8 Pages Posted: 20 Jun 2013
Date Written: April 2, 2013
We study the PnL prediction of an option from its Greeks under extreme shocks. In this situation the classical delta gamma approximation fails and adding higher order Greeks does not improve significantly the situation due to the slow convergence of the Taylor’s expansion (which even diverges in the Black-Scholes case). One obvious situation involves a far out of the money option under a scenario pushing the option in the money. Since the Greeks were almost zero initially, the delta-gamma PnL will be almost zero and off the real PnL. Another drawback of expanding further the Taylor’s expansion is the need of high order derivatives which are costly and not reliable especially at the level of the book of a large investment bank derivatives desk. Consequently we find a simple alternative solution involving only 3 deltas which we found by re-interpreting the PnL prediction as a numerical integration problem. More precisely we suggest using the Simpson’s method which predicts correctly the PnL under extreme stressed scenarios.
Keywords: PnL predict, numerical integration, delta-gamma approximation, Trapezoid’s method, Simpson's method, stress testing, scenario hedging
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