Mathematical Finance

Seminar

What is Mathematical FinanceOther Terms

• Financial Engineering

• Quantitative Finance

• Computational Finance

• Mathematical Finance

Topics Include

1. Probability / Statistics / Econometrics

2. Linear Algebra / Numerical Analysis

3. Calculus / Differential Equations

4. Stochastic Calculus

5. Programming – OOP, Data Structures, OS, Algorithms, Artificial Intelligence (Learning Algos)

6. Languages: C++ / C# / JAVA / R / Matlab / Proprietary Languages

7. Markets

1. Stock

2. Futures & Options

3. FX

4. Credit & Interest Rate Markets

Adv anced Math

ComputerScience

Financial Markets

Trading Strategies

Index Arbitrage

• What is a stock?

• What is an Index?

• How do you make money?

Stock• An instrument that signifies an ownership position (called equity) in a corporation, and represents a claim on its

proportional share in the corporation's assets and profits.

Examples of Buy and Hold Strategy

Stock: AAPL

Date: 3/21/03

Buy: $7.39

Date: Today

Value of Portfolio

Price: $205.75

Rate of Return: 2684.17%

Citigroup

Date: 3/21/03

Buy: $33.16

Date: Today

Value of Portfolio

Price: $3.42

Rate of Return: -90.47%

Investors Enjoy

• Consistent Profits

• Reduced Volatility

What is Volatility?• We define volatility as annualized standard deviation. The standard deviation of a return time

series is calculated as....

• std = sqrt[ {1/ n} * sum[ {r(t) - avr}^2 ] ]

• std... Standard deviationn... Number of returnsr(t)... Portfolio returns

avr... average portfolio return: avr = sum[r(t)]/n

Simulated Profits and Equity

Upside case simulated daily vs cum P&L

($1,000,000)

$0

$1,000,000

$2,000,000

$3,000,000

$4,000,000

$5,000,000

$6,000,000

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64

Simulated P&L Cum P&L

Sample Strategy for Achieving Steady Profits

Index Arbitrage• What is an Index?

– Constituent of Stocks

– Current Price in Market

– Index X $101

• Stock A – 40% $10

• Stock B – 30% $10

• Stock C – 30% $10

• Fair Value: $100

– What if I buy all the 3 stocks and sell the Index X at the same time?– Profit: $101 – Sum($40 + $30 + $30) = $1– How about we do this million times a day?– Examples of trade-able securities:

• S&P 500, Russell 2000, Russell 3000, DOW 30

• Exchanged Traded Funds – OIH, XLF etc

What do we need to implement this strategy in the real world

• Fast Computer Program– C++

• Index Arbitrage Formula– Dividends, Interest Rates, Bad prices

• Risk Measurement and Management?– What if we don’t get all the legs of the trade?

• Pros of the Strategy– Small Consistent Profits, Profits are exponentially multiplied during financial crisis such as 2008– Does not require a lot of manual efforts once the software is developed– No emotions involved except when managing risk

• Cons– Need to have sophisticated technology– Limitation on how much capital can be deployed– Examples of an architecture (NEXT PAGE)

TILE GX• Massively Scalable Performance• Array of 16 to 100 general-purpose processor cores (tiles)

• 64-bit VLIW processors with 64-bit instruction bundle• 3-deep pipeline with up to 3 instructions per cycle• 32K L1i cache, 32K L1d cache, 256K L2 cache per tile• Up to 750 billion operations per second (BOPS)• Up to 200 Tbps of on-chip mesh interconnect• Over 500 Gbps memory bandwidth with four 64-bit DDR3 controllers• 40 - 80 Gbps Snort® processing• 40 - 80 Gbps nProbe• H.264 HD video encode: dozens of streams of 1080p (baseline profile)• 64+ channels of OFDM baseband receiver processing (wireless)

Pairs Trading Strategy

• XOM vs CVX

Pairs Trading Strategy

• Whats the trade?

Here is the trade

• Sell CVX• Buy XOM• Relative Value Trade / Mean Reversion • If the stocks revert, I will make a profit or else not• Tools employed to measure this relationship• How do we measure relationships in statistics?

– Regression Analysis / Correlation Analysis• How do we decide that this pair is tradeable?

– Co-integration test and Hypothesis testing

• How do we build confidence in our model?– Back-test using historical data in C++ / C# / R / Matlab

• Past performance is not always a representative of the future– Market Experience

– Model Breakdown parameters

• Advanced Optimization– Using AI – Machines learn about these relationships on the fly

Options Trading

• Call Option• Contract between 2 parties Buyer and Seller

• It is the option to buy shares of stock at a specified time in the future

• Buyer has the right but no obligation

• Wants the underlying (stock) price of to rise

• Seller bets price wont rise

• Buyer Pays a fee called as the premium (Think of it as an insurance bet)

Risk / Reward AnalysisExample

Stock Price: $100

Strike: $100

Time: 1 year

Call Price: $1.00 Stock Price: $100

Dollar Invested: $1.00 Dollar Invested: $100

A] Stock goes up 10%

1 year from now:

Stock Price: $110

What if the bought 1 share

Return to Call Option Buyer Return to the Stock Buyer

Return = ($110 - $100) / $1 = 1000% Return = ($110 - $100) / $100 = 10%

B] Stock goes down 50%

1 year from now:

Stock Price: $50

Return to Call Option Buyer Return to the Stock Buyer

Return = -100% Return = -50%

Options provide leverage – Higher Risk / Higher Reward

Options Trading

• What if I want to bet the price of the stock will fall?

Put Option• Contract between 2 parties Buyer and Seller• It is the option to sell shares of stock at a specified time in the future• Buyer has the right but no obligation• Wants the underlying (stock) price of to fall• Seller bets price wont fall • Buyer Pays a fee called as the premium (Think of it as an insurance bet)

Fair Value of Option

• Call Option– Payoff Formula

c(t; St) = max(0; St ¡ K)

– Black Scholes Formula for pricing a Call Option

• Derived by Black Scholes and Merton ( 3 Mathematicians )• *** Formula does not always work in the real world• *** Many different variations of the formula can be learned in Stochastic Calculus,

Financial Modelling Class

Books

• Paul Wilmott on Quantitative Finance, by Paul Wilmott

• Options, Futures, and Other Derivatives, by John C. Hull

• More books:– http://www.quantster.com/books.html

Masters Levels Programs

Mathematical Finance / Quantitative Finance/ Operations Research / Computational Finance

1. NYU

2. Carnegie Mellon

3. Columbia

4. Stanford University

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