Analysis > Methodologies

Correlated Delta-Gamma

The Correlated Delta-Gamma is a superior implementation of the Riskmetrics™ or Parametric Model.

The RiskMetric methodology is a short term (i.e. Overnight) mean-zero static model. 

This model computes the portfolio mean-zero volatility from Today up to Tomorrow.

This model implies:

  1. The timeline is static. Positions are never aged.
  2. There are no Forward Projections.  Assets are used with the Business Date's conditions (coupons, option time-to-expiration, etc), which simplifies  calculations. Forward Volatilities are therefore only needed when computing risk on fixed income derivatives.
  3. A 1 day volatility is applied to the Position As of Business Date and then adjusted for correlation between Risk Factors.
  4. Any Horizon beyond 1 day multiplies results by the square root of time.
The Correlated Delta-Gamma model computes the portfolio's daily correlated price sensitivity (dP) with regards to market risk factor changes (dV).
In other terms it measures, the change in portfolio value according to changes in correlated market risk factors by performing a Taylor series expansion of price change.

For further details see the parametric calculation principles.

The superior implementation called Correlated Delta-Gamma is completely Generic. Instead of taking the volatility of the Risk Factors multiplied by the theoretical sensitivity to price  (i.e. the delta) which in the RiskMetrics implementation is simply 0.01 (i.e. 1 percent) times the price  volatility (except for interest rates that are multiplied by the vertex duration).

The superior model take the much more complicated route of applying a full Taylor series expansion. br>
Instead of Multiplying the Price times one percent, we shift each risk factor individually and then perform a full revaluation to obtain the trade's effective fair value change rather than the theoretical change. This operation is performed up and down for both delta and Gamma in order to account for convexity.
Moreover interest rate vertices are not time adjusted shifts on the zero coupon price to convert rate to price, but instead we shift each underlying par yield, revalue the yield curve's  underlying instrument to which the vertex belongs to (Money Market, Forward Rate Agreement, Futures, Swaps, or Bonds), we then bootstrap the entire curve and extract zero coupon price changes for up and down movements

Correlated Delta-Gamma assumes a default 1% (one percent) Relative Shift on all risk factors (DV01). You can however request any shift size either  

You can also control individually each factor and impose the type and size of shift you are after.

Note a large change might cause negative forward rates to be caught by the engine when the yield curve is near-flat or downward sloping.

Negative Forward Rates can be very tricky and depend on:
 
  • The slop and curvature of the yield curve.
  • The type of instruments that are stripped and bootstrapped to build the curve
  • The forward daycount convention used for Swaps, Futures or FRAs (i.e. the number of days between End and Begin period divided by the year basis.
  • The discount factor daycount convention (days between coupon payment dates divided by days per annum, etc).
  • etc.
Correlated Delta-Gamma seems to be more sensitive to negative forward rates when the dv/dP is absolute rather than relative. This seems less problematic when yield curves are defined as prices, especially zero coupon bond prices. (not to be confused with daily zero coupon and pure zero coupon which will not test for narbtirage violations.

The Parametric model presents the advantage of being simple, fast and easy to compute. and yet acceptably accurate when dealing with linear instruments over short horizons (usually 24 hours).


The parametric model fits very nicely into the mean zero framework.

Parametric Horizon and Market Data Sampling:

The Parametric Model is a simple model that works well with linear instruments over a one day horizon. Some model implementation  actually go beyond the one day horizon and compute quarterly (balance sheet risk) monthly (pension funds) or weekly frequencies

As a rule of thumb, market data should be sampled at a frequency that is:

  1. In line with the Manager's reporting frequency.
  2. That offers reliable and commonly available market data for ALL Risk factors in the portfolio.

The Parametric Model relies heavily on the assumptions that volatility is Stationary. Hence in the Parametric Framework, portfolio volatility does NOT vary over time. 

In the case of daily returns, the portfolio's daily volatility is scaled up to the horizon sought in the analysis by simple square root of time multiplication. 

In general terms, this means volatility is constant throughout time, which is an acceptable compromise for short term horizons and portfolios that only contain linear instruments.
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When dealing with longer horizons, Autocorrelation shows up in either one of two forms: