P. Ekkehard Kopp

Stock Price Returns and the Joseph Effect: A Fractional Version of the Black-Scholes Model

In mathematical finance, semimartingales are traditionally viewed as the largest class of stochastic processes which are economically reasonable models for stock price movements. This is mainly because stochastic integrals play a crucial role in the modern theory of finance, and semimartingales represent the largest class of stochastic processes for which a general theory of stochastic integration exists. However, some empirical evidence from actual stock price data suggests stochastic models that are not covered by the semimartingale setting.

Martingale representation and hedging policies

The integrand, when a martingale under an equivalent measure is represented as a stochastic integral, is determined by elementary methods in the Markov situation. Applications to hedging portfolios in finance are described.

From discrete to continuous stochastic calculus

We formulate the notion of D2-convergence of discrete time entities (based on simple random walks) to their continuous counterparts on Wiener space, and show that D2-convergence is preserved under stochastic integration and differentiation, and for chaos decomposition.

Option pricing and hedge portfolios for poisson progresses

Price processes influenced by Poisson processes are considered and the value of a sum of European call options obtained. The novel feature of the paper is the use of analogues of strochastic flows to derive a martingale representation resut and the heding portfolio

Direct solutions of Kolmogorov's equations by stochastic flows

Abstract Solutions of Kolmogorovs forward and backward equations are obtained by considering a family of conditional expectations and the use of stochastic flows to justify differentiation in the time variable.

Pricing by Arbitrage

The ‘unreasonable effectiveness’ of mathematics is evidenced by the frequency with which mathematical techniques that were developed without thought for practical applications find unexpected new domains of applicability in various spheres of life. This phenomenon has customarily been observed in the physical sciences; in the social sciences its impact has perhaps been less evident. One of the more remarkable examples of simultaneous revolutions in economic theory and market practice is provided by the opening of the world’s first options exchange in Chicago in 1973, and the ground-breaking theoretical papers on preference-free option pricing by Black and Scholes [18] (quickly extended by Merton [188]) which appeared in the same year, thus providing a workable model for the ‘rational’ market pricing of traded options.

Hyperfinite Mathematical Finance

Financial markets have provided one of the most remarkable growth industries in the past two decades, and now constitute a major source of employment for graduates with high levels of mathematical expertise. The principal reason for this phenomenon lies in the explosive growth of the market in derivatives, whose levels of activity now frequently exceed the underlying markets on which their products are based. The variety and complexity of new financial instruments is often bewildering, and much effort goes into the analysis of the mathematical models on which their existence is predicated.

The Fundamental Theorem of Asset Pricing

We saw in the previous chapter that the existence of a probability measure Q ~ P under which the (discounted) stock price process is a martingale is sufficient to ensure that the market model is viable; that is, it contains no arbitrage opportunities. We now address the converse: whether for every viable model one can construct an equivalent martingale measure for S, so that the price of a contingent claim can be found as an expectation relative to Q.

Consumption-Investment Strategies

The results of this chapter are a presentation of the comprehensive, fundamental, and elegant contributions of Karatzas, Lehoczky, Sethi, and Shreve. See, for example, the papers [157] through [161].

Bonds and Term Structure

Suppose (Ω, F, P) is a probability space and B t , 0 ≤ t ≤ T, is a Brownian motion, {F t } denotes the (complete, right-continuous) filtration generated by B. We first review the martingale pricing results of Chapter 7.