Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

This paper investigates the information contained in the S\&P 500 and VIX markets. We estimate a flexible affine model over a time series of underlying indices and option prices on both markets simultaneously. We find that the S\&P 500 and VIX derivatives markets contain conflicting information on variance, especially during market distress. We analyze the information spanned by each market on the distributions and tail properties of S\&P 500 returns and their variance process. An extensive model specification analysis reveals that jumps and two variance factors help reproduce these distributions and play a significant role in the variance risk premium.

/pdf/inferring-volatility-dynamics-and-risk-premia-from-the-s-p-5fby9h13oc.pdf

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

http://www.elisegourier.com/uploads/3/7/9/6/37964671/bardgett_gourier_leippold_vix_paper.pdf

Inferring volatility dynamics and risk premia from the S&P 500 and VIX markets

The aim of the BENCHOP project is to provide the finance community with a common suite of benchmark problems for option pricing. We provide a detailed description of the six benchmark problems together with methods to compute reference solutions. We have implemented fifteen different numerical methods for these problems, and compare their relative performance. All implementations are available on line and can be used for future development and comparisons.

/pdf/benchop-the-benchmarking-project-in-option-pricing-33lw1i2jud.pdf

BENCHOP – The BENCHmarking project in option pricing

In the search for robust, accurate, and highly efficient financial option valuation techniques, we here present the SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper size of the computational interval. Numerical experiments on European-style options show exponential convergence and confirm the bounds, robustness, and efficiency.

/pdf/a-highly-efficient-shannon-wavelet-inverse-fourier-technique-1e4nmmb58a.pdf

A highly efficient Shannon wavelet inverse Fourier technique for pricing European options

Sequential calibration of options

This paper concerns the use of the expectation-maximisation (EM) algorithm for inference in partially observed diffusion processes. In this context, a well known problem is that all except a few diffusion processes lack closed-form expressions of the transition densities. Thus, in order to estimate efficiently the EM intermediate quantity we construct, using novel techniques for unbiased estimation of diffusion transition densities, a random weight fixed-lag auxiliary particle smoother, which avoids the well known problem of particle trajectory degeneracy in the smoothing mode. The estimator is justified theoretically and demonstrated on a simulated example.

/pdf/particle-based-likelihood-inference-in-partially-observed-1itdgshpl6.pdf

Jonas Ströjby

Papers

Sequential calibration of options

Particle-based likelihood inference in partially observed diffusion processes using generalised Poisson estimators

Efficient Particle-Based Likelihood Estimation in Partially Observed Diffusion Processes

Likelihood Inference in Jump Diffusion driven SDE's

Calibration of Option Valuation Models using Sequential Monte Carlo Methods