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Rutgers Business School

Financial Modeling II: Syllabus

Oleg Sokolinskiy

September 3, 2011

1 The Course

What you will learn

The sources of value of various derivative securities. You will form the correct intuition

and avoid common mistakes.

Solid understanding of the math language used to describe the derivative securities mod-

els.

Practical implementation of models - numerical techniques.

The course has four parts:

I. INTRODUCTION TO THE MATHEMATICS OF CONTINUOUS TIME FINANCE

II. DERIVATIVES ON EQUITY

III. FIXED INCOME - INTEREST RATE MODELS

IV. CREDIT RISK

However, we will focus on I. and II., as they form the foundations for more specialized fields

of financial modeling like III. and IV.

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1 THE COURSE 2

The structure of the course (preliminary and subject to change (!) - the pace of the course

and the extent of its coverage will be adjusted to optimally suit your needs!)

1. Modeling philosophy. Stochastic calculus I: process for the underlying asset; Brownian

motion: quadratic variation.

2. Trading strategies and accumulated portfolio gains - Ito integral; Ito (stochastic) calculus;

language of SDEs.

3. Our first derivation of the Black-Scholes - PDE. Solving the BS PDE (Fourier transform):

diffusion equation and the contagion phenomenon. BS intuition about the source of

value of derivative securities. Greeks and their dynamics. Hedging techniques: delta,

gamma, vega - hedging. Beating the market: first introduction to the concept of implied

volatility; measuring realized volatility, hedging using realized or implied volatil-

ity. Black-Scholes in practice: discrete hedging and transaction costs. [First computer

assignment: hedging in the B-S world.]

4. Risk neutral pricing. Binomial model illustration of risk neutral pricing. Spotting a Brow-

nian motion: the Levy theorem. Changing the probability measure - Girsanov theorem.

Justifying risk neutral pricing: the martingale representation theorem.

5. Fundamental asset pricing theorems. What if we cannot perfectly replicate? Risk-neutral

pricing as an approximation. The pricing kernel - risk neutral density representation of

a derivatives price. Monte Carlo pricing. [Second computer assignment: Monte Carlo

pricing.]

6. Linking risk-neutral pricing to partial differential equations: the Feynman-Kac theorem.

PDE approach and finite difference methods.

7. American derivative securities: possibility of early exercise. [Third computer assignment

(part 1): numerically solution of pricing PDEs]

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2 GRADING 3

8. The smile. Local volatility surface: mere tool for studying models or a consistent pricing

mechanism? Discussion: excesses of calibration. [Third computer assignment (part 2):

incorporating LVS.]

9. Stochastic volatility - the Heston model. [Third computer assignment (part 3): SV model.]

10. Jumps and market incompleteness. Jump-diffusion models. Mertons model. (Advanced

material: SV-J. Characteristic function methods.) [Fourth computer assignment: Heston

- closed form (numerical integration).]

11. Exotics. Closed-form solutions. Pricing PDEs. Static replication.

12. Fixed income. Short rate models. One factor. Two factor Gaussian.

13. HJM framework. Why LIBOR? Blacks formula. LIBOR forward and swap rate models:

calibration.

14. Credit risk I: Overview of products. Jump-to-ruin model. Reduced form intensity mod-

els. Brief description of firm-value-based models. Pricing CDS, defaultable bonds. (Ad-

vanced material:Credit risk II: default risk of multiple names. Dependence of default

events - copula functions. Pricing CDOs.)

2 Grading

The final grade will be the result of (a weighted average of):

home assignments (each home assignment receives either 0 (fail) or 1 (pass)) and brief

quizzes (designed to test your understanding of the basic concepts taught during the pre-

vious lectures) [weight - 30%]

computer assignments designed to teach you the numerical implementation of the models

and provide a practical feel for the models [weight - 40%]

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3 LITERATURE 4

final exam: two parts - fundamental and advanced. The exam material will be covered

in the lecture notes and in the book by Shreve (see Literature below) [attention! possible

minimum requirements based on the fundamental part of the exam; weight - 30%]

3 Literature

The course will utilize numerous textbooks and articles. Here I list the most important sources.

The lecture notes will be as self-contained as possible. But for the purely theoretical aspects

of the course I will also rely heavily on:

Shreve, S., 2004. Stochastic calculus for finance II: continuous-time models. Springer.

ONLY THE BOOK by Shreve and the lecture notes are obligatory reading! The rest is for your

information and advanced studies.

The following are the additional textbooks that I will draw upon:

Volatility smile:

Gatheral, J., 2006. The volatility surface: a practitioners guide. John Whiley & Sons

Inc., Hoboken, New Jersey.

Monte Carlo pricing:

Paul Glasserman. Monte Carlo Methods in Financial Engineering: 53 (Stochastic Mod-

elling and Applied Probability).

Finite Difference Approach:

Domingo Tavella, Curt Randall. Pricing Financial Instruments: The Finite Difference

Method (Wiley Series in Financial Engineering).

Interest rates and credit risk:

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