Showing papers in "Journal of Computational Finance in 1998"

Pricing Continuous Asian Options: A Comparison of Monte Carlo and Laplace Transform Inversion Methods

Abstract: In this paper, we investigate two numerical methods for pricing Asian options: Laplace transform inversion and Monte Carlo simulation. In attempting to numerically invert the Laplace transform of the Asian call option that has been derived previously in the literature, we point out some of the potential difficulties inherent in this approach. We investigate the effectiveness of two easy-to-implement algorithms, which not only provide a cross-check for accuracy, but also demonstrate superior precision to two alternatives proposed in the literature for the Asian pricing problem. We then extend the theory of Laplace transforms for this problem by deriving the double Laplace transform of the continuous arithmetic Asian option in both its strike and maturity. We contrast the numerical inversion approach with Monte Carlo simulation, one of the most widely used techniques, especially by practitioners, for the valuation of derivative securities. For the Asian option pricing problem, we show that this approach will be effective for cases when numerical inversion is likely to be problematic. We then investigate ways to improve the precision of the simulation estimates through the judicious use of control variates. In particular, in the problem of correcting the discretization bias inherent in simulation when pricing continuous-time contracts, we find that the use of suitably biased control variates can be beneficial. This approach is also compared with the use of Richardson extrapolation.

Computation of Deterministic Volatility Surfaces

Abstract: The 'volatility smile' is one of the well-known biases of Black-Scholes models for pricing options. In this paper, we introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility $\sigma = \sigma (S,t)$. This instantaneous volatility is chosen as a spline whose weights are determined by a regularised numerical strategy that approximately minimises the difference between Black-Scholes vanilla prices and known market vanilla prices over a range of strikes and maturities; these Black-Scholes prices are calculated by solving the relevant partial differential equation numerically using finite element methods. The instantaneous volatility generated from vanilla options can be used to price exotic options where the skew and term-structure of volatility are important, and we illustrate the application to barrier options.

The pricing of discretely sampled Asian and lookback options: a change of numeraire approach

Abstract: This paper considers the pricing of discretely sampled Asian and lookback options with floating and fixed strikes. In the modelling framework of Black and Scholes (1973), it is shown that a change of numeraire of the martingale measure can be used to reduce the dimension of these path-dependent option pricing problems to one in addition time. This means that the pricing problems can be solved by numerically solving onedimensional partial differential equations. The author demonstrates how a Crank‐ Nicolson scheme can be applied to the numerical solution. Finally, the methodology is extended to the case when the underlying stock exhibits discontinuous returns, and it is shown that in this case the Asian and lookback option pricing problems can be solved by numerically solving one-dimensional partial integrodifferential equations.

A Series Expansion for the Bivariate Normal Integral

Abstract: All other trademarks are the property of their respective owners. Abstract An infinite series expansion is given for the bivariate normal cumulative distribution function. This expansion converges as a series of powers of 1 2 − ρ d i , where ρ is the correlation coefficient, and thus represents a good alternative to the tetrachoric series when ρ is large in absolute value.

LP valuation of exotic American options exploiting structure

Abstract: We investigate the numerical solution of American nancial option pricing problems, using a novel formulation of the valuation problem as a linear programme (LP) introduced in [14, 21]. By exploiting the structure of the constraint matrices derived from standard Black-Scholes \\vanilla\" problems we obtain a fast and accurate revised simplex method which performs at most a linear number of pivots in the temporal discretization. When empirically compared with projected successive overrelaxation (PSOR) or a commercial LP solver the new method is faster for all the vanilla problems tested. Utilising this method we value discretely-sampled Asian and lookback American options and show that path-dependent PDE problems can be solved in `desktop' solution times. We conclude that LP solution techniques { which are robust to parameter changes [15] { can be tuned to provide fast eÆcient valuation methods for nitedi erence approximations to many vanilla and exotic option valuation problems.

Pricing of interest rate contingent claims: implementing a simulation approach

Abstract: This paper is an empirical study of the Heath-Jarrow-Morton model using time stamped transactions data of screen traded Danish bond and option prices. The paper shows how to implement a simulation approach to price contingent claims written on purely interest rate-dependent securities fulfilling the Heath-Jarrow-Morton model. This method implies simulation of solutions of stochastic differential equations since the pricing model is too complicated to give closed form pricing formulas. Therefore, parameters of the volatility of the Heath-Jarrow-Morton model is estimated using Simulated Moments Estimation. Estimated prices of the model are mostly within the bid-ask spread.