**Tools of Financial Mathematics**

Past

Present

Future

**Pricing a Binary Contract**

**Round 1**

Round 2

"Free-Lunch" Case

"Free-Lunch" (Up)

"Free-Lunch" (Down)

"No-Free-Lunch"

"No-Free-Lunch" (Up)

"No-Free-Lunch" (Down)

Replicating Portfolio

Let us find a portfolio in stock (x share ) and cash (y dollars) whose value is equal to the payoff of the contract

Market moves up

Market moves down

Replicating Portfolio

By solving these equations

we obtain the portfolio

The # of shares

The cash amount

The today value of the portfolio gives the fair ("no-arbitrage") value of the contract is $ 5 = $ 2 x 5 - $ 5

**What is the fair price P?**

Louis Bachelier (1900)

Bachelier wrote "The Theory Of Speculation" in 1900 and presented it as his doctoral thesis to the faculty of the Academy of Paris.

Bachelier discussed the use of Random Walk to evaluate stock options. The Random Walk Theory describes the way stock prices change unpredictably as a result of unexpected information appearing in the market. However, it hardly caught any attention outside academia.

Harry Markowitz (1952)

Nobel Prize in Economic Sciences, 1990

Efficient Diversification

The theory developed in Portfolio Selection was a theory for optimal investment in stocks that differ in regard to their expected return and risk.

Markowitz showed an investor's portfolio choice can be reduced to balancing two dimensions: the expected return on the portfolio, and its variance or standard deviation

Paul Samuelson (1965)

Nobel Prize in Economics, 1970

Paul Samuelson’s findings can be summarized as follows: Market prices are the best estimates of value, price changes follow random patterns, and future stock prices are unpredictable.

**People of Financial Mathematics**

**Financial Mathematics comprises the branches of applied mathematics concerned with the financial markets.**

Buffon's Needle Animation

Buffon's Needle Problem

Buffon's Needle is one of the oldest problems in the field of geometrical probability. It was first stated in 1777.

It involves dropping a needle on a lined sheet of paper and determining the probability of the needle crossing one of the lines on the page

The remarkable result is that the probability is directly related to the value of Pi

Hit-or-Miss Method

Bloomberg Uses GPUs

to Speed Up Bond Pricing

A graphics processing unit or GPU is a specialized microprocessor that offloads and accelerates 3D or 2D graphics rendering from the microprocessor

Bloomberg went live in 2009 running its two-factor models on a farm of traditional servers paired with nVidia Tesla GPUs.

Instead of having to scale up to 1,000 servers, Bloomberg is using 48 server/GPU pairs.

Hit-Or-Miss Method

Binomial Tree Simulation

Suppose that we need to compute the area of a plane figure, which is contained completely within a square.

Choose at random N points in the square

The ratio of the area of the figure to the area of square is approximately equal to the ratio of the number points inside the figure to the total numer of points

**Financial Mathematics**

at Wilfrid Laurier University

at Wilfrid Laurier University

**Let us start with a definition**

**Let me give an example...**

**You are playing a stock game**

The stock price is random

If the stock goes up, then you get $10

If the stock goes down, then you get $0

It is an example of a cash-or-nothing contract

What is the fair price of such a contract?

The stock price is random

If the stock goes up, then you get $10

If the stock goes down, then you get $0

It is an example of a cash-or-nothing contract

What is the fair price of such a contract?

Math-Finance People at Laurier

Prof. Roman Makarov

Prof. George Lai

Prof. Joe Campolieti

Sample paths of a random process. Does it look familiar?

**Cool Research in Financial Math**

Development of Asset Price Models:

Hierarchy of Exactly Solvable Models

Model Validation:

Smiling Implied Volatility

**Model Calibration:**

Fitting parameters

to historical data

Fitting parameters

to historical data

Data Analysis

Portfolio Optimization

Risk Management

Interest Rate Modelling

**Stock Game**

The interesting fact is that

the result does not depend on

the probabilities !

Prof. Adam Metzler