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

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

No Arbitrage Relationships• If there is no arbitrage, then:

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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

The Put/Call Parity RelationshipEquilibrium 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?

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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 = $4.75 - $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

The Put/Call Parity Relationship• To exploit the arbitrage, arbitrageurs would:

– Write 1 call @ $4.75

– Buy 1 put @ $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

The Put/Call Parity RelationshipPROFIT Stock Price at Option Expiration

$0 $50 $100

From call 4.75 4.75 (45.25)

From put 47.00 (3.00) (3.00)

From loan (1.44) (1.44) (1.44)

From stock (50.00) 0.00 50.00

Total $0.31 $0.31 $0.31

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

$75

$50

$100

Today One Year Later

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

$75 – (N)($C)

$50

$100 - $25N

Today One Year Later

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

2N50$N25$100$

77.14$C45.45$C275$

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

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 0

Write 75 put +14.77 0 -$25

Sell stock short +75.00 -100 -50

Invest $68.18 in T-bills

-68.18 75.00 75.00

Total $6.82 $0.00 $0.00

Exploiting ArbitrageArbitrage 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 $29.00Buy 1 put ($2.23)Buy 1 share ($75.00)Borrow $60e-(.10)(2) $49.12

$0.89The value of the portfolio in two years will be worthless, regardless of the path the stock

takes over the two-year period.

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

95.7$P91.90$P275$

$75

$50 + N($75 - $50)

$100

Today One Year Later

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

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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 = $1.1052 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%

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:

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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:

45.16$)0$3422.0()25$6578.0(

88.14$1052.1/45.16$

Extension to Two Periods

$75

$50

$100

Today One Year Later

$133.33 (UU)

$66.67 (UD = DU)

$33.33 (DD)

Two Years Later

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?

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:

72.34$1052.1/37.38$

37.38$)0$3422.0()33.58$6578.0(

66.20$1052.1/84.22$

84.22$)0$3422.0()72.34$6578.0(

In sum…

$20.66

$0

$34.72

Today One Year Later

$58.33 (UU)

$0 (UD = DU)

$0 (DD)

Two Years Later

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:

– With lognormal returns, the size of the upward movement U equals:

– The probability of an up movement is:

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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

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

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