Columbia University in the City of New York | New York, N.Y. 10027
DEPARTMENT OF STATISTICS Mathematics Building
Phone: (212) 854-3652/3
Fax: (212) 663-2454
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.
D. LAMBERTON & B. LAPEYRE (1995) Introduction to Stochastic Calculus Applied to Finance. Chapman & Hall, New York and London.
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
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
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
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