Transcript Engines of the Economy or Tools of Mass Destruction?

The Math and Magic of
Financial Derivatives
Klaus Volpert
Villanova University
March 31, 2008
Financial Derivatives have
been called. . .
 . . .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)
Famous Calamities
 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
 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:
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 percentage return of S
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 derivatives correspond to different
boundary conditions on the PDE.
 for the value of European Call and Putoptions, 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 MSFT-example
 S=19.40 (the current stock-price)
E=20
(the `strike-price’)
r=3.5%
t=12 months
and. . . 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 . . .
4-month volatilities
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
7/2/2006
1/18/2007
8/6/2007
Let’s use σ= 40%
2/22/2008
9/9/2008
3/28/2009
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=102
S=101
S=100
S=100
S=99
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.
Monte-Carlo-Methods
 For our MSFT-call-option (with 3000 walks),
we get $3.10
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.