Hopscotch methods for two-state financial models

ANNIVERSARY ARTICLE: Option Pricing: Valuation Models and Applications

“…American and path-dependent options require numerical solutions of the pricing PDE. Examples include Forsyth et al (1999), who present a finite element approach to the pricing of lookback options with SV; Kurpiel and Roncalli (2000), who develop hopscotch methods for SV and other two-state models; and Apel et al (2002), who develop a finite element method for an SV model.…”

Option Hedging with Stochastic Volatility

“…In order to compare equation (14) with equation (16) suppose first that ρ = 0. In this case, the third term in equation (14) vanishes.…”

“…Generally, researchers have used Monte Carlo or finite difference methods to solve stochastic volatility option pricing problems. Kurpiel and Roncalli [1998] show how to apply Hopscotch methods, a class of finite difference algorithms introduced initially by Gourlay [1970], to two-state financial models. Unlike Monte Carlo, Hopscotch methods are very useful for American option pricing and easy greeks computing in a stochastic volatility framework.…”

Pricing European and American Options in the Heston Model With Accelerated Explicit Finite Differencing Methods

“…Other approaches used to numerically solve the option pricing PDE under the Heston model include: an incomplete LU (ILU) pre-conditioned conjugate gradient method with a penalty term devised by [49] to handle the early exercise feature of American options; multigrid methods used by [8] and [30] to price American options; a hopscotch scheme applied by [19] to price American options; a finite element approach constructed by [47] to price European and barrier options; an exponential fitting in combination with splitting and Yanenko method by [36]; various ADI schemes augmented to include the cross derivative correlation term by [22] and an ADI predictor-corrector scheme by [48]. A number of schemes used to solve LCPs were also compared by [20,21] including a PSOR method, a projected multigrid method, an operator splitting method and a component wise splitting method.…”