Stochastic Methods in Economics and Finance

Malliaris and Brock is a classic of financial engineering. During the formative days of the OTC derivatives markets, it was from this book that early financial engineers learned stochastic calculus. It remains an excellent book today.

 

 

The book is written for readers with some experience with measure theory and measure-theoretic probability. You don't have to be an expert! However, some practical familiarity with concepts covered by Bartle (1966) and Resnick (1999) (or similar books) is essential.

The authors balance the competing goals of accessibility, rigor and practical usefulness. Focusing on results, they define concepts and specify theorems rigorously. Few derivations are provided. Instead, readers are directed to more advanced books on probability or stochastic calculus. At points, concepts are motivated intuitively before being formalized. This is especially useful when the authors introduce stochastic integrals and differentials.

The book comprises four long chapters:

Results from probability

Stochastic calculus

Applications in economics, and

Applications in finance.

The first chapter is an excellent review of relevant concepts from measure-theoretic probability—measurable spaces, sigma fields, random variables, Radon-Nikodym derivatives, stochastic processes, martingales, etc. Don't expect to learn these concepts here—the discussion is far too cryptic. However, for synthesis and review, this is a very nice chapter.

The second chapter is the meat of the book. Here, the notions of Ito processes, stochastic differentials, stochastic integrals and stochastic differential equations are formalized. Ito's lemma is presented. A number of mathematical applications are presented.

The closing two chapters cover applications from, respectively, economics and finance. Depending upon your interest, you can skip one and focus on the other. The financial applications chapter covers the Black-Scholes option pricing formula and jump-diffusion processes, among other things. The applications are dated, but this doesn't detract from their educational content.

This book is appealing because it is rigorous while remaining practitioner oriented. It isn't an easy read. If you have the necessary mathematical background and some financial sophistication, your will find the effort well rewarded. I recommend the book to budding financial engineers or highly quantitative financial risk managers.

Contents

1. Results from probability

probability spaces

Random variables

Expectation

Conditional probability

Martingales and applications

Stochastic processes

Optimal stopping

Miscellaneous applications and exercises

Further remarks and references

2. Stochastic calculus

Modeling uncertainty

Stochastic integration

Ito's lemma

Examples

Stochastic differential equations

Properties of solutions

Point equilibrium and stability

Existence of stationary distribution

Stochastic control

Bismut's approach

Jump processes

Optimal stopping and free boundary problems

Miscellaneous applications and exercises

Further remarks and references

3. Applications in economics

Neoclassic economic growth under uncertainty

Growth in an open economy under uncertainty

Growth under uncertainty: Properties of solutions

Growth under uncertainty: Stationary distribution

The stochastic Ramsey problem

Bismut on optimal growth

The rational expectations hypothesis

Investment under uncertainty

Competitive processes, transversality condition and convergence

Rational expectations equilibrium

Linear quadratic objective function

State valuation functions of exponential form

Money, prices and inflation

An N-sector discrete growth model

Competitive firm under price uncertainty

Stabilization in the presence of disturbances

Stochastic capital theory in continuous time

Miscellaneous applications and exercises

Further remarks and references

4. Applications in finance

Stochastic rate of inflation

The Black-Scholes option pricing model

Consumption and portfolio rules

Hyperbolic absolute risk aversion functions

Portfolio jump processes

The demand for index bonds

Term structure in an efficient market

Market risk adjustment in project valuation

Demand for cash balances

The price of systematic risk

An asset pricing model

Existence of an asset pricing model

Certainty equivalence formulae

A testable formula

An example

Miscellaneous applications and exercises

Further remarks and references

 

 

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