# PSTAT 222 - FINANCIAL MODELING - an engineering approach

#### OUTLINE

This is an introduction to stochastic models in finance. The course develops some basic stochastic models and applied them to understand how prices are determined for stocks, bonds, derivative securities, and the term structure of interest rates. Portfolio issues, including optimal portfolio theory, hedging, risk management and financial engineering are discussed. The course is necessarily quantitative. Students should be comfortable with basic probability and differential equations. Students should also be computer-literate. Numerical Methods will be heavily emphasized.

PSTAT 222A - PSTAT 222B - PSTAT 222C

### PSTAT 222A - Introduction to Mathematical Finance

Course Description
Primary Asset Valuation:
Discrete and Continuous Models
random walks,
continuous time limits - diffusions,
Forward and backward equations,
Frequently Occurring Continuous Stochastic Processes;
Arithmetic Brownian Motion;
Geometric Brownian Motion;
Mean-Reverting Process;
Ito's Lemma and Its Multivariate Extensions;
Introduction to Jump Processes;
Limitations of Difussions as Models;
Simple Financial Applications of Ito's Lemma;
Time-Inhomogeneous Cases;
Separability and Homogeneity of Cash Flows;
Discount Rate for Financial Assets;
Recursive Techniques in Asset Valuation.
Pricing Derivative Securities:
Basic Concepts
Underlying Asset Prices,
Futures,
Options and Other Derivatives,
Interest Rates and Bonds;
Black-Scholes Theory
Arbitrage Opportunities,
Pricing Forward and Futures Contracts,
European Option Pricing,
the Black-Scholes PDE, and Risk-Neutrality,
Closed Form Solutions,
Extensions to Path-Dependent Options;
Numerical Methods of Option Pricing
Finite Difference Methods,
Monte Carlo methods,
Exotic Options,
Limitations of the Black-Scholes Theory - Imperfect Markets
Arbitrage Pricing and Equivalent Martingale Measures,
Asset Price Models for which Arbitrage Pricing Fails,
Transaction Costs

Textbooks:

"Finance in Continuous Time", by David C. Shimko, 1992, Kolb Publ. Co.

"The Mathematics of Financial Derivatives", by P. Wilmott, S. Howison and H. Dewynne, 1995, Cambridge University Press

Complementary Text Book:

"Options, Futures and Other Derivative Securities", by John C. Hull, 1993, Prentice-Hall.

Outline of PSTAT222 series

### PSTAT222B - Term-Structure Models and Portfolio Theory (4-4-4)

Course Description
Term-Structure Models
One-Factor Term-Structure Models,
Cox-Ingersoll-Ross Model,
Affine Single-Factor Models,
Term-Structure Derivatives,
Hedging,
Green's Function and Term Structure,
Multifactor Models,
The Multifactor CIR Term-Structure Model,
Mortgage Backed Securities,
The Heath-Jarrow-Morton Model of Forward Rates
Optimal Portfolio and Consumption Choice
Stochastic Control,
Merton's Problem,
Solution to Merton's Problem,
The Infinite Horizon Case,
The Martingale Solution,
Equilibrium
The Primitives,
Security-Spot Market Equilibrium,
Arrow-Debrew Equilibrium,
Real Security Prices,
The Consumption -Based CAPM,
The CIR Term Structure,
The CCAPM without Dynamic Spanning

Textbook:

"Dynamic Asset Pricing Theory", by D. Duffie, 1996, Princeton Univ. Press.

Complementary Textbooks:

"The Mathematics of Financial Derivatives", by P. Wilmott, S. Howison and H. Dewynne, 1995, Cambridge University Press

"Security Markets, Stochastic Models, by D. Duffie, 1988, Academic Press

"Continuous-Time Finance", by R. Merton, 1990, Blackwell

Outline of PSTAT222 series

### PSTAT 222C - Introduction to Risk Management and Financial Engineering

Course Description
Financial Optimization and Risk Management,
Risk Indicators at Instrumental Level
Single Fixed Flow,
Fixed Rate Bond,
FRA,
Interest Rate Futures,
Interest Rate Swaps,
FX Spot,
FX Forward,
Plain Vanilla Options
Credit Risk,
Risk Indicators at the Portfolio Level
Pricing Environment,
Interest Rate Factors,
FX Factors
Value-at-Risk (VAR) and Asset-Liability Management,
Risk Metrics - Market Risk in a Single Position,
Measures of Market Risk:
Linear and Non-linear Positions
Market Risk Limits,
Calibrating Valuation and Risk Models Performance Evaluation,
Probability Distributions and Statistical Assumptions
Are the returns normally distributed? Are returns jointly normal? Are the returns serially correlated? Is the volatility stable over time?,
Forecasting Volatilities and Correlations
Basic Design,
Ex-post Estimation,
Ex-ante Estimation - Forecasting,
Defining the Optimal Decay Factor,
Assessing Performance
Univariate and Multivariate Tail Probabilities
Mathematics of Structures Monte Carlo
Generating Statistics,
Properties of the Correlation Matrix,
Mapping Algorithms
Fixed Income,
Foreign Exchange,
Commodities,
Options.

Textbooks:

"Financial Optimization", edited by A. Zenios, 1993, Cambridge University Press.

"The Handbook of Interest Rate Risk Management", edited by Jack Clark Francis and Avner Wolf, 1994, IRWIN

Complementary Textbook:

"Managing Financial Risk", by Charles W. Smithson and Clifford W. Smith, Jr, 1995, IRWIN

Outline of PSTAT222 series

Professor Zari Rachev rachev@pstat.ucsb.edu