Sequential calibration of options

Abstract

Robust calibration of option valuation models to quoted option prices is non-trivial but crucial for good performance. A framework based on the state-space formulation of the option valuation model is introduced. Non-linear (Kalman) filters are needed to do inference since the models have latent variables (e.g. volatility). The statistical framework is made adaptive by introducing stochastic dynamics for the parameters. This allows the parameters to change over time, while treating the measurement noise in a statistically consistent way and using all data efficiently. The performance and computational efficiency of standard and iterated extended Kalman filters (EKF and IEKF) are investigated. These methods are compared to common calibration such as weighted least squares (WLS) and penalized weighted least squares (PWLS). A simulation study, using the Bates model, shows that the adaptive framework is capable of tracking time varying parameters and latent processes such as stochastic volatility processes. It is found that the filter estimates are the most accurate, followed by the PWLS estimates. The estimates from all of the advanced methods are significantly closer to the true parameters than the WLS estimates which overfits data. The filters are also faster than least squares methods. All calibration methods are also applied to daily European option data on the S&P 500 index, where the Heston, Bates and NIG-CIR models are considered. The results are similar to the simulation study and it can be seen that the overfitting is a real problem for the WLS estimator when applied complex models.

Inferring Volatility Dynamics and Risk Premia from the S&P 500 and VIX Markets

“…10 Parameters obtained when calibrating to daily options prices are not stable over time, as explained in Broadie et al (2007) and Lindström et al (2008).…”

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

“…To eliminate randomness, each code was run 10 times, then the two longest and the two shortest times were removed and the mean over the remaining six times was measured. [12] in the single-asset case, and the COS method [16] in the double-asset case. For the RBF-PUM experiments we select the multiquadric basis function φ = √ 1 + ε 2 r 2 .…”

“…The FGL (Fourier-Gauss-Laguerre) method [12] is the reference solution. The second column displays Figure 1: Left: The non-uniform grid with 5776 computational nodes (76 per dimension) that was used in the two-dimensional OS experiments.…”

A Highly Efficient Pricing Method for European-Style Options Based on Shannon Wavelets

“…For the computation of the expectation we require knowledge about the probability density function governing the stochastic asset price process, which is typically not available for relevant price processes. Methods based on quadrature and the Fast Fourier Transform (FFT) [1,6,7], methods based on Fourier cosine expansions [4,12] and methods based on wavelets [8,9] have therefore been developed because for relevant log-asset price processes the characteristic function appears to be available. The characteristic function is defined as the Fourier transform of the density function.…”