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Value at risk of an instrument for time $T$ following a density with Fourier transform $\hat{q}(\xi,t)$ can be calculated by a put option price as follows.  Let’s say the portfolio is a single stock with current value $x_0$, and we are interested in the loss beyond 2% at time $T$.  Then the put option cost struck at $0.98 x_0$ will give us the value at risk of loss beyond 2% and this can be computed using the Fourier transform $f(\xi) = \int_{-\infty}^\infty e^{ix\xi} (.98x_0 - x)_+ d\xi$ as
$VAR = \int_{-\infty}^\infty \hat{q}(\xi,T) f(\xi) d\xi$
This is useful when our model of return density is not standard but given by a Fourier transform (such as the solution of a fractional diffusion and $\hat{q}$ gives the Fourier transform of the equivalent martingale measure).