Rapid globalization, innovation in the design of derivative securities, and examples of spectacular losses associated with derivatives over the past decade have made firms recognize the growing importance of risk management. This increased focus on risk management has led to the development of various methods and tools to measure the risks firms face. One popular risk-measurement tool is Value at Risk(VaR) which is defined as the maximal amount that may be lost in a portfolio over a given period of time, at a certain confidence level. In this thesis we compute VaR using the Dirichlet process based on the article by Mahmoud Zarepour, Thierry Bedard, Andre R.Dabrowski(june, 2008). Statistically speaking, the VaR of a portfolio is the quantile of the distribution of returns. Return is defined as the logarithm of ratio of the asset prices. The fact that returns need not follow the distribution specified in any specific model is a significant limitation. The goal of this thesis is to develop an adaptive method of estimating VaR in financial markets for a random walk setting i.i.d returns, but where we avoid specifying the distribution the returns. We introduce an alternative methodology to simulate these returns by using a Montecarlo simulation of the Dirichlet process and estimate the quantile of these simulated returns. This is a Bayesian approach that uses the past data directly in the simulation of asset price evolution, and thereby avoids choosing a particular model for the asset price. The analyst’s experience enters into the method through a prior guess for the distribution of Y and the degree of confidence he has in the prior distribution. As with all Bayesian procedures, past data is used to update the prior distribution. The posterior distribution is then used to simulate future market behavior and perhaps to produce a credible confidence interval for VaR.We then estimate VaR produced by this method, particularly for cases where tails of the distribution for returns are assumed to be heavy (for example stable). Finally we establish that the produced estimation is a consistent estimator of VaR.
