Estimation of smooth volatility functions in option pricing models

Volatility is the most difficult parameter to estimate in option pricing. We present a method in which volatility is represented as a cubic spline and estimated from market option prices. Representing volatility as a spline has two benefits. First, it is a flexible extension of the classical Black-Scholes model with constant volatility. Second, the finite number of knots in a spline enable us to overcome the ill-posedness of the local volatility function estimation problem.; Volatility estimation is crucial in computing hedge factors such as delta and gamma for risk management. We compare the constant/implied volatility approach, which is popular among practitioners, and the local volatility function approach in terms of hedging performance. We apply our methodology to S&P 500 index options and S&P 500 futures options and illustrate that the local volatility function model outperforms the constant and implied volatility models in terms of absolute delta hedge errors for hedge periods longer than 17 days and 8 days with S&P 500 index options and S&P 500 futures options, respectively.; Drift as well as volatility need to be estimated in short-term interest rate models. Now we represent both functions as cubic splines and compute them from market option data. We calibrate four classical short-term interest rate models such as Vasicek, CIR, Brennan and Schwartz, and Chan et al. as well as our extended interest rate model with spline-based drift and volatility functions to Eurodolloar futures option prices. Our empirical work shows that the 1-dimensional spline interest rate model outperforms the four classical models in terms of 1-day and 5-day prediction results as well as 1-day and 5-day delta hedge results. In addition, we show how interest rate derivatives can be priced using Green's function.