Sequential Monte Carlo Methods for Option Pricing

In the following paper we provide a review and development of sequential Monte Carlo (SMC) methods for option pricing. SMC are a class of Monte Carlo-based algorithms, that are designed to approximate expectations w.r.t a sequence of related probability measures. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research. SMC methods are highly suited to many option pricing problems and sensitivity/Greek calculations due to the nature of the sequential simulation. However, it is seldom the case that such ideas are explicitly used in the option pricing literature. This article provides an up-to date review of SMC methods, which are appropriate for option pricing. In addition, it is illustrated how a number of existing approaches for option pricing can be enhanced via SMC. Specifically, when pricing the arithmetic Asian option w.r.t a complex stochastic volatility model, it is shown that SMC methods provide additional strategies to improve estimation.Comment: 37 Pages, 2 Figure

Financial Securities Under Nonlinear Diffusion Asset Pricing Model

In this thesis we investigate two pricing models for valuing financial derivatives. Both models are diffusion processes with a linear drift and nonlinear diffusion coefficient. The forward price process of these models is a martingale under an assumed risk-neutral measure and the transition probability densities are given in analytically closed form. Specifically, we study and calibrate two different families of models that are constructed based on a so-called diffusion canonical transformation. One family follows from the Ornstein-Uhlenbeck diffusion (the UOU family) and the other—from the Cox-Ingersoll-Ross process (the Confluent-U family). The first part of the thesis considers single-asset and multi-asset modeling under the ∪O∪ model. By applying a Gaussian copula, a multivariate UOU model is constructed whereby all discounted asset (forward) prices are martingales. We succeed in calibrating the ∪O∪ model to market call option prices for various companies. Moreover, the multivariate ∪O∪ model is calibrated to historical return data and captures the correlations for a pool of 4 assets. In the second part of the thesis we examine the application of the Confluent-U model to the credit risk modeling. An equity-based structural first-passage time default model is constructed based on the Confluent-U model with efficient closed-form (i.e. spectral expansions) formulas for default probabilities. The model robustness is tested by its calibration to the credit default swap (CDS) spreads for companies with various credit ratings. It is shown that the model can be accurately calibrated to the credit spreads with a piecewise default barrier level. Finally, we investigate the linkage between CDS spreads and out-of-the-money put options

Numerical methods for option pricing.

This thesis aims to introduce some fundamental concepts underlying option valuation theory including implementation of computational tools. In many cases analytical solution for option pricing does not exist, thus the following numerical methods are used: binomial trees, Monte Carlo simulations and finite difference methods. First, an algorithm based on Hull and Wilmott is written for every method. Then these algorithms are improved in different ways. For the binomial tree both speed and memory usage is significantly improved by using only one vector instead of a whole price storing matrix. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm (in C) which is capable of improving speed by a factor which equals the number of processors used. Furthermore, MatLab code for Monte Carlo was made faster by vectorizing simulation process. Finally, obtained option values are compared to those obtained with popular finite difference methods, and it is discussed which of the algorithms is more appropriate for which purpose

Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis

We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimization for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.Comment: 43 pages, 21 figures, 6 table

A Sequential Monte Carlo Approach for the pricing of barrier option in a Stochastic Volatility Model

In this paper we propose a numerical scheme to estimate the price of a barrier option in a general framework. More precisely, we extend a classical Sequential Monte Carlo approach, developed under the hypothesis of deterministic volatility, to Stochastic Volatility models, in order to improve the efficiency of Standard Monte Carlo techniques in the case of barrier options whose underlying approaches the barriers. The paper concludes with the application of our procedure to two case studies in a SABR model