Discontinuous payoff option pricing by Mellin transform: A probabilistic approach

The Mellin transform technique is applied for solving the Black-Scholes equation with time-dependent parameters and discontinuous payoff. We show that the option pricing is equivalent to recovering a probability density function on the positive real axis based on its moments, which are integer or fractional Mellin transform values. Then the Mellin transform can be effectively inverted from a collection of appropriately chosen fractional (i.e. non-integer) moments by means of the Maximum Entropy (MaxEnt) method. An accurate option pricing is guaranteed by previous theoretical results about MaxEnt distributions constrained by fractional moments. We prove that typical drawbacks of other numerical techniques, such as Finite Difference schemes, are bypassed exploiting the Mellin transform properties. An example involving discretely monitored barrier options is illustrated and the accuracy, efficiency and time consuming are discussed.