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# Financial Mathematics at Laurier

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by Roman Makarov on 20 March 2014

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#### Transcript of Financial Mathematics at Laurier

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

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?

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

Data Analysis
Portfolio Optimization
Risk Management
Interest Rate Modelling
Stock Game
The interesting fact is that
the result does not depend on
the probabilities !