The Math and Magic of Financial Derivatives

1. The Math and Magic of Financial Derivatives Klaus Volpert Villanova UniversityMarch 31, 2008

2. 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)

3. 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

4. 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

5. On the Other Hand • 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.

6. So, what is a Financial Derivative? • 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.

7. An Example: A Call-option on Oil • 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?

8. Other such Derivatives can be written on underlying assets such as • Coffee, Wheat, and other `commodities’ • Stocks • Currency exchange rates • Interest Rates • Credit risks (subprime mortgages. . . ) • Even the Weather!

9. Fundamental Question: • 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?

10. Test your intuition: a concrete example • 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?

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

12. Price can be determined by • 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.

13. They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation:

14. That means they follow a Geometric Brownian Motion Model: 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

15. The Black-Scholes PDE V =value of derivativeS =price of the underlyingr =riskless interest ratσ=volatilityt =time

16. 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:

17. 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

18. For our MSFT-example • 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

19. Volatility is not as constant as one would wish . . . Let’s use σ= 40%

20. Put all this into Maple: • 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

21. Discussion of the PDE-Method • 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

22. S=102 S=101 S=100 S=100 S=99 S=98 Cox-Ross-Rubinstein (1979) 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.

23. Monte-Carlo-Methods • 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.

24. Monte-Carlo-Methods • For our MSFT-call-option (with 3000 walks), we get $3.10

25. Summary • 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.