Math86 Mathematical Finance I, Fall 2012     

 strategies (revised on 30 Aug, 2012.


Meifang Chu

Office      318 Kemeny
Tel           646-1614

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
Option-Pricing Theory and its applications to the valuation and
risk management of financial derivatives
products. Topics includes:

Prerequisites: CS 1 (computer science)
                          Math 23 (Differential Equations),
                          Math 20 (Discrete Probability) or Math 60 (Probability)
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)
  hull8 or  hull6 

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

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
(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
(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
(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  

(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 

(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

(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
    by Steven E. Shreve  *
    Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit
    by  Damiano Brigo and Fabio Mercurio *
   The Mathematics of Financial Derivatives : A Student Introduction (Paperback)
   by Paul Wilmott,  Sam Howison  and  Jeff Dewynne *          
    Option Pricing: Mathematical Models and Computation                               
    by Paul Wilmott, Jeff Dewynne, Sam Howison   * 
    Monte Carlo Methods in Financial Engineering
     (Stochastic Modelling and Applied Probability)
     by Paul Glasserman *
   Financial Calculus
    by Martin Baxter and Andrew Rennie  *
    Credit Derivatives Pricing Models: Model, Pricing and Implementation
     by Philipp J. Schönbucher
    Credit Derivatives: A Primer on Credit Risk, Modeling and Instrument
    by George Chacko, Anders Sjoman, Dideto Motohashi, Vincent Dessain
   Salomon Smith Barney Guide to Mortgage-Backed and Asset-Backed Securities
Lakhbir Hayre
   Stochastic Differential Equations

     by Bernt K. Oksendal   *
    Probability with Martingales
     by David Williams    
    The Theory of Stochastic Processes
     by D.R. Cox and H.D. Miller   
    Continuous-Time Finance
    by Robert Merton
     Principles od Corporate Finance
     by R. A. Brealey and  S. C. Myers
      Time Series Analysis
     by James D. Hamilton
       How I became a Quant: Insights from 25 of Wall Street's Elite 
     by Barry Schachter and Richard Lindsey    
      The Black Swan (2nd edition): The Impact of the Highly Improbable
     by Nassim Nicholas Taleb 

Link to the   Mathematics Department.