The Pricing of Stock Options and other Financial Derivatives

Klaus Volpert, PhDVillanova UniversityFeb 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)

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

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)

Famous Calamities

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 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

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 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:

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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 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*Sthe 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 =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 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:

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 APPLE-example S=343 (current stock price)

E=360 (strike price)r=1% (current riskless interest rate)t=5.5 months (time to maturity)and σ=27% (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 $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, Lookback-options, 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)

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.

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stockprice = caseIndex

exp

Randomw alk Line Scatter Plot

The Monte-Carlo-Method

For our Apple-call-option (with 10000 walks and 50 subdivisions), we get a mean payoff of $____

with a standard error of $____

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payoff

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= 21.30

Measures from Randomwalk Histogram