Chapter 1: Expectation And Arbitrage

Question 1

Long term winnings via law of large numbers

Answer 1

Theorem 1.1.1

Let (X_i) for i in N be a sequence of random variables that are independent and identically distributed. Suppose that E[X ₁] = µ and var(X ₁) <

S_n = (X ₁+ X₂ + … + X_n ) /n

Then, with probability one, S_n tends to µ as n tends to ∞

Question 2

Expected return of playing a coin toss

Answer 2

If I guess correctly I win $1

E[X]= ½$1 + ½ $0 = $0.50

Single play is the amount to buy a random quantity

If we played the game a large number n of times and on play i our winnings are X_i

Then our average winnings would be S_n ≈ E[X ₁] = ½

So we might regard 50 cents as a fair price for a single play. If we paid less long run we’d make money and more long run lose money.

EXPECTED VALUE IS NOT CORRECT

Question 3

Market

Answer 3

A market is any setting where it is possible to but and sell one or more quantities.

Question 4

Commodity

Answer 4

An object that can be bought or sold

The value of the commodity usually changes with time t

Money is a commodity.

Question 5

Cash

Answer 5

Cash or cash bond when referring to money as a commodity

Traded and sold

Question 6

Chapter one simple market

Answer 6

In chapter one define simple market with only two commodities, as time passes money earns interest or money owed will have interest required to be paid.

One step of time in our simple market ->

t=0 and t=1

Question 7

One-period market

Answer 7

One-period market

When only one step of time in our market : t=0 and t=1

Question 8

Interest rate

Answer 8

Let r>0 be a deterministic constant known as the interest rate.

If we put an amount x of cash into the bank at time t=0 and leave it there until t=1 the bank will then contain

x(1+r) in cash

The same formula applies if x is a LOAN ( x is negative) borrowing C_0 from the bank and the bank requires interest to be paid

Question 9

Stock

Answer 9

A collection of shares in a company is known as stock

Amount of stock can be any real number - can own fractional amounts and negative amount eg borrowing stock without having to pay interest

Question 10

Share

Answer 10

A share us a proportion of the rights of ownership of a company - the value of a share varies over time

Question 11

The value or worth of a stock

Answer 11

Value or worth of any commodity is the amount of cash requires to buy a single unit of stock. This changes randomly.

Let u>d>0 and s>0 be deterministic constants. At time t=0, it costs S_0 = s cash to buy one unit of stock. At time t=1, one unit of stock becomes worth

S_1 =.
{ su with probability p_u
{ sd with probability p_d

Of cash

( p_u and p_d bigger than 0 and p_u + p_d =1)

And one unit of cash became 1+r worth

Question 12

Changes in value of cash and stocks as tree

Answer 12

Stock

           (Su)
           / p_u
(S) <
         \ p_d
         (Sd)

Cash (1) ____1_____(1+r)

Commodities exchanged based on current price in one-period market

Question 13

x unit of cash at t=0 becomes worth

Answer 13

x unit of cash at t=0 becomes worth x(1+r) at t=1

Question 14

y unit of stock at t=0 becomes worth

Answer 14

y unit of stock at t=0

Is worth
yS_0 =S_y
becomes worth

yS_1 =
{ySu with prob p_u
{ySd with prob p_d

At t=1

Question 15

Liquid market

Answer 15

Possible to buy/sell/trade unlimited amounts of commodities

Question 16

Example: given r bigger than 0, S bigger than 0

u=3/2
d=1/3

p_u= p_d = 0.5

at t=0 portfolio is 5 cash and 8 stock

At t=1?

Answer 16

At t=1 expected value is 5(1+r) cash and stock worth

{8Su with prob 0.5
{8Sd with prob 0.5

So expected value at time t=1 is

E[V_1] = 5(1+r) + ( 3/2)(8S)(0.5) + (1/3)(8S)(0.5) = 5 + 5r + (22/3)S

( we used 8Sup_u + 8Sdp_d)

Question 17

Arbitrage

Answer 17

We assume no arbitrage can occur in market

When it’s possible to make free money without risk

( ie we might expect to make money on average without risk)

Question 18

In a one period market, offer single unit stock at s/2

Answer 18

An arbitrage strategy is possible:

We could take out a loan of s/2 , BUY STOCK , SELL @ market rate for a cash and repay loan all at time t=0, pocketing s/2

Question 19

Contract

Answer 19

Agreement between two parties

Question 20

Forward contract

Answer 20

Future contract, nothing happens at time t=0 but at some time in the future something happens for agreed price ( strike price)

Question 21

Strike price

Answer 21

A possible value ( due to no arbitrage in market) is

K= S(1+r)

Where S is the value of stock at t=0

Proof:

Suppose k is less than S(1+r)
Then at t=0 enter forward contract as SELLER. BORROW S from bank and BUY single unit of stock. At t=1 sell as agreed and pay loan S(1+r), pocket K-S(1+r) bigger than 0 (profit)

Suppose K is bigger than S(1+r)
Then at t=0 enter forward contract as BUYER. BORROW single unit of stock from stockbroker and SELL for cash S. wait for t=1 then BUY the stock paying K out of cash S(1+r). Use stock to pay back stockbroker and profit S(1+r) -K bigger than 0 cash.

Question 22

Expectation as a price for forward contract?

Answer 22

If we were to co sided pricing by expectation the value of forward contract at t=1 from the point of view of the buyer is S_1 -K.

By pricing by expectation this implies it costs nothing so E(S_1 -K) =0 implies E(S_1) - K =0 implies
K = E(S_1) = Sup_u + Sdp_d which is not equal to K= S(1+r)

Hence expectation/average winnings don’t matter and market adjusts to equilibrium in which no arbitrage is possible

(This would be sensible if markets weren’t always changing)

• Also as time passes we gain information
• and this stochastic process cannot be modelled by only 2 branches ( insufficient)

Question 23

Expectation vs arbitrage

Answer 23

We tried to find K using pricing by expectation

The value of the contract to the buyer at time 1 is V_1 = S_1 - K

The contract costs nothing at time 0 and so by pricing by expectation 0 = E(V_1) = E(S_1) -K gives K = Sup_u + Sdp_d

Which is NOT the same as K=S(1+r)

Therefore pricing by expectation is not used in market without arbitrage

Pricing by expectation would mean that the law of large numbers would give some value gained or lost overall which wouldn’t work in a real market as the law of large numbers is applied to something that happens again and again (not real)