pricing derivatives feb 2011

8/10/2019 Pricing Derivatives Feb 2011

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The Pricing of Stock Optionsand other Financial Derivatives

Klaus Volpert, PhD

Villanova University

Feb 3, 2011


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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)


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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 $2billion

Sep 2006: the Hedge Fund Amaranthcloses after losing $6billion in energy derivatives.

January 2007: Reading(PA) School District has to pay $230,000to 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 tomortgages and loans (CDOs) due to rising foreclosures

13 September 2008: Lehman Brothers fails, setting off a massivefinancial crisis

Oct 2008: AIGgets a massive government bail-out ($180 billion)

Famous Calamities
http://localhost/var/www/apps/conversion/tmp/Research/Reading.dochttp://localhost/var/www/apps/conversion/tmp/Research/Reading.doc


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On the Other Hand

August 2010: BHP, the worlds largest mining company,proposes to buy-out PotashInc, 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 JohnPaulson, have made billions a year. . .

John Paulsons take-home pay in 2010 of $5 Billion exceededthe take home pay of all PhD math professors in the countrycombined.

Gold rushes in Stock options (beginning in the early 1990s)

Mortgage-backed-securities (early 2000s)

Collateralized Debt Obligations (CDOs)

Credit Default Swaps (CDSs)


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So, what is a Financial Derivative?

Typically it is a contractbetween two parties A and B,stipulating that, - depending on the performance ofan under ly ing assetover a predetermined time - ,

a certain amount of money will change hands.


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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 priceper barrel is above $100 on Sep 1st2011, then B will

pay A the differencebetween that price and $100. To enter into this contract, A pays B apremium

A is called theholder of the contract, B is thewriter.

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 Questions:

What premium should the buyer (`holder`) pay to theseller (`writer), so that the writer enters into thatcontract??

Later on, if the holder wants to sell the contract toanother party, what is the contract worth?

i.e., as the price of the underlying changes, howdoes the value of the contract change?


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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 $360and 5.5-month maturity would paythe difference between the stock priceon July 15, 2011 and thestrike(as long the stock price is higherthan the strike.)

So if Apple is worth$400 then, this option would pay $40. If thestock is below$360 at maturity, the contract expires worthless. .. . . .

So, what would youpayto hold this contract?

What would you wantfor 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 stock prices of Apple over

the last two years:
http://finance.yahoo.com/chartshttp://finance.yahoo.com/charts


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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 thefoundation 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:

dSdt dX

S

whereS = price of underlyingdt = infinitesimal time perioddS= change in S over period dtdX = random variable with N(0,dt)= volatility of S= average return of S (`drift)


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Using this assumption, Black and

Scholes showed that

By setting up a portfolio consisting of the derivative Vand a counter position of a -number of stocks S:

V - *Sthe portfolio can be made riskless, i.e. have aconstant return regardless of what happens to S.(turns out to be =dV/dS and it is constantlychanging ->strategy of dynamic hedging)

This allows us to compare the portfolio to a risklessasset and be priced accordingly.

This eventually implies that V has to satisfy thedynamic condition given by the PDE.


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The Black-Scholes PDE

Vt+1=22S22VS2+rSVSrV=0





V =value of derivative

S =price of the underlying

r =riskless interest rat

=volatilityt =time

22 2

2

10

2

V V VS rS rV

t S S


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Different Derivative Contracts correspond todifferent boundary conditionson the PDE.

for the value of European Call -op t ion, Blackand Scholes solved the PDE to get a closedformula:


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Where N is the cumulative distributionfunction for a standard normal randomvariable, and d1 and d2 are parametersdepending on S, E, r, t,

This formula is easily programmed into Mapleor other programs

1 2( ) ( )rt

C SN d Ee N d


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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 =27% (historic volatility of Apple)

put into Maple: with(finance);

blackscholes(343, 360, .01, 5.5/12, .27));And the output is . . . .


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$18.60


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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!

Thats an almost 50% gain!

Thats called Leverage, and Thats the power ofoptions!


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Discussion of the PDE-Method

There are a few other types of derivative contracts, for whichclosed formulas have been found. (Barrier-options, Lookback-options, Cash-or-Nothing Options). Those are accessible on theweb at sitmo.com

Others need numerical PDE-methods.

Or entirely different methods:

Cox-Ross-Rubinstein Binomial Trees (1979)

Monte Carlo Methods! (1977)
http://sitmo.com/http://sitmo.com/


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The Monte-Carlo-Method

Assume=r.

For a given contract, simulate random walks (basedon the geometric Brownian Motion), keeping track ofthe pay-offs for the contract for each path.

Calculate the average payoff, discount it to presenttime, for an estimate of the present value of thecontract.

Calculate the standard error, for an estimate of theaccuracy of the value-estimate.


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The Monte-

Carlo-Method

For our Apple-call-option (with 10000walks and 50subdivisions), we

get a mean payoffof $____

with a standarderror of $____

500

1000

1500

2000

2500

3000

3500

payoff

0 100 200 300 400 500

= 21.30

Measures from Randomwalk Histogram

pricing derivatives feb 2011

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