Omega {PerformanceAnalytics} R Documentation

## calculate Omega for a return series

### Description

Keating and Shadwick (2002) proposed Omega (referred to as Gamma in their original paper) as a way to capture all of the higher moments of the returns distribution.

### Usage

```Omega(R, L = 0, method = c("simple", "interp", "binomial", "blackscholes"), output = c("point", "full"), Rf = 0, ...)
```

### Arguments

 `R` an xts, vector, matrix, data frame, timeSeries or zoo object of asset returns `L` L is the loss threshold that can be specified as zero, return from a benchmark index, or an absolute rate of return - any specified level `method` one of: simple, interp, binomial, blackscholes `output` one of: point (in time), or full (distribution of Omega) `Rf` risk free rate, in same period as your returns `...` any other passthru parameters

### Details

Mathematically, Omega is: integral[L to b](1 - F(r))dr / integral[a to L](F(r))dr

where the cumulative distribution F is defined on the interval (a,b). L is the loss threshold that can be specified as zero, return from a benchmark index, or an absolute rate of return - any specified level. When comparing alternatives using Omega, L should be common.

Input data can be transformed prior to calculation, which may be useful for introducing risk aversion.

This function returns a vector of Omega, useful for plotting. The steeper, the less risky. Above it's mean, a steeply sloped Omega also implies a very limited potential for further gain.

Omega has a value of 1 at the mean of the distribution.

Omega is sub-additive. The ratio is dimensionless.

Kazemi, Schneeweis, and Gupta (2003), in "Omega as a Performance Measure" show that Omega can be written as: Omega(L) = C(L)/P(L) where C(L) is essentially the price of a European call option written on the investment and P(L) is essentially the price of a European put option written on the investment. The maturity for both options is one period (e.g., one month) and L is the strike price of both options.

The numerator and the denominator can be expressed as: exp(-Rf=0) * E[max(x - L, 0)] exp(-Rf=0) * E[max(L - x, 0)] with exp(-Rf=0) calculating the present values of the two, where rf is the per-period riskless rate.

The first three methods implemented here focus on that observation. The first method takes the simplification described above. The second uses the Black-Scholes option pricing as implemented in fOptions. The third uses the binomial pricing model from fOptions. The second and third methods are not implemented here.

The fourth method, "interp", creates a linear interpolation of the cdf of returns, calculates Omega as a vector, and finally interpolates a function for Omega as a function of L. This method requires library `Hmisc`, which can be found on CRAN.

Peter Carl

### References

Keating, J. and Shadwick, W.F. The Omega Function. working paper. Finance Development Center, London. 2002. Kazemi, Schneeweis, and Gupta. Omega as a Performance Measure. 2003.

`Ecdf`
```    data(edhec)