The Math and Magic of Financial Derivatives

The Math and Magic of Financial Derivatives Klaus Volpert, PhD Villanova UniversityAugust 20, 2010

Advocates of Derivatives: • “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.“Alan Greenspan in a speech on May 8, 2003 • “. . . Engines of the Economy. . . “Alan Greenspan 1998, (the exact quote is lost)

“Greenspan warns against Rules for Derivatives”headline in Financial Times,1999

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, 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)

Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal." Warren Buffett's Annual Letter to Shareholders of Berkshire Hathaway, March 8, 2003.

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 • Sep 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 • 13 September 2008: Lehman Brothers fails, setting off a massive financial crisis • Oct 2008: AIG gets a massive government bail-out ($180 billion)

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

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.

An Example: A Call-option on Oil • Suppose, the oil price is $75 a barrel today. • Suppose that A stipulates with B, that if the oil price per barrel is above $100 on Sep 1st2011, then B will pay A the difference between that price and $100. • 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?

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!

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?I.e., as the price of the underlying changes, how does the value of the contract change?

Test your intuition: a concrete example • Current stock price of Apple is $247.00. (as of a couple of hours ago) • A call-option with strike $250and 5-month maturity would pay the difference between the stock price on Jan 21, 2011 and the strike (as long the stock price is higher than the strike.) • So if Apple is worth$300 then, this option would pay $50. If the stock is below$250 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 stock prices of Apple over the last two years:

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.

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

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 return of S

The Geometric Brownian Motion Model: Q: how do you measure μ and σ for any given stock??

Moving Averages Most recent 1-year volatility is 30.4%, But long-time volatility is 42%. It turns out that the market prices imply a current volatility of σ =35%

Using this assumption, Black and Scholes showed that • By setting up a portfolio consisting of the derivative V and a counter position of a Δ-number of stocks S:V - Δ*Sthe portfolio can be made riskless, i.e. have a constant return regardless of what happens to S. (Δ turns out to be =dV/dS and it is constantly changing ->strategy of dynamic hedging) • This allows us to compare the portfolio to a riskless asset and be priced accordingly. • This eventually implies that V has to satisfy the dynamic condition given by the PDE.

The Black-Scholes PDE 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 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

For our APPLE-example • S=247E=250r=1%t=5 monthsand σ=35% • put into Maple: with(finance); • blackscholes(247, 250, .01, 5/12, .35)); • And the output is . . . .

$21.33

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

S=249 S=248 S=247 S=247 S=246 S=245 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.

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.

? ? mean Histogram Measures from Randomwalk 3500 3000 2500 2000 1500 1000 500 0 100 200 300 400 500 payoff = 21.30 Monte-Carlo-Methods • For our Apple-call-option (with 5000 walks), we get a mean payoff of $21.30

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.

The Math and Magic of Financial Derivatives