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The Math and Magic of Financial Derivatives
Klaus VolpertVillanova UniversityMarch 31, 2008
. . .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)
Financial Derivatives have been called. . .
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
Famous Calamities
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:
85
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20 40 60 80 100 120
day
stockprice = caseIndex
exp
Randomw alk Line Scatter Plot
That means they follow a Geometric Brownian Motion Model:
dSdt dX
S
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±t+1=2¾2S2±2V±S2+rS±V±S¡rV=0
±V±t+1=2¾2S2±2V±S2+rS±V±S¡rV=0
±V±t+1=2¾2S2±2V±S2+rS±V±S¡rV=0
±V±t+1=2¾2S2±2V±S2+rS±V±S¡rV=0
±V±t+1=2¾2S2±2V±S2+rS±V±S¡rV=0
V =value of derivativeS =price of the underlyingr =riskless interest ratσ =volatilityt =time
22 2
2
10
2
V V VS rS rV
t S S
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
1 2( ) ( )rtC SN d Ee N d
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%
00.10.20.30.40.50.60.7
7/2/2006 1/18/2007 8/6/2007 2/22/2008 9/9/2008 3/28/2009
4-month volatilities
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
Cox-Ross-Rubinstein (1979)
This approach uses the discrete method of binomial trees to price derivatives
S=100
S=101
S=102
S=99
S=100
S=98
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.
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day
stockprice = caseIndex
exp
Randomw alk Line Scatter Plot
Monte-Carlo-Methods
For our MSFT-call-option (with 3000 walks), we get $3.10
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call
0 5 10 15 20 25 30 35
mean = 2.42466
Measures from Randomw alk Histogram
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.
