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