By Lars Tyge Nielsen

Textbook in continuous-time finance theory

Oxford University Press, 1999.

## Intended readership

The book is an introduction to the theory of pricing and hedging of derivative securities in continuous time for graduate and advanced undergraduate students and for researchers in both academia and the financial industry. It is suitable as a text in graduate (postgraduate) and advanced undergraduate courses not only in finance and economics programs, but also in mathematical finance, statistics and mathematics programs.

## Design

The material and the exposition have been thoroughly tested in doctoral courses at INSEAD, New York University, and Columbia University, and in executive courses in derivative securities pricing at the Amsterdam Institute of Finance. Innumerable comments and questions from students and colleagues have been incorporated, explained, and answered in succesive revisions.

The level of mathematics preparation required by the reader is even. The book does not assume that the reader is already an expert in the mathematics. The necessary mathematical machinery is developed in a precise and rigorous manner, while unnecessary mathematics is avoided. A lot of effort has gone into deciding what to include and what not to include.

Where the book does not provide complete proofs of a theoretical or mathematical result, it gives a reference to where a complete proof can be found. It defines all the necessary concepts and states all the necessary results in a precise manner. It explains the intuition behind those concepts and results, how they fit in to the finance theory, and why they are necessary.

## For students and teachers of finance and economics

The theory of continuous-time stochastic processes is an essential prerequisite for continuous-time finance. It is not easily accessible, and it has for a long time formed a barrier of entry into the field. One purpose of this book is to help break down that barrier and make it possible for the reader actually to learn this material.

Finance instructors often refuse to teach the mathematics behind derivative securities pricing. They either “assume” that the students already know it (the way economists assume a can opener), or they ask them to go take courses in the mathematics or statistics department in order to learn it.

If the math courses are not designed with a view to mathematical finance, then all but the most talented and motivated finance students will lose their motivation and their bearings.

This book offers finance instructors the opportunity to teach and learn the necessary mathematics in a way that is intimately related to the derivative securities applications. Alternatively, those who do not wish to teach mathematics can go directly to the financial economics chapters while using the mathematical chapters for review purposes.

The book has been used in one-semester courses with excellent results, but a better idea might be to stretch it over two semesters. In that case, it could be supplemented with survey-style coverage of topics that are not included in the book (survey-style material on derivative securities can be found in many other books) as well as in-depth coverage of the instructor’s own favorite issues.

## For students and teachers of mathematical finance and financial engineering

Mathematics instructors may wish to fill in some of the proofs that have been skipped in the text and to supplement with more advanced and difficult material. However, they will benefit pedagogically from sticking to the structure of the text, and they are advised to exercise restraint in supplementing it.

Once the students have learned the mathematics and the applications in the book, they will be much better motivated and prepared to study the most advanced material, both in mathematics and in finance.

## For students and teachers of stochastic calculus

Teachers of stochastic calculus who recognize the pedagogical and motivational value of applications might consider structuring a course around this book or using the book as a supplement to their primary materials.

## The fundamental material is covered

Two theoretical chapters cover price processes and trading strategies, prices of risk and state price processes, arbitrage, replication, delta hedging, dynamic market completeness, and the martingale valuation principle, with examples and exercises scattered throughout. The treatment of the fundamental theory of derivative securities pricing is detailed and extensive.

Two applications chapters analyze the Black-Scholes model and the one-factor Gaussian term structure models in detail.

Every chapter has a summary which explains and reviews the chapter and the intuition behind it in a mix of common sense and technical terms. There are also critical notes on the literature at the end of each chapter.

There are exercises scattered throughout the book, and there are suggested solutions of all of them. Doing exercises is a very helpful, even indispensable, part of the learning process.

## You get the necessary background in stochastic process theory

Dynamic information structures, measurable and adapted processes, Wiener processes, geometric Brownian motion, stochastic integrals, Ito processes, Ito calculus with plenty of examples, Girsanov’s Theorem, the Martingale Representation Theorem, Gaussian processes such as Ornstein-Uhlenbeck processes and Brownian bridges.

All this mathematics is absolutely necessary for mastery of the pricing and hedging of derivative securities. The book explains why it is necessary and makes it easy for you to learn.

There are also two appendixes about measure and integration theory and one about the heat equation. You can read them or just use them as a reference.

## The book answers all these intriguing questions about the theory

- Do trading strategies really have to satisfy some kind of square integrability condition?
- How exactly are the state price process, the interest rate, the prices

of risk, and the risk adjusted probabilities related to each other? - What is the relationship between the state price process and the Hansen-Jagannathan bounds?
- How do we deal with the Harrison-Kreps doubling strategy, which shows that arbitrage is always possible in the Black-Scholes model, multitudinous claims to the contrary notwithstanding?
- Why did Harrison and Pliska abandon their idea of requiring the value processes of self-financing trading strategies to be bounded below?
- When Cheng showed that the Ball-Torous model is not arbitrage free, was that the same point that Harrison and Kreps made or was it a different one?
- What was Merton’s Nobel prize winning argument about absence of arbitrage, and how does it square with Harrison and Kreps?
- What was the subtle point about complete markets made by Mueller and elaborated by Jarrow and Madan?
- Does the risk adjusted probability distribution have to be uniformly absolutely continuous with respect to the original probabilities?
- Is it necessary to assume that the density of the risk adjusted probability distribution is square integrable?
- Does the Black-Scholes partial differential equation have a unique solution?

## Of course, you will also get the standard stuff

Price processes, trading strategies, the budget constraint, interest rates, prices of risk, their existence and uniqueness, state price processes, arbitrage, changing the unit of account or the numeraire, replication of claims, delta hedging, dynamic market completeness and the complete markets theorem, the martingale valuation principle using either the state price process or the risk adjusted probabilities, and lots of examples.

## Comprehensive analysis of two important applications

Would you like to see, once and for all, a comprehensive analysis of the

Black-Scholes model and the Black-Scholes Formula?

- Do we really know that the value function of a claim is sufficiently differentiable to apply Ito’s Lemma?
- How is the volatility of a claim related to its elasticity?
- How can the elasticity be interpreted in terms of the leverage of the replicating portfolio?
- Do these observations apply only to standard options or also to more fancy claims?
- In what sense does the value of a cash-or-nothing call option converge

to the payoff as the time to maturity goes to zero, and in what sense does it not so converge? - What is the trick that allows easy calculation of the partial derivatives

(the Greeks) of the Black-Scholes Formula? - Where does gamma of a call option reach its maximum, at the money, at the money relative to the forward price, or somewhere else?
- Why does the value of the call option approach the value of the underlying when volatility goes to infinity?
- How can you go back and forth between the Black-Scholes partial differential equation and the heat equation?
- What is the boundary condition that should be imposed, and how should it be imposed?

And how about a detailed exposition of all the one-factor Gaussian term

structure models? A derivation of the models based on the martingale valuation principle, and a mathematical analysis of their qualitative features, illustrated by plenty of graphs? Get to understand these models really well, so that you can use them as a reference point when you deal with more complicated models.

- How can the extended Vasicek model, the simplified Hull-White model and the continuous-time Ho-Lee model be calibrated?
- What is the relation between the forward rate risk premium and the yield risk premium?
- What is the shape of the bond price volatility curve in the Vasicek model and in the simplified Hull-White model?
- What about the Merton model and the continuous time Ho-Lee model?
- What are the shapes of the forward rate standard deviation curves in these models?
- The zero coupon yield standard deviation curves?
- What are the possible shapes of the yield curves in the Vasicek model and in the Merton model?
- How are these shapes affected by the model parameters?

Amish HariaHi Lars

I am a masters student in applied finance in Australia and I would just like to show you my gratitude as you have successfully showed me how to understand nd1 and nd2 in the BS model. Many papers and sites are too hard to understand but your brief paper has finally done it it a very simplistic nature. You have save me a lot of time and I thank you.

Keep up the good work.

Amish

LTNPost authorThank you, Amish!

—Lars