Counting Money

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COUNTING MONEY

AMBER HABIB MATHEMATICAL SCIENCES FOUNDATION NEW DELHI

Lecture at Alpha, Mathematics Society festival, Hindu College, Delhi – Nov 4, 2009

Where does Maths come from? 2

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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,… 

How do we estimate “Value”? 3

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What is Value of an item in terms of money?

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One answer: What we will get if we sell it.

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Problem: How do you estimate value without selling the item?

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This will obviously involve uncertainty and probability. In fact, a very large chunk of modern mathematics is now applied to this problem.

Mathematics of Finance 4

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Probability & Statistics (Partial) Differential Equations Stochastic Differential Equations Stochastic Calculus Measure Theory Functional Analysis Optimization Numerical Analysis

Is This Profit? 5

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You invest $100 today and get back $120 after a week. Is this a profit?

Are you sure?

Is This Profit? 6



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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

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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 Example 8

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? 

Arbitrage 9

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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 Principle 10

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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)

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Thus, a “correct” value is one which prevents the possibility of arbitrage.

Continuously Compounded Interest 11

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Recall that if interest is compounded, the growth over n periods is given by

A  P(1 r)n 

For convenience, we replace this by continuous compounding:

nr A  Pe

Risk-free Rate of Interest 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

Futures 13

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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 Futures 14

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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

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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 Futures 16

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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.

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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 Shares 17

Exercise Price 18



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. 

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If X< SerT the holder can earn arbitrage in a similar fashion. So No Arbitrage Principle ⇒ X= SerT

Futures Price 19

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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:

V  St  Xe 

Remark: Xe

r(T t)

r(T  t)

is the present value of X.

Generalizations 20

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

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).

Options 21

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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 Option 22

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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 Option 23

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Main Q: How to determine C?

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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.

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Binomial Model 24

SU Suppose the price starts at S and over time T can go up by factor U or down by factor D.

S SD

Then the option also has two possible final values.

CU = (SU-X)+

 x, x  0 x   0, x  0 

C t=0

CD = (SD-X)+ T=T

Binomial Model 25

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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,

S(U  D) h CU  CD

Binomial Model 26

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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

erT  D q UD

Binomial Options Pricing Model 27

We make the model realistic by letting the asset price evolve over many steps:

SU3 SU2 SU S

SUD SD

SD2

SU2D SUD2 SD3

Binomial Options Pricing Model 28

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The tree for the call prices:

CUUU=(SU3-X)+ CUU CU C

CD

CUD CDD

CUUD =(SU2D-X)+ CUDD

CDDD

BOPM 29

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Working back from the end of the tree to its root, over n steps of length T/n each, we get:

Ce

rT

n

 k 0

n

k

n-k

k

nk

Ck q (1- q) (SU D

 X)

e D q UD rT/n

where 

The proof is by mathematical induction.



Features of BOPM 30

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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 Probability 31

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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

n

 k 0

n

Ck qk (1 q)nk SUkDnk  S(qU  (1 q)D)n  SerT



Under q, the expected value grows at the risk free rate. We call such a probability risk neutral.

BOPM in Action 32

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 Derivatives 33

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 Model 34

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By letting n→∞ we transform BOPM into a continuous model. The binomial distribution becomes normal. The BOPM formula becomes rT

C  S(w)  e X(w  σ T ) where  is the cdf of the standard normal distribution and w is a known function of r, T, X, S and .

Some History 35

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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 History 36

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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 BlackScholes from it.

What Next? 37

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Create models which are not restricted by BlackScholes’ 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

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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 Studies 39

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Rabindranath Chatterjee 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  MSc

Two Case Studies 40

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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 Prizes 41

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