The pricing problem for a continuous path-dependent option results in a path integral which can be recast into an infinite-dimensional integration problem. We study ANOVA decomposition of a function of infinitely many variables arising from the Brownian bridge formulation of the continuous option pricing problem. We show that all resulting ANOVA terms can be smooth in this infinite-dimensional ...

Combinatorial methods prove extremely useful towards designing blazingly fast yet simple algorithms for pricing European-style barrier options. Closed-form formulae to standard European-style barrier options can then be easily derived. Combinatorial formulae under the trinomial model are also presented. The common practice in the literature compares algorithms based on their respective numbers ...

We describe and implement a novel methodology for Monte Carlo simulation of one-dimensional killed diffusions. The proposed estimators represent an unbiased and efficient alternative to current Monte Carlo estimators based on discretization methods for the cases when the finitedimensional distributions of the process are unknown. For barrier option pricing in finance, we design a suitable Monte...

The goal of this paper is to show that the jump-diffusion models are an essential and easy-to-learn tool for option pricing and risk management, and that they provide an adequate description of stock price fluctuations and market risks. We try to give an overview of the field without focusing on technical details. After introducing several widely used jump-diffusion models, we discuss Fourier t...

Markov-modulated models for equity prices have recently been extensively studied in the literature. In this paper, we apply some old results on the Wiener–Hopf factorization of Markov processes to a range of option-pricing problems for such models. The first example is the perpetual American put, where the exact (numerical) solution is obtained without discretizing any PDE. We then show how the...

A semi-static replication method is introduced for pricing discretely sampled path-dependent options. It depends upon buying and selling options at the reset times of the option but does not involve trading at intervening times. The method is model independent in that it only depends upon the existence of a pricing function for vanilla call options which depends purely on current time, time to ...

The stock model and option pricing problem are central contents in modern finance. In this paper, generalized stock model for financial market is proposed and the European option pricing formula for the generalized stock model is computed.

This paper describes and analyses the history of the fundamental equation of modern financial economics: the Black-Scholes (or Black-Scholes-Merton) option pricing equation. In that history, several themes of potentially general importance are revealed. First, the key mathematical work was not rule-following but bricolage, creative tinkering. Second, it was, however, bricolage guided by the goa...