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The Math and Magic of Financial Derivatives

The Math and Magic of Financial Derivatives. Klaus Volpert, PhD Villanova University February 4, 2013. Derivatives are controversial . Influential proponents include Alan Greenspan : ( chairman of the Federal Reserve 1987-2006 ).

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The Math and Magic of Financial Derivatives

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  1. The Math and Magic of Financial Derivatives Klaus Volpert, PhD Villanova UniversityFebruary 4, 2013

  2. Derivatives are controversial.Influential proponents include Alan Greenspan:(chairman of the Federal Reserve 1987-2006) • “Although the benefits and costs of derivatives remain the subject of spirited debate, the performance of the economy and the financial system in recent years suggests that those benefits have materially exceeded the costs.“in a speech to Congresson May 8, 2003

  3. I can think of no other area that has the potential of creating greater havoc on a global basis if something goes wrong Critics Dr. Henry Kaufman, economist, May 1992 Derivatives are the dynamite for financial crises and the fuse-wire for international transmission at the same time. Alfred Steinherr, author of Derivatives: The Wild Beast of Finance (1998)

  4. Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal." Warren Buffettin his Annual Letter to Shareholders ofBerkshire Hathaway, March 8, 2003.

  5. Famous Calamities • 1994: Orange County, CA: bankrupt after losses of $1.5 billion • 1995: Barings Bank: bankrupt after losses of $1.5 billion • 1998: LongTermCapitalManagement (LTCM) hedge fund, founded by Meriwether, Merton and Scholes. Losses of over $2 billion • Sep 2006: the Hedge Fund Amaranth closes after losing $6 billion in energy derivatives.

  6. 2002-2012: Philadelphia(PA) School District loses $331 million due to bad interest rate swaps(according to a Jan 2012 report by PA Budget and Policy Center) • 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 • 15 September 2008: Lehman Brothers fails, setting off a massive financial crisis • Oct 2008: AIG needs a massive government bail-out ($180 billion) due to its losses in Credit Default Swaps (CDS’s)

  7. On the Other Hand • August 2010: BHP, the worlds largest mining company, proposed to buy-out Potash Inc, a Canadian mining company, for $38 billion. The CEO of Potash, Bill Doyle, stood to make $350 million due to his stock options. • Hedge fund managers, such as James Simon and John Paulson, have made billions a year, often using derivatives. . .

  8. So, what is a Financial Derivative? • Typically it is a contract between two parties A and B, who agree on a future cash flow that is contingent on future developments in the price of an underlying asset or an index.

  9. An Example: A Call-option on Oil • Suppose the oil price today is $100 a barrel. • Suppose that A stipulates with B, that if the oil price per barrel is above$120 on Sep 1st2013, then B will pay A the difference between that price and $120. • 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?

  10. Reasons to trade derivatives: • Hedge (reduce) risks • Give up potential profits in exchange for the premium and higher bottom line (`yield enhancement’) • Investment • Speculation

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

  12. Fundamental Questions: • 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 then?i.e., as the price of the underlying changes, how does the value of the contract change?

  13. Test your intuition: a concrete example • Last Friday, Apple’s share price ended at $453.62. • A call-option with strike $500and 5.5-month maturity would pay the difference between the stock priceon July 19, 2013 and the strike (as long the stock price is higher than the strike.) • So if Apple is worth$600 then, this option would pay $100. If the stock is below$500 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?

  14. Want more information ? • Here is a chart of stock prices of Apple over the last two years:

  15. Want more information ? • Here is a chart of stock prices of Apple over the last two years: Please write down your estimate for a price of a 5.5-month call-option on Apple with strike $500

  16. Historically • Prices were determined by supply and demand, through a mechanism similar to an auction • In 1973, however, Fischer Black and Myron Scholes came up with a model to price options mathematically. It was very successful, won the Nobel prize in economics, and became the foundation of the options market.

  17. They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation: where riskless interest rate

  18. Aside: How do you measure σ?

  19. They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation: where riskless interest rate

  20. This implies that the probability distribution for is lognormal:

  21. Fair price=expected payoff of the option, discounted to present time • Where N is the cumulative distribution function for a standard normal random variable, and d1 and d2 are parameters depending on , K, r, T, σ • This formula is easily programmed into Maple or other programs • So for our example (Apple, $453.62 now, $500 strike, 5.5-month maturity) we get the price. . . . (drumroll).. . . .

  22. $13.62

  23. The `Nuclear Power` Effect: Leverage • So if Apple is at $453.62, the strike at $500, the time to maturity 5.5 months, volatility=.24, the price for the call-option was $13.62. • Suppose, the stock price went up 5% today,to $476.30, what would happen to the price of the option? • Answer: the option price would go to $21.97! • That’s up almost 62%, 12-fold the increase of the stock price! • That’s the power of options: a smallpercentage change in the underlying, creates a largepercentage change in the value of the derivative! • Derivatives amplify movements of the underlying • This, in part, explains both its usefulness and its destructiveness!

  24. Actually, Black and Scholes derived a much more general result that holds for any type of derivative contract with Value V V =value of derivativeS =price of the underlyingr =riskless interest ratσ=volatilityt =time Different Derivative Contracts correspond to different boundary conditions on the PDE.For call-options, they solved the PDE and obtained the previous formula.

  25. Discussion of the PDE-Method • There are many other types of derivative contracts, for which closed formulas have been found. (Barrier-options, Lookback-options, Cash-or-Nothing Options) • Others need numerical PDE-methods. • Or entirely different methods: • Cox-Ross-Rubinstein Binomial Trees • Monte Carlo Methods

  26. Monte-Carlo-Methods • On the computer, one simulates 1000’s of random walks for the same asset. One keeps track of the pay-out for each walk, and then simply averages those pay-outs, and calls that average the fair price of the option.

  27. ? ? mean Histogram Measures from Randomwalk 3500 3000 2500 2000 1500 1000 500 0 100 200 300 400 500 payoff = 21.30 Monte-Carlo-Methods (1980’s) • For our Apple-call-option (with 1,000,000 walks), we may get a mean payoff of $13.60 with a 95% confidence interval of ± $.05 • There are methods to increase accuracy, and to speed up the simulation • Very general method, but expensive.

  28. S=513 S=510 S=507 S=507 S=504 S=501 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. And it is the best method for American-type (early exercise) options.However there some derivatives (such as the lookback) where accuracy is poor.

  29. While each method has its pro’s and con’s,it is clear that there are powerful methods to value (`price’) derivatives, simulate outcomes and estimate risks. • Such knowledge is money in the bank. • Quite literally.

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