Volume 17, Issue 1

In this note, we are concerned with approximating the distribution functions of cumulative sum (cusum) and cumulative sum of square (cusumsq) test statistics using the Hansen's (Journal of Business & Economic Statistics 15: 60–67, 1997) method. The coefficients of Hansen's polynomial are estimated by least square method. Error analysis shows that approximations are accurate. We first assume that the data are independent with common normal distribution. Then extensions to non-normal and dependent observations are considered. A conclusion section is also given.

Let the (log-)prices of a collection of securities be given by a d -dimensional Lévy process X t having infinite activity and a smooth density. The value of a European contract with payoff g ( x ) maturing at T is determined by E [ g ( X T )]. Let be a finite activity approximation to X T , where diffusion is introduced to approximate jumps smaller than a given truncation level ∈ > 0. The main result of this work is a derivation of an error expansion for the resulting model error, , with computable leading order term. Our estimate depends both on the choice of truncation level ∈ and the contract payoff g , and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error. Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.

The stochastic gradient estimation method of weak derivatives (WD) is presented with the aim of constructing efficient algorithms for the estimation of the “Greeks” of financial derivatives. The key idea is to replace the derivative of the probability measure of the underlying model by its WD. The WD method has the same advantageous property of the well-known score function method that the form of the Greek estimator does not depend on the details of the payoff function but only on the probability density of the underlying model. Simulation studies indicate that the WD estimator has significantly lower variance than the score function and finite difference estimator, however, the associated computational burden in certain cases may not be negligible.

Adaptive Monte Carlo methods are recent variance reduction techniques. In this work, we propose a mathematical setting which greatly relaxes the assumptions needed by for the adaptive importance sampling techniques presented in [Arouna, Monte Carlo Methods Appl. 10: 1–24, 2004, Arouna, The Journal of Computational Finance 7: Winter 2003/2004, Su and Fu, Journal of Computational Finance 5: 27–50, 2002, Vázquez-Abad and Dufresne, Accelerated simulation for pricing asian options: 1493–1500, IEEE Computer Society Press, 1998]. We establish the convergence and asymptotic normality of the adaptive Monte Carlo estimator under local assumptions which are easily verifiable in practice. We present one way of approximating the optimal importance sampling parameter using a randomly truncated stochastic algorithm. Finally, we apply this technique to some examples of valuation of financial derivatives.