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
http://www.global-derivatives.com/index.php?option=com_content&task=view&id=54#usa