Efficient Barrier Option Greeks using Automatic Differentation

Abstract: Automatic Differentiation (AD) is an effective method for calculation of derivatives. It can evaluate an unlimited number of derivatives to a fixed cost relative to the computing time of the original function. The AD technique is used in many fields for large and complex calculations in order to get accurate values of derivatives fast. Banks and other financial institutions handle calculations of portfolio values which could depend on a large number of input variables. AD enables fast calculation and sensitivity analysis of these complex functions such as risks of day to day trading as well as XVA’s. The accuracy of AD is also better than currently used methods for derivative calculations as it is not based on approximations. In this paper, AD in a Monte Carlo setting is studied and one way of implementing AD by operator overloading in its reverse mode is presented. Different ways to handle discontinuous payoffs are tested and applied on calculations of price and derivatives of Barrier Options. It is shown that AD combined with sigmoid smoothing of discontinuous payoffs gives accurate values for option price and Greeks with fast convergence rate.