The Math and Magic of Financial Derivatives

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

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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

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• 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 (CDOs) due to rising foreclosures

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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.

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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.

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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?

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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!

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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?

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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?

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

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Price can be determined by

• The market (as in an auction)
• Or mathematical analysisin 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.

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They started with the assumption that stocks
follow a random walk on top of an intrinsic
appreciation
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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,vdt)s volatility of
Sµ average percentage return of S
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The Black-Scholes PDE
V value of derivativeS price of the
underlyingr riskless interest rats
volatilityt time
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• 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

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• 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, s
• This formula is easily programmed into Maple or
  other programs

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For our MSFT-example

• S19.40 (the current stock-price)E20
  (the strike-price)r3.5t12 monthsand. . .
  s. . .?
• Ahh, the volatility s
• Volatilitystandard deviation of (daily) returns
• Problem historic vs future volatility

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Volatility is not as constant as one would wish .
. .
Lets use s 40
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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

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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

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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.
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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.

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Monte-Carlo-Methods

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

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Summary

• While each method has its pros and cons,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 lets
  you sleep better at night.