| Abstract : | Value-at-Risk represents the maximum amount should a portfolio value fall in the next N days with α% certainty. It is one of the most common risk measures deployed in order to quantify risk, due to its practical properties. Moreover, it can be a tool not only for measuring risk but also for managing it, as illustrated by processes of VaR minimization subject to return constraints. VaR of a single stock, stock portfolio, option, option portfolio can be computed numerically under simplification assumptions, such as normality of price changes, using explicit formulae. Concerning portfolio VaR, correlations need to be taken into consideration. Apart from VaR itself, tools stemming from VaR are widely utilized. These tools allow for assessing the contribution of an asset to portfolio risk expressed in VaR terms, the extent to which VaR would change if a given component was deleted from the portfolio, the change in VaR due to a new position.VaR can also be examined under the scope of Random Walk theory, more specifically under the Generalized Geometric Brownian Motion framework. Based on the latter, Monte Carlo methods can be applied so as to derive portfolio value change analytically, which is achieved by fully reassessing portfolio value through Black-Scholes framework. However, even though this methodology can be accurate, it embodies substantial computational complexities and it is time-consuming. In order to avoid the drawbacks of the Monte Carlo method, one could approximate the portfolio value and the portfolio value change, the left tail of which would represent the VaR at a predefined confidence level, using a linear (delta) approximation, or a quadratic (delta-gamma) approximation.As far as the portfolio value change is concerned, delta approximation is based on the normal distribution and delta-gamma approximation is based on the non-central chi-square distribution.Based on three model portfolios, delta-gamma approximation is proved to perform satisfactorily and much more accurately than the delta approximation. It is therefore considered a sufficient and reliable approach.Additionally, a special case has been examined: positions which are neither convex nor concave in the underlying asset. It is demonstrated that the delta approach performs better than the quadratic, albeit neither of them is accurate enough to be adopted. Finally, Value-at-Risk at 99%, 95% and 90% confidence levels are presented and compared to the estimations of the above mentioned approximations.
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