Class 5

Option Contracts

Options

A call option is a contract that gives the buyer the right, but not the obligation, to buy the underlying security at a prespecified price (called the strike or exercise price) within a prespecified period of time.

A put option is a contract that gives the buyer the right, but not the obligation, to sell the underlying security at a prespecified price (called the strike or exercise price) within a prespecified period of time.

Options

European options (both calls and puts) may only be exercised at the expiration date of the option.

American options (both calls and puts) may be exercised at any time prior to the expiration date of the option.

Call Option: Payoff Diagram

Payoff

Stock Price X0

Payoff = max[0, ST - X]

Buy Call Option

Call Option: Payoff Diagram

Payoff

Stock Price

X0

Sell Call Option

Payoff = - max[0, ST - X]

Put Option: Payoff Diagram

Payoff

Stock Price X0

Payoff = max[0, X - ST ]

Buy Put Option

X

Put Option: Payoff Diagram

Payoff

Stock Price

X0

Sell Put Option

Payoff = - max[0, X - ST]

-X

Example

What are the payoffs on a call option and a put option if the exercise price is X=$50?

Stock Price Call Payoff Put Payoff

20 0 30

40 0 10

60 10 0

80 30 0

Option Trading Strategies: The Straddle

Buy a call and a put on the same stock with the same exercise price and time to maturity.

Appropriate when you believe the stock price will change a lot, but you are unsure of the direction.

Option Trading Strategies:The Straddle

Payoff

Stock Price X0

X

PutPayoff

CallPayoff

StraddlePayoff

Option Trading Strategies:The Spread

Buy a call and sell another call with a higher strike price on the same stock with the same time to maturity.

Appropriate when you believe the stock price will increase and you are willing to trade off some upside potential to reduce the cost of your investment.

Option Trading Strategies:The Spread

Payoff

Stock Price X1

0

Short CallPayoff

Long CallPayoff Spread

Payoff

X2

X2 -X1

Valuation of Options: Put-Call Parity

Suppose you bought a share of stock today for a price of S0 and simultaneously borrowed an amount of Xe-rT. How much would your portfolio be worth at the end of T years? Assume that the stock does not pay a dividend.

Position 0 T

Buy Stock -S0 ST

Borrow Xe-rT

-X

Net Position Xe-rT

- S0 ST - X

Put-Call Parity

Payoff

Stock Price

ST

-X

ST - X

0

Payoff on Stock

Payoff on Borrowing

Net Payoff

X

Put-Call Parity

Now assume you buy a call option and sell a put option with a maturity date of T and an exercise price of X. How much will your options be worth at the end of T years?

Position 0 T

Buy Call -CE max[0,ST-X]

Sell Put PE -max[0,X-ST]

Net Position PE-CE ST - X

Put-Call Parity

Payoff

Stock Price

-X

X

ST - X

0

Payoff on short put

Payoff on long call

Net Payoff

Put-Call Parity Since the two portfolios have the same payoffs

at date T, they must have the same price today. The put-call parity relationship is:

CE - PE = S0 - Xe-rT

Example

A stock is currently selling for $100. A call option with an exercise price of $90 and maturity of 3 months has a price of $12. A put option with an exercise price of $90 and maturity of 3 months has a price of $2. The one-year T-bill rate is 5.0%. Is there an arbitrage opportunity available in these prices?

Example

From Put-Call Parity, the price of the call option should be equal to: CE = PE+ S0 - Xe-rT

CE = 2.00 +100.00 -90.00 e-(0.05)0.25

CE = 13.12

Since the market price of the call is $12, it is underpriced by $1.12. We would want to buy the call, sell the put, sell the stock, and invest $90e-(0.05)0.25 for 3 months.

Example

The cash flows for this investment are outlined below:

Position 0 ST<X ST>X

Buy call -12.00 0 ST-90

Sell put 2.00 ST-90 0

Sell stock 100.00 -ST -ST

Buy T-bill -90e-(0.05)0.25

90 90

Net Position 1.12 0 0

Lower Bounds for European Option Prices

Since both put options and call options must have non-negative prices, the put-call parity relationship establishes the following lower bounds for European option prices:

CE > max[ 0, S0 - Xe-rT ]

PE > max[ 0, Xe-rT - S0 ]

Example Suppose a stock is selling for $50 per share. The

riskfree interest rate is 8%. A call option with an exercise price of $50 and 6 months to maturity is selling for $1.50. Is there an arbitrage opportunity available? CE > max[ 0, S0 - Xe-rT ]

CE > max[ 0, 50 - 50e-(0.08)0.5 ] = 1.96

Since the price is only $1.50, the call is underpriced by at least $0.46.

Example

The arbitrage involves the following cash flows.

Position 0 ST<X ST>X

Buy call -1.50 0 ST-50

Sell stock 50 -ST -ST

Buy T-bill -50e-(0.08)0.5

50 50

Net Postion 0.46 50-ST>0 0

Example

Now suppose you observe a put option with an exercise price of $55 and 6 months to maturity selling for $2.50. Does this represent an arbitrage opportunity? PE > max[ 0, Xe-rT - S0

]

PE > max[ 0, 55e-(0.08)0.5 - 50] = 2.84

Since the price is only $2.50, the put is underpriced by at least $0.34

Example

The arbitrage involves the following cash flows:

Position 0 ST<X ST>X

Buy put -2.50 55-ST 0

Buy stock -50 ST ST

Borrow 55e-(0.08)0.5

-55 -55

Net Postion 0.34 0 ST-55>0

American vs. European Options

Recall that American options allow the holder of the option to exercise at any time prior to maturity, whereas a European option only permits the holder to exercise at maturity.

Because the option to exercise early cannot have a negative value, American options must be more valuable than European options.

American Put Options

The possibility to exercise American options at any time prior to maturity allows us to derive a tighter lower bound for the price of an American put option:

PA > max[ 0,X-S0 ]

Example

Consider the previous example where the stock price is $50. What is the lower bound for the price of an American put option with an exercise price of $55? PA > max[ 0 , X - S0 ]

PA > max [ 0 , 55 - 50 ] = $5.00

Note that $5.00 is the minimum price for an American put, regardless of the time to maturity.

American Call Options

Because of the possibility of early exercise, the price of an American call option is always at least as high as the price of its European counterpart. Hence,

CA > CE > max [ 0 , S0 - Xe-rT ]

American Call Options For stocks that do not pay dividends, CA = CE. The exercise value of an American call option

is S0-X. The unexercised value of an American call

option is at least:

CA > max [ 0 , S0 - Xe-rT ] Since the unexercised value is higher than the

exercised value, it is never optimal to exercise early for non-dividend-paying stocks.

Black-Scholes Option Pricing Formula

The Black-Scholes option pricing formula prices European options on non-dividend-paying stocks.

Black-Scholes Call Option Formula:

N(d1) = cumulative normal probability distribution, or

NORMSDIST(.) in EXCEL.

dS X r T

Td d T

1

2

2 1

5

ln( / ) ( . )

CE = S N(d1) - Xe-rT N(d2)

Call Option Sensitivities

Increase In: Effect on CallPrice

S

TrX

Intuition for Black-Scholes

Payoffif if

RSTS X S X

S XT T

T0

C e E S S X S X

e X S X

r TT T T

r TT

C

C

0

| Pr

Pr

C PV E C e E CTr T

TC

0 b g

E C E S X S X S X

S XT T T T

T

| Pr

Pr 0

Intuition for Black-Scholes

Intuition for Black-Scholes

E S S X S XT T T| Pr

( . ) ( . ) ( . ) .

0 2 101 01 102 01 103 40 70

C e

e

00 10 0 5

0 10 0 5

40 70

100 0 4 0 6659

. ( . )

. ( . )

( . )

( )( . ) .

Intuition for Black-Scholes

Intuition for Black-Scholes

e E S S X S X S dr TT T T

f * *| Pr ( )N 1

Pr ( )* S X dT N 2

C e E S S X S X

e X S X

r TT T T

r TT

C

C

0

| Pr

Pr

C e E S S X S X

e X S X

r TT T T

r TT

f

f

0

* *

*

| Pr

Pr

Black-Scholes Put Option Formula

We can use the put-call parity relationship to derive the Black-Scholes put option formula:

We have used the fact that 1-N(d1) = N(-d1) and 1-N(d2) = N(-d2).

PE = CE - S + Xe-rT

PE = -SN(-d1) + Xe-rTN(-d2)

Put Option Sensitivities

Increase In: Effect on PutPrice

S

TrX

Example

On February 2, 1996, Microsoft stock closed at a price of $93 per share. Microsoft’s annual standard deviation is about 32%. The one-year T-bill rate is 4.82%. What are the Black-Scholes prices for both calls and puts with an exercise price of $100 and a maturity of April 1996 (77 days)? How do these prices compare to the actual market prices of these options?

Example The inputs for the Black-Scholes formula are:

S = $93.00 s r = 4.82% X = $100.00 s s = 32% T = 77/365

This gives d1 = -0.351 and d2 = -0.498. The cumulative normal density for these values

are N(d1) = 0.3628 and N(d2) = 0.3103. Plugging these values into the Black-Scholes

formula gives: c = $3.02 and p = $9.02.

Example

Microsoft Put and Call Options

Option B-S Prices ActualPrices

Apr. call 100 $3.02 $3.25Apr. put 100 $9.02 $9.125

Implied Volatilities

It is common for traders to quote prices in terms of implied volatilities.

This is the volatility (s) that sets the Black-Scholes price equal to the market price.

This can be computed using SOLVER in EXCEL.

Hedging with Options

Initial investment (option premium) is required

You eliminate downside risks, while retaining upside potential

Option Hedging Example

It is the end of August and we will receive 1m DM at the end of October.

At this point, we will sell DM, converting them back into dollars.

We are concerned about the price at which we will be able to sell DM.

We can lock in a minimum sale price by buying put options.

Option Hedging Example

Since the total exposure is for 1m DM and each contract is for 62,500 DM we buy 16 put option contracts.

Suppose we choose the puts struck at 0.66 - locking in a lower bound of 0.66 $/DM.

Deutschemark Falls to $0.30

We have the right to sell 1m DM for $0.66 each by exercising the put options.

Since DM’s are only worth $0.30 each we do choose to exercise.

Our cash inflow is therefore $660,000

Deutschemark Rises to $0.90

We have the right to sell 1m DM for $0.66 each by exercising the put options.

Since DM’s are worth $0.90 each we do not choose to exercise.

We sell the DM on the open market for $0.90 each.

Our cash inflow is therefore $900,000

Debt and Equity Consider a firm with zero coupon debt

outstanding with a face value of F. The debt will come due in exactly one year.

The payoff to the equityholders of this firm one year from now will be the following:

Payoff to Equity = max[0, V-F]

where V is the total value of the firm’s assets one year from now.

Debt and Equity

Similarly, the payoff to the firm’s bondholders one year from now will be:

Payoff to Bondholders = V - max[0,V-F]

Equity has a payoff like that on a call option. Risky debt has a payoff that is equal to the total value of the firm, less the payoff on a call option.

Debt and Equity

Payoffs

Firm Value0

Equityholders

Bondholders

F

Debt and Equity

Since bondholders have essentially sold a call option on the value of the firm’s assets to equityholders, conflicts of interest can arise. Payout policy. Asset substitution problem. Underinvestment problem.

These problems can be resolved to some extent with debt covenants, conversion features, callability features, and putability features.