The Math and Magic of Financial Derivatives

Klaus Volpert, PhDVillanova UniversityJanuary 12, 2015

Derivatives are controversial.

Champions of Derivatives include Alan Greenspan:(chairman of the Federal Reserve 1987-2006)

“Although the benefits and costs of derivatives remain the subject of spirited debate, the performance of the economy and the financial system in recent years suggests that those benefits have materially exceeded the costs.“ in a speech to Congresson May 8, 2003

I can think of no other area that has the potential of creating greater havoc on a global basis if something goes wrong

Dr. Henry Kaufman, economist, May 1992

Derivatives are the dynamite for financial crises and the fuse-wire for international transmission at the same time.

Alfred Steinherr, author of

Derivatives: The Wild Beast of Finance (1998)

Critics

Derivatives are financial weapons of mass destruction, carrying dangers that, while now latent, are potentially lethal."

Warren Buffettin his Annual Letter to Shareholders ofBerkshire Hathaway, March 8, 2003.

1994: Orange County, CA: bankrupt after losses of $1.5 billion

1995: Barings Bank: bankrupt after 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.

Famous Calamities

2002-2012: Philadelphia (PA) School District loses $331 million due to bad interest rate swaps(according to a Jan 2012 report by PA Budget and Policy Center)

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

15 September 2008: Lehman Brothers fails, setting off a massive financial crisis

Oct 2008: AIG needs a massive government bail-out ($180 billion) due to its losses in Credit Default Swaps (CDS’s)

On the Other Hand

August 2010: BHP, the worlds largest mining company, proposed to buy-out Potash Inc, a Canadian mining company, for $38 billion. The CEO of Potash, Bill Doyle, stood to make $350 million due to his stock options.

Hedge fund managers, such as James Simon and John Paulson, have made billions a year, usually using derivatives to leverage their bets. . .

So. Financial Derivatives. . .

have caused disastrous losses to some companies, individuals, and municipalities, while also

creating spectacular incomes for some.

However, the use of derivatives is usually very beneficial to the economy, as i would like to explain now.

So, what is a Financial Derivative?

Typically it is a contract between two parties A and B, who agree on a future cash flow that is contingent on future developments in the price of an underlying asset or an index.

examples: stock options futures on currencies, commodities etc interest rate swaps credit default swaps

An Example: A Call-option on Oil

Suppose the oil price today is $50 a barrel. Suppose that A stipulates with B, that if the oil

price per barrel is above $70 on Sep 1st 2015, then B will pay A the difference between that price and $70.

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?

An Example: A Call-option on Oil

Why might A enter into this contract? A would be a user of oil, such as an airline, needing to

reduce their risk of rising oil prices Why might B enter into this contract?

B might be a producer of oil, who is willing to forgo potential profits when prcies rise above th strike

level in order to have the premium, when prices to stay low

Of course, speculators, who want to make a bet on the oil market are allowed be either writer or holder. even speculators are usually beneficial to the market, as

they help keep the market efficient and are helpful in the `price finding’ mechanism.

sometimes a problem, as they can make the market more volatile (herd mentality, creating manias and panics etc)

Reasons to trade derivatives:

Hedge (reduce) risks Give up potential profits in exchange for the

premium and higher bottom line (`yield enhancement’)

Investment Speculation

Besides Oil, Derivatives can be written on underlying assets such as Coffee, Wheat, Gold and other `commodities’ Stocks Currency exchange rates Interest Rates Credit risks (subprime mortgages. . . ) Even the Weather!

Fundamental Questions:

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 then?

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

Test your intuition: a concrete example

Today, Jan-12-2015, Apple’s share price is hovering at $110.

A call-option with strike $130 and 6-month maturity would pay the difference between the stock price on July 17, 2015 and the strike (as long the stock price is higher than the strike.)

So if Apple is worth $200 then, this option would pay $70. If the stock is below $1300 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:

Want more information ? Here is a chart of stock prices of Apple over

the last two years:

Please write down your estimate for a price of a 6-month call-option on Apple with strike $130

Prices of options were determined by supply and demand, through a mechanism similar to an auction

In 1973, however, Fischer Black and Myron Scholes came up with a model to price options mathematically. It was very successful, won the Nobel prize in economics, and became the foundation of the options market.

Historically

They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation:

85

90

95

100

105

110

115

20 40 60 80 100 120

day

stockprice = caseIndex

exp

Randomw alk Line Scatter Plot

dSrdt dX

S where

(0, )dX N dtvolatility

r riskless interest rate

Aside: How do you measure σ?

They started with the assumption that stocks follow a random walk on top of an intrinsic appreciation:

85

90

95

100

105

110

115

20 40 60 80 100 120

day

stockprice = caseIndex

exp

Randomw alk Line Scatter Plot

dSrdt dX

S where

(0, )dX N dtvolatility

r riskless interest rate

This implies that the probability distribution for is lognormal:

2 2

02

1(ln( ) ( ) )

2

21( )

2

TSr T

S

TT

T

pdf S eS T

TS

Fair price=expected payoff of the option, discounted to present time

( )( )rTT T T

K

e pdf S S K dS

0 1 2( ) ( )rTS N d Ke N d

Where N is the cumulative distribution function for a standard normal random variable, and d1 and d2 are parameters depending on , K, r, T, σ

This formula is easily programmed into Maple or other programs

So for our example (Apple, $110 now, $130 strike, 6-month maturity) we get the price. . . . (drumroll).. . . .

$3.05

The `Nuclear Power` Effect: Leverage

So if Apple is at $110, the strike at $130, the time to maturity 6 months, volatility=.30, the price for the call-option is $3.05.

Suppose, the stock price went up 5% today,to $115.50, what would happen to the price of the option?

Answer: the option price would go to $4.64! That’s up almost 50%, 10-fold the increase of the stock price! That’s the power of options: a small percentage change in the

underlying, creates a large percentage change in the value of the derivative!

Derivatives amplify movements of the underlying

This, in part, explains both its usefulness and its destructiveness!

Four different Methods of calculating this price:

$3.05

1. Expected Payoff Method

( )( )rTT T T

K

e pdf S S K dS

0 1 2( ) ( )rTS N d Ke N d

Where N is the cumulative distribution function for a standard normal random variable, and d1 and d2 are parameters depending on , K, r, T, σ

This formula is easily programmed into Maple or other programs

Actually, Black and Scholes derived a much more general result that holds for any type of derivative contract with Value V

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 call-options, they solved the PDE and obtained the previous formula.

2. The Approach via Partial Differential Equations:

Discussion of the PDE-Method

There are many other types of derivative contracts, for which closed formulas have been found. (Barrier-options, Lookback-options, Cash-or-Nothing Options)

Others need numerical PDE-methods. Or entirely different methods:

Cox-Ross-Rubinstein Binomial Trees Monte Carlo Methods

3. Monte-Carlo-Methods On the computer, one simulates 1000’s of random

walks for the same asset. One keeps track of the pay-out for each walk, and then simply averages those pay-outs, and calls that average the fair price of the option.

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90

95

100

105

110

115

20 40 60 80 100 120

day

stockprice = caseIndex

exp

Randomw alk Line Scatter Plot

Monte-Carlo-Methods (1980’s)

For a call-option (with 1,000,000 walks), we may get a mean payoff of $21.30 with a 95% confidence interval of ± $.05

There are methods to increase accuracy, and to speed up the simulation

Very general method, but expensive.

500

1000

1500

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2500

3000

3500

payoff

0 100 200 300 400 500

= 21.30

Measures from Randomwalk Histogram

4. Binomial Trees (Cox-Ross-Rubinstein,1979):

This approach uses the discrete method of binomial trees to price derivatives

S=110

S=111

S=112

S=109

S=110

S=108

This method is mathematically much easier. It is extremely adaptable to different pay-off schemes. And it is the best method for American-type (early exercise) options.However there some derivatives (such as the lookback) where accuracy is poor.

While each method has its pro’s and con’s,it is clear that there are powerful methods to value (`price’) derivatives, simulate outcomes and estimate risks.

Such knowledge is money in the bank. Quite literally.

1997: Merton and Scholes win Nobel prize in Economics Cheers in The Economist: The professors have

turned risk management from a guessing game into a science

Jeers in Barron’s: The pair snared the rich honor, and the tidy sum that goes with it, for devising a formula to measure the worth of a stock option, thus paving the way for both the spectacular growth of options and their use as instruments of mass destruction.