Efficient Computation of Hedging Parameters for Discretely Exercisable Options

We propose an algorithm to calculate confidence intervals for the values of hedging parameters of discretely exercisable options using Monte-Carlo simulation. The algorithm is based on a combination of the duality formulation of the optimal stopping problem for pricing discretely exercisable options and Monte-Carlo estimation of hedging parameters for European options. We show that the width of the confidence interval for a hedging parameter decreases, with an increase in the computer budget, asymptotically at the same rate as the width of the confidence interval for the price of the option. The method can handle arbitrary payoff functions, general diffusion processes, and a large number of random factors. We also present a fast, heuristic, alternative method and use our method to evaluate its accuracy. Efficient Computation of Hedging Parameters for Discretely Exercisable Options The idea behind no-arbitrage option pricing is that, in a complete market, one can almost surely replicate the payoff of an option using a suitably chosen portfolio of instruments. The construction of this replicating portfolio is based on the computation of the hedging parameters, or sensitivities, of option prices with respect to parameters of the underlying process.1 Indeed, the first derivative of the option price with respect to the initial asset price, ∆, corresponds to the amount of the underlying asset held in the replicating portfolio, while the second derivative, Γ, corresponds to the characteristic time interval between rebalancings. Reliable estimation of option prices and hedging parameters, or option price sensitivities, has become very important with the ever expanding range of applications of options from, for example, problems in supply chain management, to problems in energy finance and real estate. In this paper we develop an algorithm that uses Monte-Carlo simulation to estimate option price sensitivities for options with multiple exercise dates and a potentially large number of underlying assets. The advantage of Monte-Carlo simulation and the reason it is the method of choice for problems with many assets is that, by its nature, Monte-Carlo simulation does not suffer from an exponential increase in effort for a linear increase in the number of underlying assets, a common problem with finite difference discretizations of partial differential equations and high dimensional lattice algorithms. In addition, Monte-Carlo simulation offers an estimate of its own accuracy and is relatively easy to perform in parallel, leading to significant increases in computational speed. In the literature, Monte-Carlo simulation has been used to calculate sensitivities of the option price for European options; i.e., options with a single exercise date, see Glynn (1989), Broadie and Glasserman (1996), Fournié, Lasry, Lebuchoux, Lions, and Touzi (1999), and Glasserman (2003) for an overview. We show how the simulation-based algorithms for calculating sensitivities of European options, and in particular the likelihood ratio algorithm proposed by Broadie and Glasserman (1996), can be used to compute confidence intervals for the values of sensitivities of discretely exercisable options with multiple exercise dates. To this end, we combine the likelihood ratio algorithm for estimating sensitivities of European options with an algorithm based on a dual representation of discretely exercisable option prices, originally proposed by Davis and Karatzas (1994) and further developed by Rogers (2002), Haugh and Kogan (2004), and Andersen and Broadie (2004). The duality based algorithm provides confidence intervals for the price of a discretely exercisable option. Our algorithm uses these intervals in a multi-stage algorithm, where Monte-Carlo simulation is employed at every stage. 1These sensitivities are frequently referred to as ”Greeks”.