Math86 Mathematical Finance I, Fall 2012

strategies (revised on 30 Aug, 2012.)

Lecturer Meifang Chu
Office 318 Kemeny
Tel 646-1614
meifang.chu@dartmouth.edu

Time and Place
Lectures: MWF 08:45-09:50 am (9L) at 105 Kemeny in Fall 2012
Final Exam: Take-Home, Nov 16 - 19, 2012.
Office Hour: Thursdays 09:00-12:00 (including x-hour) or by appointment.

Course Description and Requirements
This course takes a mathematically rigorous approach to understanding
the Option-Pricing Theory and its applications to the valuation and
risk management of financial derivatives products. Topics includes:

finite probability spaces, random variables, Ito calculus, Martingales and Markov processes
risk-neutral and arbitrage-free pricing theory in a complete market
binomial and trinomial tree models
Black-Scholes analysis of European equity/FX/commodity options (pde method)
trading strategy, risk measurements and management using Monte Carlo simulation
stochastic interest rates and fixed-income derivatives

Prerequisites: CS 1 (computer science)
                          Math 23 (Differential Equations),
                        Math 20 (Discrete Probability) or Math 60 (Probability)
                          and Mathematics 50 (Probability and Statistical Inference).
Dist: QDS.

Grades are determined at 65% from the homework problem sets and
35% from a take-home final exam. These problem sets involve deriving and
solving equations numerically, analytically and graphically. All the lecture notes,
problem sets, sample codes (Excel and Visual Basics) etc will be accessed from
the Dartmouth Blackboard.

Text Books

Stochastic Calculus for Finance. I : The Binomial Asset Pricing Model
by Steven E. Shreve ( required)
shreve1

Options, Futures and Other Derivatives (8th or earlier Edition)
by John C. Hull * (required)
hull8

Syllabus (S=Shreve, H=Hull, n=chapter_number)

Module	Reading
1. Introduction to the Capital Markets and Derivatives	H1-4, H7-9
(a) market dynamics and risk factors (b) contingency claims: futures, swaps, options and other derivatives (c) market completeness
2. Probability, Sigma Algebra, Random Variables and Markov Processes	H12,13, S2
(a) probability theory, sigma algebra and conditional expectations (b) Binomial, Gaussian, Poisson distributions and the tree models (c) random variables and probability density functions (d) Random Walk and Brownian motions (e) Martingales and Markov processes
3. Discrete-time Formulation: no arbitrage, risk neutral valuation, the tree models	H12,20 S1,3
(a) binomial/trinomial trees for a lognormal process-Markov chain (b) present value and the risk-free rate (c) no arbitrage argument (d) European Call/Put Options and American Call/Put Options
4. Continuous-time Formulation (Martingale method): State Prices and Risk Neutral Measure	H27, S3
(a) change of measure, Radon-Nikodym Derivative (b) Capital Asset Pricing Model (c) Risk Neutral valuation and the Martingale method (d) European options and the Feynman-Kac formula
5. Continuous-time Formulation (PDE method): Itoh Calculus and Black-Scholes Option Pricing Theory	H5,6.13,14,16
(a) Itoh Lemma and Itoh Calculus (b) 1-factor Black-Scholes model with constant volatility and interest rate (c) present value and the Feynman-Kac formula (d) valuation of vanilla options
6. Risk Management and Trading Strategies	H18,19,21
(a) valutation and management of the greeks (b) Value-at-Risk and Monte-Carlo simulations (c) trading strategies and counter party risk
7. Fixed-Income Products - Interest Rate Models	H4,6,28-32 S6
(a) short-rate model (b) multi-factor forward rate models (Heath-Jarrow-Morton & Libor models) (c) fixed income market (d) valuation of swaps, caps and floors.
8. Other Derivatives	H23-25 S4,5
(a) American options and other path-dependent options (b) stopping times (c) first passage times (d) reflection principal (e) credit risks and prepayment risks