Fast Relevant Simulation in Finance

This book introduces new Monte Carlo methods for computing Value-at-Risk(VAR) in finance. 2 major cases (i.i.d. Monte Carlo and Markov Chain Monte Carlo) are treated in this book. New i.i.d Monte Carlo technique is based on the combination of importance sampling, non-linear optimization, and newly proposed exponential twisting density. Its theoretical justification will also be given based on the Large Deviation Theory and the Laplace method. For the Markov Chain Monte Carlo, this book introduces new techniques based on Metropolis-within-Gibbs algorithm combined with Robbins-Monro algorithm from stochastic approximation theory. Its theoretical justification will be given motivated by Ergodic Theory as well. Recently (especially after the financial crisis of 2008), industry practitioners started seeking more general non-Gaussian distribution (historical simulation, etc) and Markov Chain Monte Carlo can be used to deal with such cases. Although this book deals extensively with new techniques for VaR calculation, later chapter of this book contains several examples of its application to pricing various far out of the money options/basket options/Asian options/American options.