A single number or a convergence diagram

For many applications where the practitioner has to resort to a Monte Carlo technique, one ideally wishes to obtain a single number as the answer. And there are many situations, where due to the embedding of the calculation engine one can only afford to return a single number from a calculating subroutine due to application programming interface restrictions, automation of daily reports, etc. However, in most applications, and here I speak from experience, it is possible to have a slight...

Interpolation of the term structure of implied volatility

Volatility Curve

When we value an exotic derivative contract, we will hardly ever have market information about implied volatility for all of the relevant time horizons. As a consequence, we have to use an interpolation rule to constructs paths for a Monte Carlo simulation. When practitioners require a Black-Scholes implied volatility at a point in time that is in between two maturities for which there are traded options, they frequently use linear interpolation in implied volatility over maturity. As long as...

Latin hypercube sampling

Latin Hypercube Sampling

Latin hypercube sampling isn't actually a Monte Carlo method. Latin hypercube sampling is a way to crash cars. Seriously. This technique is used when probing the sampling space is quite literally extremely expensive. Basically, a Latin hypercube sampling scheme is the attempt to place sampling points in a multi-dimensional stratification with as little overlap in all one-dimensional projections as possible. Imagine that you wish to evaluate the effect of four control parameters on the safety of...

The Brownian bridge

Brownian Bridge

Similar to the spectral path construction method, the Brownian bridge is a way to construct a discretised Wiener process path by using the first Gaussian variates in a vector draw z to shape the overall features of the path, and then add more and more of the fine structure. The very first variate z1 is used to determine the realisation of the Wiener path at the final time tn of our n-point discretisation of the path by setting Wtn v nZ. The next variate is then used to determine the value of...

Empirical discrepancies

A more thorough measure for the homogeneity properties of a low-discrepancy number generator than visual inspection of projections is the discrepancy as defined in equations 8.4 or 8.5 in section 8.1. In order to provide even harder evidence that suitably initialised Sobol' numbers are reliable indeed, even in significantly high-dimensional applications, I show in figures 8.2 to 8.9 the discrepancy with respect to the L2-norm T as defined in equation 8.4 for d 2, 3, 5,10,15, 30, 50,100 for...

Angular decomposition of lower triangular form

The form given in equation 6.3 is the most general decomposition B of any valid correlation matrix C such that C BBT. However, any so derived matrix B can be transformed by a generic orthogonal matrix O without any change of the effective correlation matrix given by the product BBT. The group of all possible orthogonal matrices O G Rnxn represents all possible rotations and reflections. Ignoring the latter and just taking into account all possible rotation matrices, we end up with rax n-1...

The Frank copula

Clayton Copula

Density of Clayton copula for 6 0.1 Density of Clayton copula for 6 0.1 1 1 1 Figure 5.7 The Clayton copula density as given by equation 5.17 . For 6 0, the density is identically 1 1 1 Figure 5.7 The Clayton copula density as given by equation 5.17 . For 6 0, the density is identically for 6 G R 0 . The Frank copula enhances the upper and lower tail dependence equally as can be seen in figure 5.8. For negative values of 6, this copula is able to produce negative codependence similar to the...

The Ali MikhailHaq copula

Ali Mikhail Haq

Density of Frank copula for 6 1 Density of Frank copula for 6 5 Density of Frank copula for 6 10 Density of Frank copula for 6 1 Density of Frank copula for 6 5 Density of Frank copula for 6 10 Figure 5.8 The Frank copula density as given by equation 5.17 . For 6 0, the density approaches the uniform density on the unit square. for 6 E -1,1 . The Ali-Mikhail-Haq copula enhances lower tail dependence for positive 6, and displays some strong negative codependence for 6 lt 0 as is shown in figure...

Introduction

We are on the verge of a new era of financial computing. With the arrival of ever faster desktop computers, clustering technology, and the tools to utilize spare cpu cycles from a variety of sources, computer processing power has expanded dramatically. This expansion, coupled with the development of new numerical methods is making techniques which were previously considered to be prohibitively computationally expensive not only feasible but the method of choice. There are countless examples of...