Columbia  University   in the City of New York        |     New York, N.Y.  10027


DEPARTMENT OF STATISTICS                                                                                Mathematics Building

                                                                                                                                                 2990 Broadway

                                                                                                                                   Phone: (212) 854-3652/3

                                                                                                                                          Fax: (212) 663-2454




Spring 2001


Professor Ioannis Karatzas








Mathematical theory and probabilistic tools for the analysis of security markets.

Lectures of 2.5 hrs. per week. 14 weeks.   Homework.  Final Examination.


Prerequisites:  A course on Stochastic Processes at the level of  G.Lawler’s book, and an introductory course on the Mathematics of Finance at the level of J. Hull’s book.





Required Text:


D. LAMBERTON & B. LAPEYRE (1995)   Introduction to Stochastic Calculus Applied to Finance.    Chapman & Hall, New York and London.



Lecture Notes:


T. BJORK (1997)  Interest Rate Theory.  In “Financial Mathematics” (W.J. Runggaldier, Ed.), Lecture  Notes in Mathematics  1656, 53-122.  Springer-Verlag, New York.

S.E. SHREVE:   Lecture Notes



Recommended Textbooks:


R.J. ELLIOTT & P.E. KOPP (1999)  Mathematics of Financial Markets. Springer-Verlag, NY.

M. MUSIELA & M. RUTKOWSKI  (1997)  Martingale Methods in Financial Modelling.   Springer-Verlag, New York.









Pricing by Arbitrage:   Pricing and Hedging, single- and multi-period models, Binomial models. Bounds on option prices.


Martingale Measures:   General discrete-time market model, trading strategies, arbitrage opportunities, martingales and risk-neutral pricing, equivalent martingale measures, Black-Scholes formula as the limit of  binomial models.


The Fundamental Theorem of Asset-Pricing:  Construction of equivalent martingale measures; local form of the “no-arbitrage” condition.


Complete Markets:  Uniqueness of the equivalent martingale measure, completeness and the martingale representation property, characterization of attainable claims.


Stopping Times and American Options:   Hedging of American claims. Optimal stopping, Snell envelope, optimal exercise time. 


Review of Stochastic Calculus:  Continuous-time processes, martingales, stochastic integrals, Ito’s rule, stochastic differential equations, Feynman-Kac formula. Martingale representation property, Girsanov’s theorem. 


European Options in Continuous-Time Models:  Dynamics, self-financing strategies, Black-Scholes formula as expectation of the claim’s discounted value under the equivalent martingale measure.  Connections with partial differential equations. Barrier options, exchange options, look-back options.


American Options:  Extended trading strategies, free boundary problems, optimal exercise time, early exercise premium.  


Bonds and Term-Structure of Interest Rates:   Market dynamics, forward-rate models. Heath-Jarrow-Morton framework, no-arbitrage condition.  Change of numeraire technique and the Forward measure. Diffusion models for the short-rate process; calibration to the initial term-structure; Gaussian and Markov-Chain models. Pricing of bond-options.  Caps, Floors, Swaps, Forward contracts.


Optimization Problems: Portfolio optimization, risk minimization, pricing in incomplete markets. 










Lecture #1: Tue  16  January

The one-period Binomial model: notions of portfolio, arbitrage, equivalent martingale measure, contingent claim, attainability. Examples: European call- and put-options. 



Lecture #2:  Thu 18  January

The one-period Binomial model: property of completeness under the condition  u<1+R<d .  The Trinomial model, failure of completeness, meaning of attanainability in this context. The many-period Binomial model: martingale property of discounted stock-prices under the equivalent martingale measure, notion of self-financed portfolio.



Lecture #3:  Tue  23  January

The many-period Binomial Model: martingale property of discounted self-financed-portfolio-values under the equivalent martingale measure, absence of arbitrage, completeness. The transform-representation property of martingales, on the filtration of the simple random walk.


Assignment  # 1:

Read Chapter 1 from Lamberton-Lapeyre (pp. 1-16), or Chapters 1-2 of Elliott-Kopp (pp. 1-43).

Do Problems 1-7, pp. 12-16 in Lamberton-Lapeyre.



Lecture #4:  Thu  25  January

Notion of value of a contingent claim in terms of the minimal amount required for super-replication. The backwards-induction, Cox-Ross-Rubinstein formula. The notions of stopping time and of American Contingent Claim: value of an American Contingent Claim in terms of the solution of an optimal stopping problem.



Lecture #5:  Tue  30  January

Brief overview of the notions and properties of martingales and stopping times: optional stopping and optional sampling theorems. Elementary theory for the optimal stopping problem in discrete-time: the Snell envelope and the Dynamic Programming Equation. Backwards induction.



Lecture #6:  Thu  1  February

Elementary theory for the optimal stopping problem in discrete-time: the Snell envelope and characterization of an optimal stopping time. The valuation of American Contingent claims, and its relation to optimal stopping. The special case of American call-option.


Assignment  # 2:

Read Chapter 2 from Lamberton-Lapeyre (pp. 17-28), or Chapter 5 of Elliott-Kopp (pp. 75-98).

Do Exercises 1-4, pp. 25-26 in Lamberton-Lapeyre.   Due  Tue. 13 February.



Lecture #7:  Tue  6  February

Conditional Expectations. Radon-Nikodym theorem, likelihood ratios of absolutely continuous probability measures, their martingale properties and explicit computations. “Bayes rule” for conditional expectations, notion and significance of state-price-densities.



Lecture #8:  Thu  8  February

Portfolio Optimization: maximization of expected utility from terminal wealth. Explicit computa-tions in the logarithmic and power-cases. Idea of partial-hedging: maximization of the probability of perfect hedge, or of the success-ratio.


Assignment  # 3:

On maximization of the probability of perfect hedge, and of the success-ratio.  Due Thu 8 March.



Lecture #9:  Tue  13  February

Continuous-time processes, Poisson process, Brownian motion as a limit of simple random Walks. Quadratic variation of the Brownian path. Markov processes and  Martingales in continuous time. Notion of stopping time.




Lecture  # 10:  Thu  15  February

Square-integrable martingales, bracket- and quadratic variation- processes. Eamples from

the Poisson and Wiener processes. P. Levy’s characterization of Brownian motion. Notion

of Ito’s Stochastic Integral, as generalization of the martingale transform. Elementary

properties. Notion and properties of local martingales.


Assignment  # 4:

Read Chapter 3 from Lamberton-Lapeyre (pp. 29-42).

Do Exercises 6, 8-13, pp. 56 – 58  in Lamberton-Lapeyre.



Lecture  # 11:  Tue  20  February

Extension of the Stochastic Integral to general processes. Stochastic Calculus; he Ito rule

and its ramifications. Examples; elementary stochastic integral equations. Proof of P. Levy’s characterization of Brownian motion.




Lecture  # 12:  Thu  22  February

Cross-variation of continuous martingales. The multi-dimensional Ito formula; integration-

by-parts. Examples. The martingale representation property of the Brownian filtration. 


Assignment  # 5:

Read Chapter 3 from Lamberton-Lapeyre (pp. 43-56).

Do Exercises 14-17, pp. 56 – 57  in Lamberton-Lapeyre.




Lecture  # 13:  Tue  27  February

The basic theory of stochastic differential equations; Ito’s existence and uniqueness

theorems. The Markov property of solutions. The Girsanov theorem.




Lecture  # 14:  Thu  1  March

The Samuelson-Merton-Black-Scholes model for a financial market. Self-financing portfolios, wealth processes, equivalent martingale measure, arbitrage. 



Lecture  # 15:  Tue  6  March

Contingent claims, upper- and lower-hedging prices. Notions of Arbitrage and Complete-

ness. Sufficient conditions for absence of Arbitrage. Necessary and sufficient conditions

for Completeness.


Assignment  # 6:

Read Chapter 4 from Lamberton-Lapeyre (pp. 63-72).

Do Exercises 19, 21, 23, 24, 27, pp. 77 – 80  in Lamberton-Lapeyre. (Not to be handed in.)




Lecture  # 16:  Thu  8  March

The Black-Scholes model; formulae for the pricing and hedging of the European Call-Option.

Robustness of Black-Scholes Hedging, under Stochastic Volatility misspecification.








Lecture  # 17:  Tue  20  March

Mid-Term Examination




Lecture  # 18:  Thu  22  March

European Put-Call Parity; Forward Contracts. Exchange Options.

The method of “change-of-numeraire”.



Lecture  # 19:  Tue  27  March

Portfolio Optimization Problems: maximizing rate-of-growth,

maximizing the probability of “reaching a goal” on a given time-horizon.




Lecture  # 20:  Thu  29  March

Portfolio Optimization: Minimizing the expected shortfall in hedging.

The Feynman-Kac formula, and some of its applications.


Assignment  # 7:

Read Chapter 5 from Lamberton-Lapeyre (pp. 95-110).




Lecture  # 21:  Tue  3  April

Introduction to Interest-Rate Models: notions of Yield Curve, Forward Rates,

Spot Rates. Relations among them.  The Heath-Jarrow-Morton framework.

The Vasicek, Cox-Ingersoll-Ross, Ho-Lee and Hull-White models.




Lecture  # 22:  Thu  5  April

Interest-Rate Models: notion of measure-valued portfolios, “absence of arbitrage”, 

and equivalent martingale measure in the Heath-Jarrow-Morton framework.


Assignment  # 8:

Read Chapter 6 from Lamberton-Lapeyre.

Do Exercises 31, 32, 33, 37, 38,  pp. 136 – 139  in Lamberton-Lapeyre. (Not to be handed in.)




Lecture  # 23:  Tue  10  April

Interest-Rate Models: the Affine Term-Structure, inversion of the

Yield-Curve. Calibration. Examples: the Ho-Lee and Hull-White models.




Lecture  # 24:  Thu  12  April

Change of Numeraire: the notion and significance of the Forward Measure.

Examples: the pricing of Caps and Floors. Explicit computations in the

framework of the Hull-White model.




Lecture  # 25:  Tue  17  April

The pricing of American contingent claims; elements of the theory of

Optimal Stopping in continuous time. The American call-option.


Assignment  # 9:

Read Chapter 4 from Lamberton-Lapeyre (pp. 72-77).

Project:  Do Problem 4 (pp. 86-88)  and  Problem 7  (pp. 91.93).



Lecture  # 26:  Thu  19  April

Distribution of the maximum of Brownian motion and its Laplace transform.

The “perpetual” American put-option; brief discussion of the finite-horizon case.

The American put-option of up-and-out barrier type; explicit computations.




Lecture  # 27:  Tue  24  April

The joint distribution of Brownian Motion and its running maximum.

Application: the pricing of a European Barrier option. Asian options.



Lecture  # 28:  Thu  26  April

Hedging and Portfolio Optimization under Portfolio Constraints.