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Mathematical Asset Management

Thomas Höglund
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
John Curran
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This book is a textbook about derivative securities and portfolio selection. It is intended for the mathematically sophisticated undergraduate or beginning graduate student. The author states that the prerequisites are “calculus including Taylor's formula in several variables and Lagrange multipliers, elementary linear algebra, and probability theory including the central limit theorem.” This list is accurate and non-negotiable: results from these areas are used freely without elaboration. The probability background should be oriented towards statistics, as the author refers to confidence intervals, convergence in probability and regression results.

The book is terse, no doubt resembling the lecture notes from which it was developed. The author does not waste time on motivation and exposition, jumping right in with derivations and analysis. Motivation and notation are often introduced via the exercises interspersed with the text. Since the exercises must be done in order to follow the book, most of the answers are provided, and there are relatively few exercises per chapter. Because of the terseness and the number of exercises, I believe this text would work better for a second course in mathematical finance, or as a supplemental text, rather than as an introduction to the subject.

There are some standard choices that an author of a book like this must make, such as how to derive the option pricing formula and whether to introduce stochastic calculus. The current author uses a replicating portfolio approach for option pricing with a binary tree model of price movements. He avoids the use of stochastic calculus, but does spend considerable time discussing stock prices as stochastic processes. The author spends more time on portfolio management than on derivative securities, which in a sense are reduced to a special case of portfolio management. The discussion of derivative securities is limited to calls and puts derived from stocks, including some simple trading strategies and the derivation of their pricing formulas.

The book has strengths and weaknesses as a reference work. The author states his assumptions clearly and tests them against real data. He usually gives a sense of what he considers best practice after presenting the theorems and data. (He is undecided about the validity of the capital asset pricing model, for example, but does develop its consequences.) He treats portfolio management in depth, including immunization, re-balancing, diversification, performance criteria, and statistical behavior of portfolios.

If the book has a weakness as a reference work it is that it focuses almost exclusively on stocks, and options derived from them. Bonds are treated in the first chapter only; futures appear in one subsection. Other asset classes are not treated. From a pedagogic point of view, it makes sense to focus on the more common kinds of assets, such as stocks, and to avoid complicating details; but one often wants reference works to be comprehensive.

A student, aspiring actuary, or entry-level banker could consolidate a lot of knowledge by working straight through the book. People with finance jobs will find a good summary of financial theory, but not an instruction manual for their spreadsheets or numerical software. The author assumes a practitioner can provide the details when numerical integration or regression is called for, and knows what a Monte Carlo method is. Quantitative analysts may desire an additional reference that treats asset classes other than stocks in more detail.

The author's English is quite clear but with occasional lapses in phrasing (such as subject-verb agreement). These lapses do not detract from the sense of the book, although some choices of wording may lead a cautious student down the wrong path (e.g., the use of “number of stocks” or even “number” to mean “number of shares in a given stock.”) A student willing to work out details should have no difficulties on this count.

John Curran is Assistant Professor of Mathematics at Eastern Michigan University, where he coordinates the actuarial science program. Curran worked for a Wall Street firm for several years before obtaining his Ph.D. in applied mathematics from Brown University.


1. Interest Rate.

1.1 Flat Rate.

1.1.1 Compound Interest.

1.1.2 Present Value.

1.1.3 Cash Streams.

1.1.4 Effective Rate.

1.1.5 Bonds.

1.1.6 The Effective Rate as a Measure of Valuation.

1.2 Dependence on the Maturity Date.

1.2.1 Zero-Coupon Bonds.

1.2.2 Arbitrage Free Cash Streams.

1.2.3 The Arbitrage Theorem.

1.2.4 The Movements of the Interest Rate Curve.

1.2.5 Sensitivity to Change of Rates.

1.2.6 Immunization.

1.3 Notes.

2. Further Financial Instruments.

2.1 Stocks.

2.1.1 Earnings, Interest Rate and Stock Price.

2.2 Forwards.

2.3 Options.

2.3.1 European Options.

2.3.2 American Options.

2.3.3 Option Strategies.

2.4 Further Exercises.

2.5 Notes.

3. Trading Strategies.

3.1 Trading Strategies.

3.1.1 Model Assumptions.

3.1.2 Interest Rate.

3.1.3 Exotic Options.

3.2 An Asymptotic Result.

3.2.1 The Model of Cox, Ross and Rubinstein.

3.2.2 An Asymptotic Result.

3.3 Implementing Trading Strategies.

3.3.1 Portfolio Insurance.

4. Stochastic Properties of Stock Prices.

4.1 Growth.

4.1.1 The Distribution of the Growth.

4.1.2 Drift and Volatility.

4.1.3 The Stability of the Volatility Estimator.

4.2 Return.

4.3 Covariation.

4.3.1 The Asymptotic Distribution of the Estimated Covariance Matrix.

5. Trading Strategies with Clock Time Horizon.

5.1 Clock Time Horizon.

5.2 Black-Scholes Pricing Formulas.

5.2.1 Sensitivity to Perturbations.

5.2.2 Hedging a Written Call.

5.2.3 Three Options Strategies Again.

5.3 The Black-Scholes Equation.

5.4 Trading Strategies for Several Assets.

5.4.1 An Unsymmetrical Formulation.

5.4.2 A Symmetrical Formulation.

5.4.3 Examples.

5.5 Notes.

6. Diversification.

6.1 Risk and Diversification.

6.1.1 The Minimum-Variance Portfolio.

6.1.2 Stability of the Estimates of the Weights.

6.2 Growth Portfolios.

6.2.1 The Auxiliary Portfolio.

6.2.2 Maximal Drift.

6.2.3 Constraint on Portfolio Volatility.

6.2.4 Constraints on Total Stock Weight.

6.2.5 Constraints on Total Stock Weight and Volatility.

6.2.6 The Efficient Frontier.

6.2.7 Summary.

6.3 Rebalancing.

6.3.1 The Portfolio Development as a Function of the Development of the Stocks.

6.3.2 Empirical Verification.

6.4 Optimal Portfolios with Positive Weights.

6.5 Notes.

7. Covariation with the Market.

7.1 Beta.

7.1.1 The Market.

7.1.2 Beta Value.

7.2 Portfolios Related to the Market.

7.2.1 The Beta Portfolio.

7.2.2 Stability of the Estimates of the Weights.

7.2.3 Market Neutral Portfolios.

7.3 Capital Asset Pricing Model.

7.3.1 The CAPM-Identity.

7.3.2 Consequences of CAPM.

7.3.3 The Market Portfolio.

7.4 Notes.

8. Performance and Risk measures.

8.1 PerformanceMeasures.

8.2 Risk Measures.

8.2.1 Value at Risk.

8.2.2 Downside Risk.

8.3 Risk Adjustment.

9. Simple Covariation.

9.1 Equal Correlations.

9.1.1 Matrix Calculations.

9.1.2 Optimal Portfolios.

9.1.3 Comparison with the General Model.

9.1.4 Positive Weights.

9.2 Multiplicative Correlations.

9.2.1 Uniqueness of the Parameters.

9.2.2 Matrix Calculations.

9.2.3 Parameter Estimation.

9.2.4 Optimal Portfolios.

9.2.5 Positive Weights.

9.3 Notes.

Appendix A: Answers and solutions to exercises.