Sum of all Black–Scholes–Merton models: An efficient pricing method for spread, basket, and Asian options

Abstract

Contrary to the common view that exact pricing is prohibitive owing to the curse of dimensionality, this study proposes an efficient and unified method for pricing options under multivariate Black–Scholes–Merton (BSM) models, such as the basket, spread, and Asian options. The option price is expressed as a quadrature integration of analytic multi‐asset BSM prices under a single Brownian motion. Then the state space is rotated in such a way that the quadrature requires much coarser nodes than it would otherwise or low varying dimensions are reduced. The accuracy and efficiency of the method is illustrated through various numerical experiments.

A Black–Scholes user's guide to the Bachelier model

“…Therefore, pricing such claims under the BS model requires either a simplifying approximation or numerical schemes. See Choi (2018) and the references therein for a review of such methods. Under the Bachelier model, however, the pricing becomes trivial because the weighted sum of the correlated arithmetic BMs remains normally distributed.…”

Leave-One-Out Least Square Monte Carlo Algorithm for Pricing American Options

“…with the parameter set tested by Krekel et al [2004] and Choi [2018] in the context of the European payoff, S j (0) = 100, σ j = 40%, r = q j = 0, ρ j =j = 0.5, t i = i 2 , and I = 10 (T = 5).…”

“…Because the underlying assets are not paying dividends, it is optimal not to exercise the option until maturity; hence, the European option price is equal to the Bermudan's. Therefore, we refer to Choi [2018] for the exact prices. For the regressors, polynomials up to degree 2 (M = 16) are used for the first experiment:…”

Asymptotic Estimation of Two Telegraph Particle Collisions and Spread Options Valuations

“…Pearson (1999) [7] published an algorithm requiring a one-dimensional numerical integration to compute the option value. Choi (2018) [8] showed that the numerical integral can be performed very efficiently by using an appropriate rotation of the domain and Gauss-Hermite quadrature. Li, Deng, and Zhou (2006) [9] published accurate approximation formulas for both spread option prices and their Greeks.…”