Download - Counting Money
COUNTING MONEY
AMBER HABIB
MATHEMATICAL SCIENCES FOUNDATION
NEW DELHI
Lecture at Alpha, Mathematics Society festival, Hindu College, Delhi – Nov 4, 2009
1
Where does Maths come from?
Pure Imagination
Practical Problems
Accounting → Arithmetic
Measuring area to estimate tax revenue → Geometry
Maps → Coordinate & Spherical Geometry
Interest & Loans → Roots of Polynomials
Gambling → Probability
Mechanics → Calculus
Heat → PDE, Harmonic Analysis, Cardinality,…
2
How do we estimate “Value”?3
What is Value of an item in terms of money?
One answer: What we will get if we sell it.
Problem: How do you estimate value without selling the item?
This will obviously involve uncertainty and probability. In fact, a very large chunk of modern mathematics is now applied to this problem.
Mathematics of Finance4
Probability & Statistics
(Partial) Differential Equations
Stochastic Differential Equations
Stochastic Calculus
Measure Theory
Functional Analysis
Optimization
Numerical Analysis
Is This Profit?5
You invest $100 today and get back $120 after
a week.
Is this a profit?
Are you sure?
Is This Profit?6
Well, what if you bought the $’s using Rupees,
and the exchange rate changed?
$100 → $120
Rs 50/$ → Rs 40/$
Rs 5000 → Rs 4800
What is Profit?7
The amount and direction of profit depends on
how we measure it.
The fact of profit is only independent of the
unit of measure when we invest zero (or less)
and get back something positive.
Certain Profit: An Example8
Bank A loans money at an annual interest rate
of 10%, while Bank B pays 15% interest
annually on deposits.
A strategy to exploit this situation:
Borrow 100 from A and deposit in B for a year.
After a year, withdraw 115 from B, use 110 to
pay off A, and pocket a profit of 5 on a zero
investment.
Can such situations exist?
Arbitrage9
Arbitrage is the technical name for certain
profit. Its general definition is:
An investment strategy is said to lead to
arbitrage if:
The initial investment is non-positive.
The final return is certainly non-negative and has
a non-zero probability of being positive. (Note its
precise value doesn’t have to be known.)
No Arbitrage Principle10
In an “efficient market” (in which communication
is instantaneous and complete), arbitrage
opportunities will not exist.
(This is an idealized situation – in real life they
should just die out quickly)
Thus, a “correct” value is one which prevents
the possibility of arbitrage.
Continuously Compounded Interest11
Recall that if interest is compounded, the
growth over n periods is given by
For convenience, we replace this by
continuous compounding:
nr)P(1A
nrPeA
12
No Arbitrage Principle ⇒
Everyone uses same r.
Suppose a portfolio has current value P and it
is certain that its value after time T will be A.
Then the growth must be at the risk free rate:
A = PerT
Risk-free Rate of Interest
Futures13
A futures contract (or just futures) is an
agreement between two parties for a future
trade.
Terminology:
Underlying Asset: The asset which will be
traded.
Spot Price: Current price of underlying asset.
Writer: Who issues the contract.
Holder: Who acquires the contract.
Terms of a Futures14
At time t=0, the holder acquires the futures
from the writer.
The futures describes the amount of the
underlying asset to be traded, the time T of
delivery (expiration date) and the price X to
be paid (exercise price).
No money exchanged at t=0.
At t=T, holder pays X to writer and acquires the
underlying asset.
Why Futures?15
A packaged food company and a farmer will trade in a certain amount of potatoes 3 months from now, after the harvest.
If the crop is poor, prices will rise, and the company will face a loss.
If there is a bumper crop, prices will fall, and it will be the farmer who will face a loss.
Both parties can mutually eliminate their risk by agreeing now on what price they will trade in 3 months time.
Trade in Futures16
Suppose, as the expiration date T approaches,
the price of the underlying asset rises above X.
Then the holder starts receiving offers to sell
the futures to a new holder.
What should be the price of the futures? What
factors may be relevant?
In the same vein, when the contract is being
written, what should be X?
Futures on Reliance Shares17
Exercise Price18
If X > SerT the writer can make an arbitrage profit:
She initially borrows S and uses it to buy the asset.
At time T she delivers the asset to the holder, earns X and uses SerT of that to pay off the loan.
She pockets a riskless profit of X − SerT.
If X< SerT the holder can earn arbitrage in a similar fashion.
So No Arbitrage Principle ⇒ X= SerT
Futures Price19
Consider a futures written at time t=0 with
exercise price X and expiration time T.
Its value V at a later time t depends on the
spot price St at time t:
Remark: is the present value of X.
t)r(T
t XeSV
t)r(TXe
Generalizations20
This simple formula is valid when interest rates are fixed and owning the asset implies no extra income or cost.
No Arbitrage arguments easily give formulas for exercise & futures price when:
Asset generates known income/cost (interest, rent, storage costs).
Asset has known dividend yield – income/cost is proportional to asset value (certain shares, stock indices, gold loans).
Options21
Futures eliminate uncertainty but not the
possibility of a felt loss – depending on the
final price of the asset either holder or writer
may get a very poor deal.
Options are contracts which allow one party to
withdraw. The one who has this right pays an
initial fee to acquire it.
European Call Option22
Like a futures, a European call option is a
contract for a future trade with expiration date
T and exercise price X. However,
The holder pays an initial call premium C to the
writer.
At time T the holder may pay X to the writer.
If the holder makes the payment, the writer must
deliver the asset.
European Call Option23
Main Q: How to determine C?
Depends on at least T, r, X and S.
In this case, No Arbitrage Principle by itself
gives some loose bounds for C but not an
exact price.
It becomes necessary to model how the asset
price may fluctuate.
Binomial Model24
S
SU
SD
C
CU = (SU-X)+
CD = (SD-X)+
0,0
0,
x
xxx
t=0 T=T
Suppose the price starts at S and
over time T can go up by factor U
or down by factor D.
Then the option also has two
possible final values.
Binomial Model25
Consider a portfolio with 1 unit of asset and h
written calls.
Final value of the portfolio:
Up move: SU-hCU
Down move: SD-hCD
We can choose h & make the portfolio risk
free: SU-hCU = SD-hCD or,
DU CC
D)S(Uh
Binomial Model26
With this value of h, the portfolio must grow at
the risk free rate:
SU-hCU = erT(S-hC)
Substitute h value and solve for C:
C = e-rT (qCU+(1-q)CD), where
DU
Deq
rT
Binomial Options Pricing Model27
We make the model realistic by letting the
asset price evolve over many steps:
S
SU
SD
SU2
SUD
SD2
SUD2
SD3
SU3
SU2D
Binomial Options Pricing Model28
The tree for the call prices:
C
CU
CD
CUU
CUD
CDD
CUDD
CDDD
CUUU=(SU3-X)+
CUUD =(SU2D-X)+
BOPM29
Working back from the end of the tree to its root, over n steps of length T/n each, we get:
where
The proof is by mathematical induction.
X)D(SUq)(1-qCeC knkk-nk
k
n
0k
nrT
DU
Deq
rT/n
Features of BOPM30
What is important is the dispersion of asset
prices (measured by U,D) not their actual
probabilities.
Yet the form is of an expectation of a future
value, if we think of q as a probability.
The model therefore treats the final asset
values as having a binomial probability
distribution and then takes the present value of
the expectation of the call prices.
Risk Neutral Probability31
What is special about q? If we treat it as the
probability of an up move, then the probability of a
final asset price of SUkDn-k is nCkqk(1-q)n-k.
So the expectation of the final price is
Under q, the expected value grows at the risk free
rate. We call such a probability risk neutral.
rT
nknkknk
k
n
0k
n
Se
q)D)(1S(qUDSUq)(1qC
BOPM in Action32
Predicted call premiums by a 10-step BOPM for calls on
Maruti shares (line), compared with actual premia (stars)
over a 1-month period. (Data from NSE)
Other Derivatives33
The BOPM approach can also be applied to
European Put Options (Writer buys asset from
holder)
American Options (Holder can exercise
contract before T)
Barrier Options (Contract expires if asset price
crosses set barriers)
Asian Options (Final payoff depends on
average of asset price over [0,T])
Black-Scholes Model34
By letting n→∞ we transform BOPM into a continuous model.
The binomial distribution becomes normal.
The BOPM formula becomes
where is the cdf of the standard normal distribution and w is a known function of r, T, X,
S and .
)()( TσwXewSC rT
Some History35
Louis Bachelier (1900, Paris) models price
fluctuations using normal distributions; applies
to pricing options on bonds; develops
Brownian motion and connects problem to
heat equation.
His work inspires development of Markov
processes by Kolmogorov and stochastic
calculus by Ito. (1930s)
Some History36
Fischer Black, Myron Scholes & Robert Merton (1973) correct Bachelier’s work by replacing real life probability with risk neutral probability. They use Ito calculus.
William Sharpe (1978) introduces BOPM as a tool to simplify exposition of ideas of Black et al.
John Cox, Stephen Ross and Mark Rubinstein (1979) extend BOPM and derive Black-Scholes from it.
What Next?37
Create models which are not restricted by Black-Scholes’ assumptions:
Asset prices modeled by Normal distribution (Symmetric, dies out quickly – so extreme events very rare). Use general heavy tailed stable distributions instead.
Constant volatility () – Models like GARCH allow for time varying volatility.
Constant risk free rate (r) – Develop probabilistic models for interest rates and incorporate them.
Who Can Do It?38
Best equipped people for modeling the
modern world of Finance are Maths and
Physics PhDs who can work with stochastic
calculus and numerical analysis.
These “quants” are the most highly paid
people on Wall Street.
Two Case Studies39
Rabindranath Chatterjee
MSc Physics – IIT Kanpur (1988)
PhD Physics – Rutgers University, New Jersey,
USA in particle physics. (1995)
First Job – Morgan Stanley, New York.
Current – Senior Vice President, Citibank, New
York
Two Case Studies40
Samarendra Sinha
MSc Maths - IIT Kanpur (1989)
PhD Maths – University of Minnesota (1995) –
algebraic geometry
Post-Doc at IAS, Princeton (1995-96)
Asst Prof, Ohio State University (1996-97)
MA Finance – Wharton (1999)
Current – “Quant Analyst” at JP Morgan, NY –
numerical PDEs
Nobel Prizes41
Nobel prizes for work in mathematical finance:
James Tobin – 1981
Franco Modigliani – 1985
Merton Miller, Harry Markowitz, William
Sharpe – 1990
Robert Merton, Myron Scholes – 1997
Robert Engle – 2003