Mathematical Finance Seminar

What is Mathematical Finance Other 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

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: $ Rate of Return: %

Citigroup Date: 3/21/03 Buy: $33.16 Date: Today Value of Portfolio Price: $3.42 Rate of Return: %

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 deviation n... Number of returns r(t)... Portfolio returns avr... average portfolio return: avr = sum[r(t)]/n

Simulated Profits and Equity

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 Gbps Snort® processing 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 Analysis Example Stock Price: $100 Strike: $100 Time: 1 year Call Price: $1.00Stock Price: $100 Dollar Invested: $1.00Dollar Invested: $100 A] Stock goes up 10% 1 year from now: Stock Price: $110 What if the bought 1 share Return to Call Option BuyerReturn 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 BuyerReturn 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: –

Masters Levels Programs Mathematical Finance / Quantitative Finance/ Operations Research / Computational Finance 1.NYU 2.Carnegie Mellon 3.Columbia 4.Stanford University derivatives.com/index.php?option=com_conte nt&task=view&id=54#usa