The use of derivative securities for hedging and alpha-enhancement is widespread in investment markets globally. In this Learning Curve we examine how derivative strategies may be weighed up against each other in terms of risk and return characteristics. To this end, we make use of Omega functions. The result is an intuitive framework for understanding and evaluating various derivative strategies.
Much of classical financial mathematics and risk theory developed over the last few decades has been plagued by the mathematically convenient, yet somewhat unrealistic assumption that asset returns are normally distributed. This assumption seems to be reasonably adequate for many conventional asset classes. Derivative securities, however, with their non-linear and sometimes complex payoff profiles have return distributions that certainly can not be modelled using normal distributions.
It is for this reason that traditional risk measures do not make sense in the context of investment strategies involving the use of derivative securities. Risk measures such as standard deviation and the Sharpe ratio, based off only the first two distributional moments--mean and variance--do not suffice. In the simplest case, using a standard deviation metric to gauge the risk characteristics of any derivative security will lead to a completely inaccurate assessment of risk.
These shortfalls have been highlighted in several papers, and many alternative measures have been proposed. One such measure is the Omega ratio. The Omega ratio and function removes any reliance on distributional assumptions in the measurement of risk. It is derived by a simple transformation of the empirical return distribution, and hence it captures all the higher order moments of this distribution.
To calculate the Omega ratio requires two steps. In the first step, we convert the empirical return distribution or PDF to a cumulative distribution function, or CDF. In the second step, a loss threshold, l, is defined. For example, l may be set at the rate of inflation, zero, or an interest rate--in this case, any return less than the return of cash in the bank, is considered to be a bad return. The CDF is then partitioned by the threshold l, and the area to the left of l, and below the CDF, and to the right of l, and above the CDF, are computed. These respective areas are identified by the letters L and G in Figure 1 below.
The area L represents the probability weighted losses of the asset, and the area G represents the probability weighted gains of the asset. The Omega ratio at l is then simply defined as the area G divided by the area L. The analytical formulation of the Omega ratio is given by the following equation.
Note that [a, b] represents the interval on which F(x) is defined.
By computing the Omega ratio for all possible values of l, we obtain the Omega function of an asset, which will always be a monotonically decreasing function of the threshold. The Omega function has many appealing properties, but simply put, investors will always prefer the area G--the gains--to be as large as possible, while keeping the area L--the losses--as small as possible. Thus, assets with higher Omega ratios for a specified loss threshold will always be preferred to assets with lower Omega ratios in terms of both risk and return.
To illustrate the utility of Omega functions within a derivative context we construct return histories for hypothetical three-month European options on the Standard & Poor's 500 for a period of 10 years, 1996-2006. A protective put with a strike of 100% and a covered call with a strike of 110% are examined. We construct Omega functions for each strategy and for the S&P 500 index over the same period. The Omega functions are illustrated in Figure 2 below.
We first compare the Omega function of the S&P 500 index to that of the 100% protective put option strategy. Below thresholds of -1.7%, the protective put strategy has higher Omega ratios than the index. This indicates that the probability of achieving returns of less than -1.7% during any three-month period is lower for the protective put than for the index. This is intuitive of course, as we are insuring an investment in the index at 100%, and paying a premium for this insurance. The omega function indicates that we should only enter into a 100% protective put option if our expectation is that the index will fall by more than 1.7% within three months.
The 110% covered call option has lower Omega ratios than the index for thresholds above 3.2%. This indicates that the probability of achieving returns upwards of 3.2% during any three-month period is greater for an investment in the index than it is for the covered call option. The Omega function indicates that we should only enter into a 110% covered call option if our expectation is that the index will not return more than 3.2% within 3 months.
The Omega functions make the assessment of risk easier in the context of derivatives. The gearing achieved through the use of derivatives is difficult to capture in conventional analyses but is easily assimilated through the use of Omega functions.
To sum up, most conventional risk measurements assume normal return distributions, and can not be used to infer the risks inherent in derivative securities. By using Omega functions, we have shown how simple derivative strategies may be investigated. Assessment of investment strategies containing derivative structures in this manner will aid in making the most suitable strategic decision which incorporates both a view on return, and a tolerance for risk.
This week's Learning Curve was written by Mark de Araújo, derivative analyst at Peregrine Securities in South Africa, and Eben Mare, professor of mathematics and applied mathematics at the University of Pretoria in South Africa.
