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.

Continuous-time stochastic processes have become central to many disciplines, yet the fact that they are approximations to physically realizable phenomena is often overlooked. We quantify one aspect of the approximation errors of continuous-time models by investigating the replication errors that arise from delta hedging derivative securities in discrete time. We characterize the asymptotic distribution of these replication errors and their joint distribution with other assets as the number of discrete time periods increases. We introduce the notion of temporal granularity for continuous-time stochastic processes, which allows us to quantify the extent to which discrete-time implementations of continuous-time models can track the payoff of a derivative security. We show that granularity is a function of the contract specifications of the derivative security, and of the degree of market completeness. We derive closed form expressions for the granularity of geometric Brownian motion and of an Ornstein–Uhlenbeck process for call and put options, and perform Monte Carlo simulations that illustrate the practical relevance of granularity.

When is time continuous

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

Sequential calibration of options

We analyze the errors arising from discrete readjustment of the hedging portfolio when hedging options in exponential Levy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases. We compare the quadratic hedging strategy with the common market practice of delta hedging, and show that for discontinuous option pay-offs the latter strategy may suffer from very large discretization errors. For options with discontinuous pay-offs, the convergence rate depends on the underlying Levy process, and we give an explicit relation between the rate and the Blumenthal-Getoor index of the process.

Tracking errors from discrete hedging in exponential L\'evy models

Tracking errors from discrete hedging in exponential Lévy models

Hedging errors induced by discrete rebalancing of the hedge portfolio of a delta-gamma hedging strategy are investigated. The rate of convergence of the expected squared hedging error as the number of adjustments of the hedge portfolio goes to infinity is analyzed. It is found that the delta-gamma strategy produces higher convergence rates than the usual delta strategy.

On the Convergence of Higher Order Hedging Schemes: The Delta-Gamma Case

Discrete time hedging in a complete diffusion market is considered. The hedge portfolio is rebalanced when the absolute difference between delta of the hedge portfolio and the derivative contract reaches a threshold level. The rate of convergence of the expected squared hedging error as the threshold level approaches zero is analyzed. The results hinge to a great extent on a theorem stating that the difference between the hedge ratios normalized by the threshold level tends to a triangular distribution as the threshold level tends to zero.

Mats Brodén

Papers

Sequential calibration of options

Tracking errors from discrete hedging in exponential L\'evy models

Tracking errors from discrete hedging in exponential Lévy models

On the Convergence of Higher Order Hedging Schemes: The Delta-Gamma Case

Hedging Errors Induced by Discrete Trading Under an Adaptive Trading Strategy