## Delta–gamma approximation of Value at Risk for delta neutral derivative allocations

##### Peltonen, Valtteri (2017-08-07)

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Peltonen, Valtteri

07.08.2017

##### Tiivistelmä

The purpose of this thesis is to study the delta-gamma approximation of Value at Risk for delta neutral derivative allocations. The performance of the approximation is examined when the underlying distribution is either the t distribution or the normal distribution. The adavantage of using the t distribution is that it can model fat tails unlike the normal distribution. Derivative allocations have been studied extensively in financial literature, since they form an important class of distributions from a risk management perspective. Typically, Value at Risk is computed by simulation for derivative portfolios, because the method universally applicable. However, simulations are usually computationally burdensome. To circumvent the issue, the portfolio loss can be estimated with the delta¬–gamma approximation if the loss function is smooth enough. Simulating the delta–gamma approximation is usually much simpler than performing a full Monte Carlo re-evaluation. Also, the delta–gamma approximation admits a semi-explicit expression for the distribution of the portfolio loss in some cases, which is a useful property, since the real loss is often intractable.

The empirical part of the study demonstrates the delta–gamma approximation for three artificially created portfolios using a method presented in Glasserman, Heidelberger and Shahabuddin (2002), which exploits the characteristic function of the portfolio loss. Specifically, the characteristic function can be inverted using an inverse Fourier transform to find the distribution of the portfolio loss. However, the method requires numerical integration, which is always subject to inaccuracies. In this thesis, the analysis in Glasserman et al. (2002) is extended by introducing a tight bound for the error in the numerical integration procedure. This improves the applicability of the method.

The comparison of the normal distribution and t distribution when fitted to real returns offered no surprises: the t distribution provided a significantly better fit than the normal distribution. This was evident already when the densities of both distributions were plotted against the real returns, but was confirmed by Akaike’s information criterion. Value at Risk was computed for three different portfolios in order to demonstrate several important aspects from the point of view of risk management, including the effect of nonlinearity and maturity structure of the portfolio. For portfolios one and two the delta–gamma approximation seemed to perform relatively well compared to the full simulation approach. The biggest issue that arose was the effect of maturity, which was demonstrated by the third portfolio. As the Value at Risk horizon approached the maturity of the options, the bias in the Value at Risk estimates grew larger. Hence, it is concluded that for such portfolios Value at Risk should be used with caution. The last section of the empirical part discusses possible issues in practical applications.

The empirical part of the study demonstrates the delta–gamma approximation for three artificially created portfolios using a method presented in Glasserman, Heidelberger and Shahabuddin (2002), which exploits the characteristic function of the portfolio loss. Specifically, the characteristic function can be inverted using an inverse Fourier transform to find the distribution of the portfolio loss. However, the method requires numerical integration, which is always subject to inaccuracies. In this thesis, the analysis in Glasserman et al. (2002) is extended by introducing a tight bound for the error in the numerical integration procedure. This improves the applicability of the method.

The comparison of the normal distribution and t distribution when fitted to real returns offered no surprises: the t distribution provided a significantly better fit than the normal distribution. This was evident already when the densities of both distributions were plotted against the real returns, but was confirmed by Akaike’s information criterion. Value at Risk was computed for three different portfolios in order to demonstrate several important aspects from the point of view of risk management, including the effect of nonlinearity and maturity structure of the portfolio. For portfolios one and two the delta–gamma approximation seemed to perform relatively well compared to the full simulation approach. The biggest issue that arose was the effect of maturity, which was demonstrated by the third portfolio. As the Value at Risk horizon approached the maturity of the options, the bias in the Value at Risk estimates grew larger. Hence, it is concluded that for such portfolios Value at Risk should be used with caution. The last section of the empirical part discusses possible issues in practical applications.

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