially problematic. Our solution to this problem, built on Tikhonov's regularization approach, suggests alternative iterative discrete linear system-based estimation procedures. To implement Contemporaneous with and subsequent to our model development, Andersen and BrothertonRatcli e[5] have re ned the Shimko, Derman-Kani and Dupire interpolation-based approach. Avellaneda et. al. [6], Bouchouev[8], Brown-Toft[9], Derman, Kani and Zou[12], Jackwerth-Rubinstein[18] and Lagnado-Oscher[21] all have implemented regularization-based non-linear estimation approaches. Among this second set of works, Avellaneda et. al. is closest in spirit to our work in that they also estimate true local volatility surface. Furthermore, their regularizer is analogous to the one that we use in estimation. The other works either explicitly or implicitly estimate volatility curves for particular maturities and then interpolate local volatilities from these estimates. As these works are, e ectively, curve tting implementations, the regularizers used are de ned by second derivatives (the Laplacian) alone. For surface estimation, such regularizers are not su cient to control poles (spikes) in the parameter estimate surface. For some related discussion, also see Silverman[26]. Work following Whaba[30] is also related. We minimize a least squares goodness of t criterion that is smoothed by Sobolev norms that are related to the parameter vector. Our treatment of a complicated nonlinear model, which involves solution of a parabolic partial di erential equation subject to boundary conditions, contributes to the statistics literature. Fisher, Nychka and Zervos[14], and Adams and Van Deventer[1] smooth a deterministic term structure. Ait-Sahalia[2] has developed a kernel-based estimator for spot interest rate-contingent volatilities.  Ait-Sahalia and Lo[3] adapt this approach to the volatility surface estimation problem under particular stationarity assumptions.