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**

** **

** **

** **

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.

- Brief review of Stochastic Calculus:
integration with respect to continuous martingales, Ito’s change of
variable formula, Girsanov theorem, stochastic differential equations and
diffusion processes.
- Discrete- and continuous-time stochastic
models for asset-prices. Notions of trading strategies, arbitrage
opportunities, contingent claims, hedging and pricing. The Fundamental
Theorem: equivalence between the absence of arbitrage opportunities and
the existence of equivalent martingale measures. Complete and incomplete
markets. Fair price as an
expectation under the equivalent martingale measure, and as the solution
to a Partial Differential Equation. Examples: Black-Scholes formula for a
European call-option; American options and stopping times; barrier,
exchange and look-back options.
- Models for the
term-structure of interest rates. Heath-Jarrow-Morton model, diffusion and
Gaussian models.
- Portfolio optimization
problems: utility maximization, maximizing the probability of reaching a
“goal”, minimizing expected shortfall, measures of risk.

*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 www.cs.cmu.edu/~chal/shreve.html

*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.

DETAILED COURSE SCHEDULE

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.

*SPRING BREAK*

*Lecture # 17: Tue 20 March*

*Lecture # 18: Thu 22 March*

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

The
method of “change-of-numeraire”.

*Lecture # 19: Tue 27 March*

*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).*

*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*

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

*Lecture # 28: Thu 26 April*

Hedging
and Portfolio Optimization under Portfolio Constraints.

*Thu
3 May: FINAL EXAMINATION*