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 Topics in Computational Finance Math 534 Days: Mon & Wed Time: 5:30 - 6:45 pm Winter 2015

Course abstract:

The financial world is continuously affected by perturbation factors that bring stochasticity to the system. The dynamics in this case can be described by stochastic differential equations, which can be solved by stochastic integration. Examples of these types of problems can arouse from the study of:

• 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.

The course starts with an introduction to stochastic models of interest rates and bond valuation when the rate is stochastic. It continues with stochastic models of stock price in the cases when the drift and the volatility are either constant or time dependent, leading to the Black-Scholes formulas for options. An entire chapter is dedicated to the risk neutral valuation method and its applications in pricing a large range of derivatives products of European type. Other related topics treated in this part of the course are the risk-neutral world and the associated martingale measure. The rest of the material deals with the Black-Scholes analysis for European and Asian derivatives. This includes finding explicit formulas for the prices of the most used derivatives. The course ends with a discussion on American options, which in general do not have close form solutions and their pricing involves numerical methods.

Course goals:

• Pricing derivatives
• Applications to Finance and Actuarial Science
• Helps the preparation for the MFE exam (the 3rd Actuarial Exam)
• Possible direction of research for your Master’s thesis.

Prerequisites:

Calculus I, II
Probability 360 or 370.

Topics Covered:

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
13.7 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
15 American Options
15.1 Exercising time as a stopping time
15.2 Numerical methods