The math and magic of financial derivatives
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The Math and Magic of Financial Derivatives. Klaus Volpert Villanova University March 31, 2008. Financial Derivatives have been called. . . .Engines of the Economy . . . Alan Greenspan (long-time chair of the Federal Reserve)

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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


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.


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 $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?


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?


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?


  • Want more information ?

  • Here is a chart of recent stock prices of Microsoft.


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 percentage return of S


The Black-Scholes PDE

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


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


Volatility is not as constant as one would wish . . .

Let’s use σ= 40%


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


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=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.


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.


Monte-Carlo-Methods

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


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.


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