Difference schemes are treated as operator equations in real linear normed space [l, 23. Twoand three-layer schemes are discussed in real Hilbert space (more precisely, in unitary space, since no use is made of completeness). Stable schemes are classified. Schemes are written in a canonical form that enables stability operators, or what will be termed regularizers, to be introduced. Sufficient ...

In these notes, I will tell how in 1953–1954 the first version of the “Godunov’s scheme” was invented and how it was modified in the subsequent works by myself (until 1969) and by others at the Institute of Applied Mathematics in Moscow (now named after its founder, academician M. V. Keldish). Parallel to the modifications which I will describe below (Sections 1, 2), other algorithms were being...

This paper addresses a finite difference approximation for an infinite dimensional Black-Scholes equation obtained in Chang and Youree [5]. The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the p...

C omplex derivatives have become accepted instruments to tailor risk coverage for risk managers and investors. Barrier-type options have become important instruments, particularly for the valuation of structured products (see Banks [1994]). They are also widely used in currency markets. The holder of a barrier option acquires option coverage on only a subset of the risky outcomes for which a pl...

Many underlying assets of option contracts, such as currencies, commodities, energy, temperature and even some stocks, exhibit both mean reversion and stochastic volatility. This paper investigates the valuation of options when the underlying asset follows a mean-reverting lognormal process with stochastic volatility. A closed-form solution is derived for European options by means of Fourier tr...

We call a given American option representable if there exists European claim which dominates the payoff at any time and such that values of two options coincide in continuation region option. This concept has interesting implications from probabilistic, analytic, financial, numeric point view. Relying on methods (Math. Finance 24 (2014) 156–172; Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (200...

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, focus are on both nonlinear and linear (BS) equations with numerical approximations. We aim to find an effective approximations for model. Several models from most relevant class European option analyzed study. The problem is approached by transforming into convection-diffusion equation later ...