**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

13.6 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

15 American Options

15.1 Exercising time as a stopping time

15.2 Numerical methods