- 1 Background Ideas
- 1.1 Brief History of Mathematical Finance
- 1.2 Options and Derivatives
- 1.3 Speculation and Hedging
- 1.4 Arbitrage
- 1.5 Mathematical Modeling
- 1.6 Randomness
- 1.7 Stochastic Processes
- tions (CDOs) 1.8 A Binomial Model of Mortgage Collateralized Debt Obliga-
- 2 Binomial Option Pricing Models
- 2.1 Single Period Binomial Models
- 2.2 Multiperiod Binomial Tree Models
- 3 First Step Analysis for Stochastic Processes
- 3.1 A Coin Tossing Experiment
- 3.2 Ruin Probabilities
- 3.3 Duration of the Gambler’s Ruin
- 3.4 A Stochastic Process Model of Cash Management
- 4 Limit Theorems for Stochastic Processes
- 4.1 Laws of Large Numbers
- 4.2 Moment Generating Functions
- 4.3 The Central Limit Theorem
- 4.4 The Absolute Excess of Heads over Tails
- 5 Brownian Motion 4 CONTENTS
- 5.1 Intuitive Introduction to Diffusions
- 5.2 The Definition of Brownian Motion and the Wiener Process
- 5.3 Approximation of Brownian Motion by Coin-Flipping Sums
- 5.4 Transformations of the Wiener Process
- 5.5 Hitting Times and Ruin Probabilities
- 5.6 Path Properties of Brownian Motion
- 5.7 Quadratic Variation of the Wiener Process
- 6 Stochastic Calculus
- 6.1 Stochastic Differential Equations and the Euler-Maruyama Method
- 6.2 Itˆo’s Formula
- 6.3 Properties of Geometric Brownian Motion
- 7 The Black-Scholes Model
- 7.1 Derivation of the Black-Scholes Equation
- 7.2 Solution of the Black-Scholes Equation
- 7.3 Put-Call Parity
- 7.4 Derivation of the Black-Scholes Equation
- 7.5 Implied Volatility
- 7.6 Sensitivity, Hedging and the “Greeks”
- 7.7 Limitations of the Black-Scholes Model
- 1.1 This isnotthe market for options! List of Figures
- 1.2 Intrinsic value of a call option
- 1.3 A diagram of the cash flow in the gold arbitrage
- 1.4 The cycle of modeling
- 1.5 Initial conditions for a coin flip, from Keller
- 1.6 Persi Diaconis’ mechanical coin flipper
- 1.7 The family tree of some stochastic processes
- 0 to 100 and the base mortgage default probability 0.01 to 0.15 1.8 Default probabilities as a function of both the tranche number
- 2.1 The single period binomial model.
- 2.2 A binomial tree
- 2.3 Pricing a European call
- 2.4 Pricing a European put
- 3.1 Welcome to my casino!
- 3.2 Welcome to my casino!
- 3.3 Several typical cycles in a model of the reserve requirement.
- 4.1 Block diagram of transform methods.
- distribution. 4.2 Approximation of the binomial distribution with the normal
- 4.3 The half-integer correction
- 4.4 Probability ofsexcess heads in 500 tosses
- 5.1 Graph of the Dow-Jones Industrial Average from August,
- increments with the same mean and variance (brown line). to August 2009 (blue line) and a random walk with normal
ben green
(Ben Green)
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