Pricing and hedging barrier options driven by a class of additive processes

In this paper we develop an algorithm to calculate prices and Greeks of barrier options driven by a class of additive processes. Additive processes are time-inhomogeneous Lévy processes, or equivalently, processes with independent but inhomogeneous increments. We obtain an explicit semi-analytical expression for the first-passage probability of an additive process with hyper-exponential jumps. The solution rests on a randomization and an explicit matrix WienerHopf factorization. Employing this result we derive explicit expressions for the Laplace(-Fourier) transforms of prices and Greeks of digital and barrier options. As numerical illustration, the model is simultaneously calibrated to Stoxx50E call options at four different maturities and subsequently prices and Greeks of down-and-in digital and down-and-in call options are calculated. Comparison with Monte Carlo simulation results shows that the method is fast, accurate, and stable.