Valuation of Performance‐Dependent Options

Abstract

Performance-dependent options are financial derivatives whose payoff depends on the performance of one asset in comparison to a set of benchmark assets. In this paper, we present a novel approach for the valuation of general performance-dependent options. To this end, we use a multidimensional Black-Scholes model to describe the temporal development of the asset prices. The martingale approach then yields the fair price of such options as a multidimensional integral whose dimension is the number of stochastic processes used in the model. The integrand is typically discontinuous which makes accurate solutions difficult to achieve by numerical approaches, though. Using tools from computational geometry, we are able to derive a pricing formula which only involves the evaluation of several smooth multivariate normal distributions. This way, performance-dependent options can efficiently be priced even for highdimensional problems as is shown by numerical results.

Efficient deterministic numerical simulation of stochastic asset-liability management models in life insurance

“…Similar results have been observed for the SG methods, which were applied successfully to the valuation of performancedependent options [17], the pricing of mortgage backed securities [14,15] and to likelihood estimation [23].…”

“…Examples are the pricing of bonds [2,32], options [1,17] and mortgage backed securities [11,14,15,37]. To our knowledge, it is so far not known if deterministic methods can also be successfully applied to ALM simulations.…”