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

The Math and Magic of Financial Derivatives

Klaus Volpert

Villanova UniversityMarch 31, 2008


The math and magic of financial derivatives

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)


The math and magic of financial derivatives

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


The math and magic of financial derivatives

  • 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

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

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

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

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

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

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?


The math and magic of financial derivatives

  • Want more information ?

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


Price can be determined by

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.


The math and magic of financial derivatives

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

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

The Black-Scholes PDE

V =value of derivativeS =price of the underlyingr =riskless interest ratσ=volatilityt =time


The math and magic of financial derivatives

  • 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:


The math and magic of financial derivatives

  • 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

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

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

Let’s use σ= 40%


Put all this into maple

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

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


Cox ross rubinstein 1979

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

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 methods1

Monte-Carlo-Methods

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


Summary

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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