Finding Gamma: A Path of Less Resistance

Peter J. Zangari, an associate at JP Morgan's risk management research, outlines an efficient method for incorporating gamma into VAR calculations for options portfolios

These days both regulators and risk managers are on a quest for efficient and cost-effective systems to compute the risk of portfolios that include nonlinear instruments-that is, options. Risk managers are looking for more accurate risk assessment tools. Regulators, meanwhile, are beginning to require firms to employ more sophisticated barometers for evaluating their exposure to market risk.

The once theoretical search for more precise measures is now being replaced with a practical need to comply with new rules. For example, according to the January 1996 "Amendment to the Basle Accord on the Capital Adequacy to Cover Market Risks," banks will be required to compute the delta, gamma and vega risk of their options positions. Whereas such calculations are relatively straightforward when applied to a single instrument, they often become ad hoc when applied to portfolios. This is especially true when value-at-risk (VAR) is computed from an estimated co-variance matrix as in the RiskMetrics framework.

Simulating the simulations

In order to properly evaluate the risk of a portfolio that contains nonlinear instruments, researchers often propose full simulation routines. According to this methodology, paths of future underlying prices are generated and the options portfolio's value is revalued at various prices along these paths.

While conceptually stimulating, this approach is both time- and computationally intensive. As part of RiskMetrics, we propose a complete and efficient methodology called "delta-gamma" to incorporate gamma risk into VAR. Delta-gamma's purpose is to provide risk managers with an adequate measure of market risk that is functional within their current risk management system.

In its standard context, VAR estimates are given by the bands of a symmetric confidence interval around zero. These bands represent the largest expected change in the value of a portfolio with a specified level of probability. For example, the VAR bands associated with a 90 percent confidence interval are given by where -/+ 1.65 are the 5th/95th percentiles of the standard normal distribution.

Accounting for gamma risk

Among other things, conditional normality implies that return distributions are symmetric. That is not the case, however, when one is analyzing portfolios containing instruments which yield nonlinear returns. Even if these returns are distributed normally, assuming a symmetric confidence interval for the VAR estimate is inappropriate. For example, portfolios that have significant gamma risk (gamma represents the sensitivity of the option's value to changes in underlying spot prices) have skewed return distributions (not bell-shaped distribution, but rather "fat" on either the right or left side). Such skewness invalidates the application of symmetry imposed by the quantiles of the standard normal distribution.

In addition, nonlinearities transform the moments in the probability analysis (such as mean, variance, skewness, etc.) of the portfolio's return distribution. Therefore, assumptions placed on the expected values of underlying returns do not necessarily carry over to a portfolio's expected values.

The chart below presents two distributions of an option's return series. One distribution is based only on the option's delta; the other is based on both delta and gamma.

A striking feature of the chart is the skewness embedded in the return distribution that includes gamma. This should not be surprising since the distribution that incorporates delta only is normal, while the gamma component involves the square of a normal random variable which is not normal.

Computing VAR of a portfolio that includes options requires finding the percentiles of that portfolio's standardized distribution. Once computed, the percentiles of this distribution are multiplied by the portfolio's standard deviation to obtain the VAR bands. Since the mathematical expression for this portfolio's return is complicated, rather than deriving its exact distribution we derive statistics that describe the exact distribution. What's convenient about this approach is that these statistics, which are called moments, depend only on the position deltas, gammas and the co-variance matrix of the underlying returns. These moments allow us to find the percentiles of a distribution that approximates the portfolio's return distribution. This is done by either matching these moments to a general family of distributions or by using normal analytical approximations.

Accuracy of delta-gamma

Delta-gamma is not only very fast in comparison to full simulation but, unlike full simulation, its speed does not depend on the number of underlying positions. Moreover, it produces reasonably accurate results. To determine the accuracy of the delta-gamma method we conducted a study that compares the VAR from delta-gamma to full simulation, using an underlying position of a FX call option and a FX put option.

The study showed that the relative error between delta-gamma and full simulation is reasonably low but becomes large as the option nears expiration and is at-the-money. The extremely large errors occur where the option is out-of-the-money, reflecting the fact that the option is valueless.

What that means for risk managers is that the usefulness of the delta-gamma method depends on the options' expiry dates versus the manager's VAR forecast horizon. Delta-gamma is most useful when the VAR forecast horizon is considerably shorter than the expiry dates on the options, and it becomes decreasingly accurate (vis-à-vis the full simulation) the closer the VAR horizon is to the option expiry date.

Overall, the usefulness of the delta-gamma method depends on how users view the trade-off between computational speed and accuracy. For risk managers seeking a quick, efficient means of computing VAR that measures gamma risk, delta-gamma offers an attractive path of less resistance.