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,
2006
ISSN 14039338

Abstract:

Robust calibration of option valuation models to quoted option prices is
nontrivial, 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 NIGCIR 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 nonlinear filtering methods to daily European call option
data on the S\&P500 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