1. Introduction Finance is sometimes called “the study of arbitrage”
Arbitrage is the existence of a riskless profit
Finance theory does not say that arbitrage will never appear
Arbitrage opportunities will be short-lived
The apparent mispricing may be so small that it is not worth the effort--E.g., pennies on the sidewalk
Arbitrage opportunities may be out of reach because of an impediment E.g., trade restrictions
Modern option pricing techniques are based on arbitrage principles
In a well-functioning marketplace, equivalent assets should sell for the same price (law of one price)
Put/call parity

2. Covered Call and Long Put
Riskless investments should earn the riskless rate of interest. If an investor can own a stock, write a call, and buy a put and make a profit, arbitrage is present
The covered call and long put position has the following characteristics:
One cash inflow from writing the call (C)
Two cash outflows from paying for the put (P) and paying interest on the bank loan (Sr)
The principal of the loan (S) comes in but is immediately spent to buy the stock
The interest on the bank loan is paid in the future

3. No Arbitrage Relationships
If there is no arbitrage, then:
If there is no arbitrage, then:
The call premium should exceed the put premium by about the riskless rate of interest
The difference will be greater as:
The stock price increases
Interest rates increase
The time to expiration increases

4. The Put/Call Parity Relationship
Equilibrium Stock Price Example
You have the following information:
Call price = $3.5
Put price = $1
Striking price = $75
Riskless interest rate = 5%
Time until option expiration = 32 days
If there are no arbitrage opportunities, what is the equilibrium stock price?

5. The Put/Call Parity Relationship
To understand why the law of one price must hold, consider the following information:
C = $4.75
P = $3
S0 = $50
K = $50
R = 6.00%
t = 6 months
Based on the provided information, the put value should be:
P = $ $50 + $50/(1.06)0.5 = $3.31
The actual call price ($4.75) is too high or the put price ($3) is too low

6. The Put/Call Parity Relationship
To exploit the arbitrage, arbitrageurs would:
Write 1 $4.75
Buy 1 $3
Buy a share of stock at $50
Borrow $48.56 at 6.00% for 6 months
These actions result in a profit of $0.31 at option expiration irrespective of the stock price at option expiration

7. The Put/Call Parity Relationship
PROFIT
Stock Price at Option Expiration $0
$50
$100
From call
4.75
(45.25)
From put
47.00
(3.00)
From loan
(1.44)
From stock
(50.00)
0.00
50.00
Total
$0.31

8. Option Pricing :The Binomial Model
Assume the following:
U.S. government securities yield 10% next year
Stock XYZ currently sells for $75 per share
There are no transaction costs or taxes
There are two possible stock prices in one year
$100
$75
$50
Today
One Year Later

9. Option Pricing :The Binomial Model
A call option on XYZ stock is available that gives its owner the right to purchase XYZ stock in one year for $75
If the stock price is $100, the option will be worth $25
If the stock price is $50, the option will be worth $0
What should be the price of this option?
Well, we can construct a portfolio of stock and options such that the portfolio has the same value regardless of the stock price after one year
Buy the stock and write N call options

10. Option Pricing :The Binomial Model
We can solve for N such that the portfolio value in one year must be $50:
If we buy one share of stock today and write two calls, we know the portfolio will be worth $50 in one year
The future value is known and riskless and must earn the riskless rate of interest (10%)
The portfolio must be worth $45.45 today
Assuming no arbitrage exists:
The option must sell for $14.77!
The option value is independent of the probabilities associated with the future stock price
The price of an option is independent of the expected return on the stock

11. Put Pricing in the Presence of Call Options
In an arbitrage-free world, the put option cannot also sell for $14.77; If it did, an astute arbitrageur would:
Buy a 75 call
Write a 75 put
Sell the stock short
Invest $68.18 in T-bills
These actions result in a cash flow of $6.82 today and a cash flow of $0 at option expiration

12. Put Pricing in the Presence of Call Options
Activity
Cash Flow Today
Portfolio Value at Option Expiration
Price = $100
Price = $50
Buy 75 call
-$14.77
$25
Write 75 put
+14.77
-$25
Sell stock short
+75.00
-100
-50
Invest $68.18 in T-bills
-68.18
75.00
Total
$6.82
$0.00

13. Exploiting Arbitrage Arbitrage Example
Binomial pricing results in a call price of $28.11 and a put price of $2.23. The interest rate is 10%, the stock price is $75, and the striking price of the call and the put is $60. The expiration date is in two years.
What actions could an arbitrageur take to make a riskless profit if the call is actually selling for $29.00?
Since the call is overvalued, and arbitrageur would want to write the call, buy the put, buy the stock, and borrow the present value of the striking price, resulting in the following cash flow today:
Write 1 call $29.00
Buy 1 put ($2.23)
Buy 1 share ($75.00)
Borrow $60e-(.10)(2) $49.12
$0.89
The value of the portfolio in two years will be worthless, regardless of the path the stock takes over the two-year period.

14. Binomial Put Pricing Priced analogously to calls
You can combine puts with stock so that the future value of the portfolio is known
Assume a value of $100
A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year

15. Option Pricing With Continuous Compounding
Continuous compounding is an assumption related to the fact that prices change in the marketplace continuously and option values change accordingly

16. Risk Neutrality and Implied Branch Probabilities
Risk neutrality is an assumption of the Black-Scholes model
For binomial pricing, this implies that the option premium contains an implied probability of the stock rising
Assume the following:
An investor is risk-neutral
He can invest funds risk free over one year at a continuously compounded rate of 10%
The stock either rises by 33.33% or falls 33.33% in one year
After one year, one dollar will be worth $1.00 x e.10 = $ for an effective annual return of 10.52%
A risk-neutral investor would be indifferent between investing in the riskless rate and investing in the stock if it also had an expected return of 10.52%
We can determine the branch probabilities that make the stock have a return of 10.52%

17. Risk Neutrality and Implied Branch Probabilities
Define the following:
U = 1 + percentage increase if the stock goes up
D = 1 – percentage decrease if the stock goes down
Pup = probability that the stock goes up
Pdown = probability that the stock goes down
ert = continuously compounded interest rate factor
The average stock return is the weighted average of the two possible price movements:

18. Risk Neutrality and Implied Branch Probabilities
If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25
If the stock goes down, the call will be worthless
The expected value of the call in one year is:
Discounted back to today, the value of the call today is:

19. Extension to Two Periods
Assume two periods, each one year long, with the stock either rising or falling by 33.33% in each period
What is the equilibrium value of a two-year European call?
$ (UU)
$100
$66.67
(UD = DU)
$75
$50
$33.33 (DD)
Today
One Year Later
Two Years Later

20. Extension to Two Periods
The option only winds up in the money when the stock advances twice (UU)
There is a 65.78% probability that the call is worth $58.33 and a 34.22% probability that the call is worthless
There is a 65.78% probability that the call is worth $34.72 in one year and a 34.22% probability that the call is worthless in one year
The expected value of the call in one year is:

21. In sum… Today One Year Later Two Years Later $58.33 (UU) $34.72
$0 (UD = DU)
$20.66 $0
$0 (DD)
Today
One Year Later
Two Years Later

22. Binomial Pricing with Lognormal Returns
If trees are recombining, this means that the up-down path and the down-up path both lead to the same point, but not necessarily the starting point
To return to the initial price, the size of the up jump must be the reciprocal of the size of the down jump
We assume that security prices follow a lognormal distribution
With lognormal returns, the size of the upward movement U equals:
The probability of an up movement is:

23. The case of r…
For a non-dividend or non-cash flow paying underlying security:
r =Rf (the risk-free rate)
For a dividend or cash flow paying underlying security:
r = b-Rf where b is the percentage return from the cash inflow portion—i.e., dividend yield or foreign interest rates

24. American Versus European Option Pricing
With an American option, the intrinsic value is a sure thing
With a European option, the intrinsic value is currently unattainable and may disappear before you can get at it
Thus, an American option should be worth more than a European option
At each node, the owner of the American has the option to exercise if the intrinsic value is greater than the current price of the option prices need to be adjusted accordingly
American calls on non-dividend paying stocks are worth more alive than dead. This is not true on dividend paying stocks.
American puts on not-dividend paying stocks are likely to be exercised. If the underlying asset pays dividend, this is not true