A graduate course in the Mathematics in Finance program at the Courarant Institute of Mathematical Sciences of New York University.

**Instructor: **Professor Jonathan
Goodman,

**Meeting:** Warren Weaver Hall , room 1302, Wednesday evenings 7
to 9 pm.

**First Class: **September 3, 1997

**Prerequisites: **Scientific
Computing, Mathematics
of Finance II. This these courses cover basic numerical analysis and
mathematical finance. If you have not taken the courses but think you have
the background for the course, contact the instructor.

**Grading: **The grade will be based on weekly assignments, mostly
computational.

**Course description**: Computational techniques for solving mathematical
problems arising in finance. Numerical solution of parabolic partial differential
equations, basic schemes, general theory, relation to binomial and trinomial
trees, boundary conditions for American options, computation of sensitivities,
application to one factor and multi factor models. Stochastic simulation
and Monte Carlo. Pseudo random number generators, generating random variables
with specified distributions, statistical analysis of simulation data and
error bars. Numerical solution of stochastic differential equations. Application
to pricing, hedging, and portfolio management. Path dependent options.
Model calibration and hypothesis testing. Value at risk.

**Course Outline (tentative)**

Week 1: Finite difference methods for the Black-Scholes equation, forward and backward Euler methods, order of accuracy and convergence, the relation to trees, artificial far field boundaries.

Week 2: Advanced topics: high order methods (e.g. trapezoid rule), computing sensitivities, handeling American and knockout features, interpolation and extrapolation.

Week 3: Finite differences and trees for multifactor models.

Week 4: Model calibration via least squares and nonlinear optimization: review of optimization methods and aplication.

Week 5: Review of the direct Monte Carlo method, random number generators, the Box-Muller method, error bars and the role of variance.

Week 6: Correlated multivariate normals with application to value at risk (VAR), the difficulty of computing probabilities of rare events, application of constrained minimization to finding most likely large loss scenario (time permitting).

Week 7: Review of stochastic differential equations and the forward Euler method for them, notions of accuracy (weak and strong).

Week 8: Application to many factor models, Asian and lookback options, and yield curve modeling.

Week 9: Advanced Monte Carlo techniques: variance reduction through antithetic sampling and control variates, rare event sampling through importance functions.

Week 10: Methods for estimating sensitivities in Monte Carlo.

Week 11: Model calibration via stochastic optimization, the Robins-Monroe algorithm and its generalizations.

Week 12: Autoregression models, ARCH, and its extensions.

Week 13: Monte Carlo valuation of American options.

More materials will be posted when the course is going. In the mean time, you can look at materials from Fall 1995, or Fall 1996.

**Homework assignments**

Assignment 1, in postscript format, or LaTeX .

Assignment 2, in postscript format, or LaTeX .

Assignment 3, in postscript format, or LaTeX.

Assignment 4, in postscript format, or LaTeX.

Assignment 5, in postscript format, or LaTeX.

Assignment 6, in postscript format or LaTeX. There is a data set that goes with this assignment. Notice that this data has bursts of high volatility and periods of relative calm.

Make sure to check the homework comments and hints page often while doing assignments.

**Lecture notes**

Lecture 1, in postscript format, or LaTeX . Preliminary version with a section on examples to be added.

Lecture 2, in postscript format, or LaTeX . Also very preliminary. A major typo in (2) and (3) fixed 10/1/97.

Lecture 3, in postscript format or LaTeX . Even more preliminary.

Lecture 4, in postscript format or LaTeX. Also preliminary.

The next few lectures are on Monte Carlo. Some notes on Monte Carlo
are on the web page from last spring's Monte
Carlo class.

Lecture 5, in postscript format or LaTeX. Just a few pages on importance sampling.

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