The famous Black-Sholes (BS) and Cox-Ross-Rubinstein (CRR) formulas are basic results in the modern theory of option pricing in financial mathematics. They are usually deduced by means of stochastic analysis; various generalisations of these formulas were proposed using more sophisticated stochastic models for common stocks pricing evolution. In this paper we develop systematically a determinis...

The simplest model for pricing d erivative securities is the binomial model. It generalizes the o n e period \up-down" model of Chapte r 1 t o a m ulti-period setting, assuming t hat t he price of the u nderlying asset follows a random walk. In the binomial model, there are N trading periods and N+1 trading d ates, t 0 t 1 ::: t N when it is possible to i n vest in a risky security with p r i c...

This paper presents an overview of techniques for improving the efficiency of option pricing simulations, including quasiMonte Carlo methods, variance reduction, and methods for dealing with discretization error.

After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additio...

The risk-neutral process is modeled by a four parameter self-similar process of independent increments with a self-decomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all six models are capable of adequately synthesizing European option prices across the spectrum of strikes and mat...

The complexity of financial products significantly increased in the past ten years. In this paper we investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. Our approach assumes that the underlying assets evolve as dependent GARCH(1,1) processes and it involves to model the dependency among the assets using a copula based on pair-c...

The purpose of this paper is to examine whether equity options traded on individual firms are sensitive to the firm’s leverage, and to see if adding leverage to the option model improves its pricing accuracy. In a hitherto unexamined economic approach to option valuation, we use reported leverage from financial statements, and a compound option (CO) model, for valuing stock options on individua...

In this paper, we modeled an artificial European option market under unknown volatility with liquidity costs using an agent-based modeling and simulation approach. The option price in the presence of liquidity costs is given by solving a partial differential equation. We proved that both unknown volatility and the unknown drift have significant effects in the pricing bias. Moreover, pricing bia...

Background material on measure-theoretic probability theory and stochastic calculus is provided in order to clarify notation and inform the reader unfamiliar with these concepts. These fields are then employed in exploring two distinct but related approaches to fair option pricing: developing a partial differential equation whose solution, given specified boundary conditions, is the desired fai...