Download 7 Stochastic Differential Equations

Stochastic Calculus and Applications
Course abstract:
In an ideal world the laws of Physics and Finance would be governed by laws modeled
only by the Classical Calculus. However, in real world the perturbation factors bring
stochasticity to the system. The dynamics of the system is described by stochastic
differential equations, which can be solved by stochastic integration. Examples of these
types of problems can arouse from the study of:
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The evolution of stock prices
Pricing financial instruments depending on stocks
Stochastic interest rates or stochastic volatility
Pricing interest rate options
Pricing Asian options and European plain vanilla options.
Pricing some exotic options.
Course goals:
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Description of the important stochastic processes
Hands on solving explicitly stochastic differential equations
Present techniques of stochastic integration
Applications to Finance and Actuarial Science
Prepares for the MFE exam
Possible direction of research for your Master’s thesis.
Prerequisites:
Calculus I, II
Classical Probability
Topics Covered:
Part I Stochastic Calculus
1 Basic Notions
1.1 Probability Space
1.1.1 Sample Space
1.1.2 Events and Probability
1.1.3 Random Variables
1.1.4 Distribution Functions
1.1.5 Basic Distributions
1.1.6 Independent Random Variables
1.1.7 Expectation
1.1.8 Radon-Nikodym's Theorem
1.1.9 Conditional Expectation
1.1.10 Inequalities of Random Variables
1.1.11 Limits of Sequences of Random Variables
1.2 Properties of Limits
1.3 Stochastic Processes
2 Useful Stochastic Processes
2.1 The Brownian Motion
2.2 Geometric Brownian Motion
2.3 Integrated Brownian Motion
2.4 Exponential Integrated Brownian Motion
2.5 Brownian Bridge
2.6 Brownian Motion with Drift
2.7 Bessel Process
2.8 The Poisson Process
2.8.1 Definition and Properties
2.8.2 Interarrival times
2.8.3 Waiting times
2.8.4 The Integrated Poisson Process
2.8.5 The Fundamental Relation
2.8.6 The Relations dt dMt = 0, dWt dMt = 0
i
3 Properties of Stochastic Processes 47
3.1 Hitting Times
3.2 Limits of Stochastic Processes
3.3 Convergence Theorems
3.3.1 The Martingale Convergence Theorem
3.3.2 The Squeeze Theorem
4 Stochastic Integration
4.0.3 Nonanticipating Processes
4.0.4 Increments of Brownian Motions
4.1 The Ito Integral
4.2 Examples of Ito integrals
4.2.1 The case Ft = c, constant
4.2.2 The case Ft = Wt
4.3 The Fundamental Relation
4.4 Properties of the Ito Integral
4.5 The Wiener Integral
4.6 Poisson Integration
4.6.1 An Workout Example: the case Ft = Mt
5 Stochastic Differentiation
5.1 Differentiation Rules
5.2 Basic Rules
5.3 Ito's Formula
5.3.1 Ito's formula for diffusions
5.3.2 Ito's formula for Poisson processes
5.3.3 Ito's multidimensional formula
6 Stochastic Integration Techniques
6.0.4 Fundamental Theorem of Stochastic Calculus
6.0.5 Stochastic Integration by Parts
6.0.6 The Heat Equation Method
7 Stochastic Differential Equations
7.1 Definitions and Examples
7.2 Finding Mean and Variance
7.3 The Integration Technique
7.4 Exact Stochastic Equations
7.5 Integration by Inspection
7.6 Linear Stochastic Equations
7.7 The Method of Variation of Parameters
7.8 Integrating Factors
7.9 Existence and Uniqueness
8 Martingales
8.1 Examples of Martingales
8.2 Girsanov's Theorem
Part II Applications to Finance
9 Modeling Stochastic Rates
9.1 An Introductory Problem
9.2 Langevin's Equation
9.3 Equilibrium Models
9.4 The Rendleman and Bartter Model
9.4.1 The Vasicek Model
9.4.2 The Cox-Ingersoll-Ross Model
9.5 No-arbitrage Models
9.5.1 The Ho and Lee Model
9.5.2 The Hull and White Model
9.6 Nonstationary Models
9.6.1 Black, Derman and Toy Model
9.6.2 Black and Karasinski Model
10 Modeling Stock Prices
10.1 Constant Drift and Volatility Model
10.2 Time-dependent Drift and Volatility Model
10.3 Models for Stock Price Averages
10.4 Stock Prices with Rare Events
10.5 Modeling other Asset Prices
11 Risk-Neutral Valuation
11.1 The Method of Risk-Neutral Valuation
11.2 Call option
11.3 Cash-or-nothing
11.4 Log-contract
11.5 Power-contract
11.6 Forward contract
11.7 The Superposition Principle
11.8 Call Option
11.9 Asian Forward Contracts
11.10 Asian Options
11.11 Forward Contracts with Rare Events
12 Martingale Measures
12.1 Martingale Measures
12.1.1 Is the stock price St a martingale?
12.1.2 Risk-neutral World and Martingale Measure
12.1.3 Finding the Risk-Neutral Measure
12.2 Risk-neutral World Density Functions
12.3 Correlation of Stocks
12.4 The Sharpe Ratio
12.5 Risk-neutral Valuation for Derivatives
13 Black-Scholes Analysis
13.1 Heat Equation
13.2 What is a Portfolio?
13.3 Risk-less Portfolios
13.4 Black-Scholes Equation
13.5 Delta Hedging
13.6 Tradable securities
13.7 Risk-less investment revised
13.8 Solving Black-Scholes
13.9 Black-Scholes and Risk-neutral Valuation
13.9.1 Risk-less Portfolios for Rare Events
13.10.Future research directions
14 Black-Scholes for Asian Derivatives
14.0 Weighted averages
14.1 Setting up the Black-Scholes Equation
14.2 Weighted Average Strike Call Option
14.3 Boundary Conditions
14.4 Asian Forward Contracts on Weighted Averages