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

Question

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 1

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.