Sequential Calibration of Options

Mats Brodén, Jan Holst, Erik Lindström, Jonas Ströjby and Magnus Wiktorsson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1403-9338
Robust calibration of option valuation models to quoted option prices is non-trivial, but as important for good performance as the valuation model itself. The standard textbook approach to option calibration is minimization of a suitably chosen measure of the prediction error, e.g. least squares minimization.
This paper interpretes the total prediction error as a sum of the measurement errors and effects from the parameter dynamics. We introduce an apriori dynamics for the parameters. This will allow the parameters to change over time, while treating the measurement noise in a statistically consistent way and using all data efficiently.
We here use models for which closed form expressions or Fourier transform methods are available, e.g. exponential Levy processes or certain stochastic volatility models. We use the Heston, Bates and NIG-CIR models in this paper.
We investigate the performance and computational efficiency of standard and iterated Extended Kalman filters (EKF and IEKF) as well as Particle Filters (PF).These methods are then compared to the common practice calibration method of Weighted Least Squares (WLS) and penalised Weighted Least Squares (PWLS).
Our simulation study, using the Heston model, has shown that the introduced filter framework is capable of tracking time varying parameters and latent processes such as stochastic volatility processes. We find that the filter estimates and the PWLS estimates are much closer to the true parameters than the WLS estimates. When we use the same parameters to price Binary options we find that the prices obtained using the WLS estimates are inferior to those obtained by the other methods. When pricing exotic derivatives using models calibrated on European call option data we believe that the filter methods will perform better than the WLS.
We also apply non-linear filtering methods to daily European call option data on the S\&P-500 stock index. If we price options, using yesterdays estimated parameters and todays estimated latent process and today's stock price there is a general decrease in global fit over all models and all methods. This tendency is most pronounced for the WLS method indicating overfitting. We therefore find that there is need for adaptive calibration methods to handle time varying parameters in an accurate and robust way.
Key words:
Sequential Monte Carlo filters, Calibration, Option Valuation