# Math86 Mathematical Finance I, Fall 2012 (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) Options, Futures and Other Derivatives (8th or earlier Edition)
by John C. Hull
*  (required) or 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

# (3) Optional Readings (* reserved in the Baker Library)

Stochastic Calculus for Finance. II : Continuous Time Model
Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit
The Mathematics of Financial Derivatives : A Student Introduction (Paperback)
Option Pricing: Mathematical Models and Computation
Monte Carlo Methods in Financial Engineering
(Stochastic Modelling and Applied Probability)
Financial Calculus
Credit Derivatives Pricing Models: Model, Pricing and Implementation
Credit Derivatives: A Primer on Credit Risk, Modeling and Instrument
Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities
by
Lakhbir Hayre
Stochastic Differential Equations

Probability with Martingales
The Theory of Stochastic Processes
Continuous-Time Finance
Principles od Corporate Finance
Time Series Analysis
How I became a Quant: Insights from 25 of Wall Street's Elite
The Black Swan (2nd edition): The Impact of the Highly Improbable
Link to the   Mathematics Department.