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Register a new userMonte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space
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In mathematical finance and other applications of stochastic processes, it is frequently the case that the characteristic function may be known but explicit forms for density functions are not available. The simulation of any distribution is greatly facilitated by a knowledge of the quantile function, by which uniformly distributed samples may be converted to samples of the given distribution. This article analyzes the calculation of a quantile function direct from the characteristic function of a probability distribution, without explicit knowledge of the density. We form a non-linear integro-differential equation that despite its complexity admits an iterative solution for the power series of the quantile about the median. We give some examples including tail models and show how to generate C-code for examples.
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This article presents differential equations and solution methods for the functions of the form $Q(x) = F^{-1}(G(x))$, where $F$ and $G$ are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish-Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. Examples are given of equations for converting normal samples to Student t, and converting exponential to hyperbolic, variance gamma and normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of the sample space. The avoidance of any branching statement is of use in optimal GPU computations as it avoids the effect of {it warp divergence}, and we give examples of branch-free normal quantiles that offer performance improvements in a GPU environment, while retaining the best precision characteristics of well-known methods. We also offer models based on a low-probability of warp divergence. Comparisons of new and old forms are made on the Nvidia Quadro 4000, GTX 285 and 480, and Tesla C2050 GPUs. We argue that in single-precision mode, the change-of-variables approach offers performance competitive with the fastest existing scheme while substantially improving precision, and that in double-precision mode, this approach offers the most GPU-optimal Gaussian quantile yet, and without compromise on precision for Monte Carlo applications, working twice as fast as the CUDA 4 library function with increased precision.
We propose a novel algorithm which allows to sample paths from an underlying price process in a local volatility model and to achieve a substantial variance reduction when pricing exotic options. The new algorithm relies on the construction of a discrete multinomial tree. The crucial feature of our approach is that -- in a similar spirit to the Brownian Bridge -- each random path runs backward from a terminal fixed point to the initial spot price. We characterize the tree in two alternative ways: in terms of the optimal grids originating from the Recursive Marginal Quantization algorithm and following an approach inspired by the finite difference approximation of the diffusions infinitesimal generator. We assess the reliability of the new methodology comparing the performance of both approaches and benchmarking them with competitor Monte Carlo methods.
This paper sets up a methodology for approximately solving optimal investment problems using duality methods combined with Monte Carlo simulations. In particular, we show how to tackle high dimensional problems in incomplete markets, where traditional methods fail due to the curse of dimensionality.
Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions, and then establish a deterministic error bound of $O(N^{-1/d})$ for the quantile estimates, where $d$ is the dimension of the QMC point sets used in the simulation and $N$ is the sample size. Under certain conditions, we show that the mean squared error (MSE) of the randomized QMC estimate for expected shortfall is $o(N^{-1})$. Moreover, under stronger conditions the MSE can be improved to $O(N^{-1-1/(2d-1)+epsilon})$ for arbitrarily small $epsilon>0$.
We revisit and demonstrate the Epps effect using two well-known non-parametric covariance estimators; the Malliavin and Mancino (MM), and Hayashi and Yoshida (HY) estimators. We show the existence of the Epps effect in the top 10 stocks from the Johannesburg Stock Exchange (JSE) by various methods of aggregating Trade and Quote (TAQ) data. Concretely, we compare calendar time sampling with two volume time sampling methods: asset intrinsic volume time averaging, and volume time averaging synchronised in volume time across assets relative to the least and most liquid asset clocks. We reaffirm the argument made in much of the literature that the MM estimator is more representative of trade time reality because it does not over-estimate short-term correlations in an asynchronous event driven world. We confirm well known market phenomenology with the aim of providing some standardised R based simulation tools.
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