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Using Current Greeks in Delta-Gamma Historical VaR

Article Quant Q&A · Author: tweedi
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Summary

The document answers which sensitivities to use when estimating historical VaR with a delta-gamma approximation. For a portfolio containing an option, the method applies the option's current delta and gamma to a series of historical underlying price changes. Each historical move produces an approximate portfolio profit or loss through a linear delta term and a quadratic gamma term.

The accepted answer clarifies that this setup does not require reconstructing historical delta and gamma values for every observation: current sensitivities are held fixed across the lookback shocks. The result is a set of approximate P&Ls that can be compared with a full repricing approach. This is a local sensitivity approximation, so its usefulness depends on how well today's Greeks represent the option's response to the historical moves; the document does not discuss model error or changing volatility and other risk factors.

Key ideas

  • Delta-gamma VaR applies current delta and gamma to historical underlying price changes.
  • Each historical price shock generates an approximate P&L with linear and quadratic sensitivity terms.
  • Historical Greeks are not required for the method described.
  • The calculation approximates repricing by holding current sensitivities fixed.

Tags

OptionsRisk managementBacktestingDerivatives pricing
Full text
# Delta-Gamma VaR question on inputs (do I need only one delta or the delta for the past 500 days?)


# Delta-Gamma VaR question on inputs (do I need only one delta or the delta for the past 500 days?)












When calculating a VaR using the Delta-Gamma approach, I am supposed to use this formula:

∆(V) = Delta * ∆(X) + 0.5 * Gamma * ∆(X)²

where:

∆(V) = change in value of the asset in portfolio ∆(X) = change in price of the underlying

I am trying to compare the full re-pricing VaR (500 days lookback period) of a 1 portfolio asset (an option, repriced using Black-Scholes) and the Delta-Gamma VaR of the same portfolio.

What exactly do I need for the Delta-Gamma VaR? Do I need the 500 daily returns of the underlying AND the past 500 delta and gamma values? Or do I simply use today's delta and gamma and the past 500 daily returns of the underlying?

## Answer by userPrimeNumber (score 1, accepted)

https://quant.stackexchange.com/a/42424

To calculate Delta-Gamma VaR you take today's delta and gamma and then multiply them by a vector of daily changes in the price of the underlying using the formula you provided to get 500 P&Ls: ∆(V) = Delta * ∆(X) + 0.5 * Gamma * ∆(X)² I.e. no need to know historical delta/gamma.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.