The Black–Scholes Equation with Impulses at Random Times Via Generalized Riemann Integral

The classical pricing theory requires that the simple sets of outcomes are extended, using the Kolmogorov Extension Theorem, to a sigma-algebra of measurable sets in an infinite-dimensional sample space whose elements are continuous paths; the process involved are represented by appropriate stochastic differential equations (using Itô calculus); a suitable measure for the sample space can be found by means of the Girsanov and Radon–Nikodym Theorems; the derivative asset valuation is determined by means of an expression using Lebesgue integration. It is known that if we replace Lebesgue’s by the generalized Riemann integration to obtain the expectation, the same result can be achieved by elementary methods. In this paper, we consider the Black–Scholes PDE subject to impulse action. We replace the process which follows a geometric Brownian motion by a process which has additional impulsive displacements at random times. Instead of constants, the volatility and the risk-free interest rate are considered as continuous functions which can vary in time. Using the Feynman–Ka[Formula: see text] formulation based on generalized Riemann integration, we obtain a pricing formula for a European call option which copes with many discontinuities. This paper seeks to develop techniques of mathematical analysis in derivative pricing theory which are less constrained by the standard assumption of lognormality of prices. Accordingly, the paper is aimed primarily at analysis rather than finance. An example is given to illustrate the main results.

Brownian Motion at the Speed of Light: A New Lorentz Invariant Family of Processes

We consider here a new family of processes which describe particles which only can move at the speed of light c in the ordinary 3D physical space. The velocity, which randomly changes direction, can be represented as a point on the surface of a sphere of radius c and its trajectories only may connect points of this variety. A process can be constructed both by considering jumps from one point to another (velocity changes discontinuously) and by continuous velocity trajectories on the surface. We recently proposed to follow this second strategy assuming that the velocity is described by a Wiener process (which is isotropic only in the ’rest frame’) on the surface of the sphere. Using both Ito calculus and Lorentz boost rules, we succeed here in characterizing the entire Lorentz-invariant family of processes. Moreover, we highlight and describe the short-term ballistic behavior versus the long-term diffusive behavior of the particles in the 3D physical space.

Robert C. Merton and the Science of Finance

Starting with his 1970 doctoral dissertation and continuing to today, Robert C. Merton has revolutionized the theory and practice of finance. In 1997, Merton shared a Nobel Prize in Economics “for a new method to determine the value of derivatives.” His contributions to the science of finance, however, go far beyond that. In this article I describe Merton's main contributions. They include the following: 1. The introduction of continuous-time stochastic models (the Ito calculus) to the theory of household consumption and investment decisions. Merton's technique of dynamic hedging in continuous time provided a bridge between the theoretical complete-markets equilibrium model of Kenneth Arrow and the real world of personal financial planning and management. 2. The derivation of the multifactor Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM generalizes the single-factor CAPM and explains why that model might fail to properly account for observed market excess returns. It also provides a theory to identify potential forward-looking risk premia for use in factor-based investment strategies. It is therefore both a positive and normative theory. 3. The invention of Contingent Claims Analysis (CCA) as a generalization of option pricing theory. CCA applies the technique of dynamic replication to the valuation and risk management of a wide range of corporate and government liabilities. Merton's CCA model for the valuation and analysis of risky debt is known among scholars and practitioners alike as the Merton Model. 4. The development of financial engineering, which employs CCA to design and produce new financial products. Merton was the first to apply CCA to analyze government guaranty programs such as deposit insurance, and to suggest improvements in the way those programs are managed. He and his students have applied his insights at both the micro and macro policy levels. 5. And finally, the development of a theory of financial intermediation that explains and predicts how financial systems differ across countries and change over time. Merton has applied that theory, called functional and structural finance, to guide the design and regulation of financial systems at the levels of the firm, the industry, and the nation. He has also used it to propose reforms in pensions, sovereign wealth funds, and macrostabilization policy. This article is one of a pair of articles published in this volume about Robert C. Merton's contributions to the science of financial economics. This article was originally published in Volume 11 of the Annual Review of Financial Economics. The other article in this pair is “ Robert C. Merton: The First Financial Engineer ” by Andrew W. Lo.

Robert C. Merton and the Science of Finance

Starting with his 1970 doctoral dissertation and continuing to today, Robert C. Merton has revolutionized the theory and practice of finance. In 1997, Merton shared a Nobel Prize in Economics “for a new method to determine the value of derivatives.” His contributions to the science of finance, however, go far beyond that. In this article I describe Merton's main contributions. They include the following: 1. The introduction of continuous-time stochastic models (the Ito calculus) to the theory of household consumption and investment decisions. Merton's technique of dynamic hedging in continuous time provided a bridge between the theoretical complete-markets equilibrium model of Kenneth Arrow and the real world of personal financial planning and management. 2. The derivation of the multifactor Intertemporal Capital Asset Pricing Model (ICAPM). The ICAPM generalizes the single-factor CAPM and explains why that model might fail to properly account for observed market excess returns. It also provides a theory to identify potential forward-looking risk premia for use in factor-based investment strategies. It is therefore both a positive and normative theory. 3. The invention of Contingent Claims Analysis (CCA) as a generalization of option pricing theory. CCA applies the technique of dynamic replication to the valuation and risk management of a wide range of corporate and government liabilities. Merton's CCA model for the valuation and analysis of risky debt is known among scholars and practitioners alike as the Merton Model. 4. The development of financial engineering, which employs CCA to design and produce new financial products. Merton was the first to apply CCA to analyze government guaranty programs such as deposit insurance, and to suggest improvements in the way those programs are managed. He and his students have applied his insights at both the micro and macro policy levels. 5. And finally, the development of a theory of financial intermediation that explains and predicts how financial systems differ across countries and change over time. Merton has applied that theory, called functional and structural finance, to guide the design and regulation of financial systems at the levels of the firm, the industry, and the nation. He has also used it to propose reforms in pensions, sovereign wealth funds, and macrostabilization policy.