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The Math and the Magic of Financial Derivatives. Klaus Volpert, PhD Villanova University August 27, 2012. Derivatives are controversial:. “. . . Derivatives are Engines of the Economy. . . “ Fed Chairman Alan Greenspan 1998, (the exact quote is lost)

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the math and the magic of financial derivatives

The Math and the Magic of Financial Derivatives

Klaus Volpert, PhD

Villanova UniversityAugust 27, 2012

derivatives are controversial
Derivatives are controversial:
  • “. . . Derivatives are Engines of the Economy. . . “Fed Chairman Alan Greenspan 1998, (the exact quote is lost)
  • “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 to Congress on May 8, 2003
I can think of no other area that has the potential of creating greater havoc on a global basis if something goes wrong


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 in his 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 needs a massive government bail-out ($180 billion) due to its losses in Credit Default Swaps (CDS’s)
on the other hand
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 due to his stock options.
  • Hedge fund managers, such as James Simon and John Paulson, have made billions a year, often using derivatives. . .
so what is a financial derivative
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 –, a certain amount of money will change hands.
an example a call option on oil
An Example: A Call-option on Oil
  • Suppose, the oil price is $90 a barrel today.
  • Suppose that A stipulates with B, that if the oil price per barrel is above $100 on Sep 1st2013, 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?
reasons to trade derivatives
Reasons to trade derivatives:
  • Hedge (reduce) risks
  • Give up potential profits in exchange for the premium and higher bottom line (`yield enhancement’)
  • Investment
  • Speculation
other such derivatives can be written on underlying assets such as
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!
fundamental question
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 then?i.e., as the price of the underlying changes, how does the value of the contract change?
test your intuition a concrete example
Test your intuition: a concrete example
  • Friday’s stock price of Apple was $663.00.
  • A call-option with strike $700and 6-month maturity would pay the difference between the stock priceon Feb 25, 2013 and the strike (as long the stock price is higher than the strike.)
  • So if Apple is worth$800 then, this option would pay $100. If the stock is below$700 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 five years:
Please write down your estimate for a price of a 6-month call-option on Apple with strike $700
  • 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.
They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation:
and ended with the black scholes pde
And ended with 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
  • So for our example (Apple, $663 now, $700 strike, 6-month maturity) we get the price. . . . (drumroll).. . . .
discussion of the pde method
Discussion of the PDE-Method
  • There are a few 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
cox ross rubinstein 1979







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 still the best method for American-type options

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.
monte carlo methods 1980 s





Measures from Randomwalk















= 21.30

Monte-Carlo-Methods (1980’s)
  • For our Apple-call-option (with 5000 walks), we get a mean payoff of $31.30
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.
some characteristics of call options
Some characteristics of Call-options
  • So if Apple is at $663, the strike at $700, the time to maturity 6 months, the price for the call-option was $30.52.
  • Suppose, the time to maturity was 1 year instead of 6 month, what would happen to the price of the option?
  • Suppose the volatility of the underlying would increase, what would happen to the price of the option?
  • Suppose, the stock price went up today, what would happen to the price of the option?
  • Let’s be precise. . .
the most important characteristic of options gearing or leverage
The most important characteristic of options:Gearingor Leverage
  • So if Apple is at $663, the strike at $700, the time to maturity 6 months, the price for the call-option was $30.52.
  • Now today (August 27) Apple jumped to $677 (that’s up 2%)
  • What happens to the value of the option?
  • Yes, it also goes up. How much?
  • A: from 30.52 to 36.64!
  • That’s a 20% gain! (ten-fold of the stock-gain)
  • That’s the power of options: a small change in the underlying, creates a large change in the value of the derivative!
  • Derivatives amplify movements of the underlying