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

DEPARTMENT OF MATHEMATICS                                                                 508 Mathematics Building
DEPARTMENT OF STATISTICS                                                                        618 Mathematics Building
                                                                                                                                                 2990 Broadway

Professor Ioannis Karatzas

G8273: TOPICS IN THE MATHEMATICS OF FINANCE

COURSE SYLLABUS

Prerequisites: Stochastic Differential Equations G8263.

Rapid Overview of Stochastic Analysis: Integration with respect to continuous martingales, Ito’s change-of- variable formula, Girsanov’s theorem. 
The martingale representation property for the Simple Random Walk 
and for the Brownian filtration, the Clark-Ocone formula. 
Stochastic Differential Equations and Diffusion processes: the martingale problem of Stroock and Varadhan, one-dimensional diffusions. 

Applications to Finance:  Notions of trading strategies, arbitrage opportunities, contingent claims, hedging and pricing. The Fundamental Theorem: relations between the absence of arbitrage opportunities and the existence of equivalent martingale measures. 

. Complete Markets:  Uniqueness of the equivalent martingale measure. 
Fair price as expectation under the equivalent martingale measure, and 
as the solution to a Partial Differential Equation. 
Examples: Black-Scholes formula for a European call-option. Optimal 
stopping and the pricing of American options. Free-boundary problems. 
Barrier, exchange, Asian and look-back options.

. The Consumption/Investment Optimization problem: general theory, 
explicit solutions, examples.

. The problem of Equilibrium: Representative agent, the Capital-Asset-
Pricing Model; existence, uniqueness and explicit computations of 
equilibria.

. Diverse Markets: Notion of relative arbitrage, ramifications for option
pricing and portfolio optimization.

. Incomplete markets: General convex constraints on portfolios, notions 
of support functions and polar cones. Hedging and consumption/invest-
ment optimization in incomplete markets. Fair pricing via utility maximi-
zation and entropy. Mean-variance hedging, notions of variance-optimal martingale measure and efficient frontier. Transaction costs. 
Goal problems; dynamic Measures-of-Risk. 

Open problems.

RECOMMENDED TEXTS:

I. KARATZAS  & S.E. SHREVE (1998) Methods of Mathematical Finance. Springer-Verlag, New York.

R.E. FERNHOLZ (2002)  Stochastic Portfolio Theory. Springer-Verlag, New York.

H. FOELLMER  & A. SCHIED (2003)  Stochastic Finance.  Walter de Gruyter, New York.