Computation of the effects of uncertainty in volatility on option pricing and hedging

Abstract

We quantify the effect of uncertainty in the volatility parameter σ on the Black–Scholes price of the European and American put. We apply probabilistic uncertainty analysis to the Black–Scholes model and compare the results with those of the Uncertain Volatility model. From historical data, we calibrate a probability distribution for the volatility. We then use Monte Carlo (MC) and a surrogate Polynomial Chaos (PC)/MC method to compute uncertainty bounds. The calibrated probability distribution is not one related to a standard orthogonal basis, so a basis is constructed numerically for the PC approximation. We show how to construct one stably from the probability distribution. We show that both methods give the same results, and quantify the relative speedup of the surrogate method. Finally, we investigate the effect of the parametric uncertainty, and show, for example, that the presence of uncertainty smoothes out the optimal exercise boundary of the American put.

Keywords:

• option pricing
• uncertainty
• polynomial chaos
• hedging
• Monte Carlo

2010 AMS Subject Classifications::

• 65N35
• 65N75
• 91G60
• 91G80

Acknowledgements

The second author was supported in part by the NSF grant DMS-0810925.