Chapter 1: Expectation And Arbitrage Flashcards
Long term winnings via law of large numbers
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 ∞
Expected return of playing a coin toss
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
Market
A market is any setting where it is possible to but and sell one or more quantities.
Commodity
An object that can be bought or sold
The value of the commodity usually changes with time t
Money is a commodity.
Cash
Cash or cash bond when referring to money as a commodity
Traded and sold
Chapter one simple market
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
One-period market
One-period market
When only one step of time in our market : t=0 and t=1
Interest rate
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
Stock
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
Share
A share us a proportion of the rights of ownership of a company - the value of a share varies over time
The value or worth of a stock
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
Changes in value of cash and stocks as tree
Stock
(Su)
/ p_u
(S) <
\ p_d
(Sd)
Cash (1) ____1_____(1+r)
Commodities exchanged based on current price in one-period market
x unit of cash at t=0 becomes worth
x unit of cash at t=0 becomes worth x(1+r) at t=1
y unit of stock at t=0 becomes worth
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
Liquid market
Possible to buy/sell/trade unlimited amounts of commodities
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?
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)
Arbitrage
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)
In a one period market, offer single unit stock at s/2
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
Contract
Agreement between two parties
Forward contract
Future contract, nothing happens at time t=0 but at some time in the future something happens for agreed price ( strike price)
Strike price
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.
Expectation as a price for forward contract?
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)
Expectation vs arbitrage
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)
