Transcript Engines of the Economy or Tools of Mass Destruction?

The Pricing of Stock Options
and other Financial Derivatives
Klaus Volpert, PhD
Villanova University
Feb 3, 2011
Financial Derivatives are Controversial!
 “. . . Engines of the Economy. . . “
Alan Greenspan 1998, (the exact quote is lost)
 “Derivatives are financial weapons of mass destruction, carrying
dangers that, while now latent, are potentially lethal.“
Warren Buffett's Annual Letter to Shareholders of Berkshire Hathaway, March
8, 2003.
 “Derivatives are the dynamite for financial crises and the fuse-
wire for international transmission at the same time”.
Alfred Steinherr, in Derivatives: The Wild Beast of Finance (1998)
Famous Calamities
 1994: Orange County, CA: losses of $1.7 billion
 1995: Barings Bank: losses of $1.5 billion
 1998: LongTermCapitalManagement (LTCM) hedge fund,
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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 gets a massive government bail-out ($180 billion)
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 in stock options.
 Hedge fund managers, such as James Simon and John
Paulson, have made billions a year. . .
 John Paulson’s take-home pay in 2010 of $5 Billion exceeded
the take home pay of all PhD math professors in the country
combined.
 ‘Gold rushes’ in
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Stock options (beginning in the early 1990’s)
Mortgage-backed-securities (early 2000’s)
Collateralized Debt Obligations (CDO’s)
Credit Default Swaps (CDS’s)
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
 Suppose, the oil price is $90 a barrel today.
 Suppose that A stipulates with B, that if the oil price
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per barrel is above $100 on Sep 1st 2011, 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?
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 Questions:
 What premium should the buyer (`holder`) pay to the
seller (`writer’), so that the writer enters into that
contract??
 Later on, if the holder wants to sell the contract to
another party, what is the contract worth?
i.e., as the price of the underlying changes, how
does the value of the contract change?
Test your intuition: a concrete example
 Current stock price of Apple is $343.00. (as of a couple of hours ago)
 A call-option with strike $360 and 5.5-month maturity would pay
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the difference between the stock price on July 15, 2011 and the
strike (as long the stock price is higher than the strike.)
So if Apple is worth $400 then, this option would pay $40. If the
stock is below $360 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 two years:
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:
dS
   dt    dX
S
where
S = price of underlying
dt = infinitesimal time period
dS= change in S over period dt
dX = random variable with N(0,√dt)
σ = volatility of S
μ = average return of S (`drift’)
Using this assumption, Black and
Scholes showed that
 By setting up a portfolio consisting of the derivative V
and a counter position of a Δ-number of stocks S:
V - Δ*S
the portfolio can be made riskless, i.e. have a
constant return regardless of what happens to S.
(Δ turns out to be =dV/dS and it is constantly
changing ->strategy of dynamic hedging)
 This allows us to compare the portfolio to a riskless
asset and be priced accordingly.
 This eventually implies that V has to satisfy the
dynamic condition given by the PDE.
The Black-Scholes PDE
V 1 2 2  V
V
  S

rS

rV

0
2
t 2
S
S
2
V =value of derivative
S =price of the underlying
r =riskless interest rat
σ =volatility
t =time
 Different Derivative Contracts correspond to
different boundary conditions on the PDE.
 for the value of European Call-option, Black
and Scholes solved the PDE to get a closed
formula:
 rt
C  SN (d1 )  Ee N (d2 )
 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 APPLE-example
 S=343
E=360
r=1%
t=5.5 months
and σ=27%
(current stock price)
(strike price)
(current riskless interest rate)
(time to maturity)
(historic volatility of Apple)
 put into Maple: with(finance);
 blackscholes(343, 360, .01, 5.5/12, .27));
 And the output is . . . .
$18.60
Q: How sensitive is this price to the
input parameters?
 Now suppose Apple jumps a full 5 % tomorrow. From
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$343 to $360.
What happens to the value of the option?
Yes, it goes up. How much?
A: from $18.60 to $27.00!
That’s an almost 50% gain!
That’s called Leverage, and That’s the power of
options!
Discussion of the PDE-Method
 There are a few other types of derivative contracts, for which
closed formulas have been found. (Barrier-options, Lookbackoptions, Cash-or-Nothing Options). Those are accessible on the
web at sitmo.com
 Others need numerical PDE-methods.
 Or entirely different methods:
 Cox-Ross-Rubinstein Binomial Trees (1979)
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Monte Carlo Methods! (1977)
The Monte-Carlo-Method
 Assume μ=r.
 For a given contract, simulate random walks (based
on the geometric Brownian Motion), keeping track of
the pay-offs for the contract for each path.
 Calculate the average payoff, discount it to present
time, for an estimate of the present value of the
contract.
 Calculate the standard error, for an estimate of the
accuracy of the value-estimate.
The MonteCarlo-Method
Histogram
Measures from Randomwalk
3500
3000
 For our Apple-call-
option (with 10000
walks and 50
subdivisions), we
get a mean payoff
of $____
 with a standard
error of $____
2500
2000
1500
1000
500
0
= 21.30
100
200
payoff
300
400
500