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The Math and Magic of Financial Derivatives

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The Math and Magic of Financial Derivatives

Klaus Volpert

Villanova UniversityMarch 31, 2008

Financial Derivatives have been called. . .

- . . .Engines of the Economy. . .Alan Greenspan(long-time chair of the Federal Reserve)
- . . .Weapons of Mass Destruction. . .Warren Buffett(chair of investment fund Berkshire Hathaway)

Famous Calamities

- 1994: Orange County, CA: losses of $1.7 billion
- 1995: Barings Bank: losses of $1.5 billion
- 1998: LongTermCapitalManagement (LTCM) hedge fund, founded by Meriwether, Merton and Scholes. Losses of over $2 billion

- September 2006: the Hedge Fund Amaranth closes after losing $6 billion in energy derivatives.
- January 2007: Reading (PA) School District has to pay $230,000 to Deutsche Bank because of a bad derivative investment
- October 2007: Citigroup, Merrill Lynch, Bear Stearns, Lehman Brothers, all declare billions in losses in derivatives related to mortgages and loans (CDO’s) due to rising foreclosures

- In November 2006, a hedge fund with a large stake (stocks and options) in a company, which was being bought out, and whose stock price jumped 20%, made $500 million for the fund in the process
- The head trader, who takes 20% in fees, earned $100 million in one weekend.

- Typically it is a contract between two parties A and B, stipulating that, - depending on the performance of an underlying asset over a predetermined time - , so-and-so much money will change hands.

- Suppose, the oil price is $40 a barrel today.
- Suppose that A stipulates with B, that if the oil price per barrel is above $40 on Aug 1st 2009, then B will pay A the difference between that price and $40.
- To enter into this contract, A pays B a premium
- A is called the holder of the contract, B is the writer.
- Why might A enter into this contract?
- Why might B enter into this contract?

- Coffee, Wheat, and other `commodities’
- Stocks
- Currency exchange rates
- Interest Rates
- Credit risks (subprime mortgages. . . )
- Even the Weather!

- What premium should A pay to B, so that B enters into that contract??
- Later on, if A wants to sell the contract to a party C, what is the contract worth?

- Current stock price of Microsoft is $19.40. (as of last night)
- A call-option with strike $20 and 1-year maturity would pay the difference between the stock price on January 22, 2009 and the strike (as long the stock price is higher than the strike.)
- So if MSFT is worth $30 then, this option would pay $10. If the stock is below $20 at maturity, the contract expires worthless. . . . . .
- So, what would you pay to hold this contract?
- What would you want for it if you were the writer?
- I.e., what is a fair price for it?

- Want more information ?
- Here is a chart of recent stock prices of Microsoft.

- The market (as in an auction)
- Or mathematical analysis:in 1973, Fischer Black and Myron Scholes came up with a model to price options.It was an instant hit, and became the foundation of the options market.

whereS = price of underlyingdt = infinitesimal time perioddS= change in S over period dtdX = random variable with N(0,√dt)σ = volatility of Sμ = average percentage return of S

V =value of derivativeS =price of the underlyingr =riskless interest ratσ=volatilityt =time

- Different derivatives correspond to different boundary conditions on the PDE.
- for the value of European Call and Put-options, Black and Scholes solved the PDE to get a closed formula:

- Where N is the cumulative distribution function for a standard normal random variable, and d1 and d2 are parameters depending on S, E, r, t, σ
- This formula is easily programmed into Maple or other programs

- S=19.40 (the current stock-price)E=20 (the `strike-price’)r=3.5%t=12 monthsand. . . σ=. . .?
- Ahh, the volatility σ
- Volatility=standard deviation of (daily) returns
- Problem: historic vs future volatility

Let’s use σ= 40%

- with(finance);
- evalf(blackscholes(19.40, 20, .035, 1, .40));
- And the output is . . . .
- $3.11
- The market on the other hand trades it
- $3.10

- There are only a few other types of derivative contracts, for which closed formulas have been found
- Others need numerical PDE-methods
- Or . . . .
- Entirely different methods:
- Cox-Ross-Rubinstein Binomial Trees
- Monte Carlo Methods

S=102

S=101

S=100

S=100

S=99

S=98

This approach uses the discrete method of binomial trees to price derivatives

This method is mathematically much easier. It is extremely adaptable to different pay-off schemes.

- Instead of counting all paths, one starts to sample paths (random walks based on the geometric Brownian Motion), averaging the pay-offs for each path.

- For our MSFT-call-option (with 3000 walks), we get $3.10

- While each method has its pro’s and con’s,it is clear that there are powerful methods to analytically price derivatives, simulate outcomes and estimate risks.
- Such knowledge is money in the bank, and let’s you sleep better at night.