| Περίληψη : | The main objective of this paper is to demonstrate different methodologies for calculating Value-at-Risk, or “VaR”, for portfolios that include financial instruments with non-linear payoffs, like options. Risk measurement for options is more complex than is for linear positions, since movements in the underlying risk factor (stock-prices) have a non-linear impact on option prices and option prices themselves depend on volatility,which is not directly observable on capital markets.We propose the “Delta-Gamma method” for fast and accurate computation of Value-at-Risk in large complex portfolios. We focus on the quadratic portfolio model, also known as “Delta-Gamma” that is based on a second order Taylor-series approximation of the nonlinear option pricing relationship. The main difficulty to this method is the estimation of the required quantile of the profit and loss distribution, since there exists no analytical representation of this distribution.In our analysis we examine three model portfolios and it is proved that the accuracy of the delta approximation, which is based on the normal distribution, is rather poor when assessing Value-at-Risk in portfolios containing non-linear assets. On the contrary, the delta-gamma approximation, which is based on the non-central chi-square distribution,performs much more accurately and satisfactorily. For realistic nonlinear portfolios, such work is often carried out using Monte Carlo trials. Monte Carlo methods can be applied so as to derive portfolio value change analytically,which is achieved by fully reassessing portfolio value through the Black-Scholes framework. However, such computations can be very time and resourse consuming. The delta-gamma method developed in this paper is not subject to the statistic uncertainty of the Monte Carlo method and it is substantially less resource intensive and time consuming than full-revaluation.Furthermore, in our analysis, we examine a special case in which option positions have both concave and convex regions in the range of likely prices of the underlying asset. In this case, we observe that the delta-gamma method provides a poor approximation to the distribution of actual portfolio value and that the delta–only approach performs much better.In the paper, VaR estimations derived by the linear (delta-only)approximation and the delta-gamma (linear-quadratic) approximation are compared to the simulated actual losses derived from the Monte Carlo simulation method. One of our main conclusions is that the results from deltanormal method, for portfolios that include options, sometimes differ greatly from full valuation approaches. The differences can become unacceptably large, particularly for short-maturity option positions, for highly nonlinear instruments, for long VaR horizons and high confidence levels.
|
|---|
Copyright © 2013 Βιβλιοθήκη Ο.Π.Α. Με την επιφύλαξη παντός δικαιώματος.