55-S T S T -55

0 S T >X

0 S T -55

S T -55>0

American vs. European Options n 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.

n Because the option to exercise early cannot have a negative value, American options must be

more valuable

than European options.

American Put Options n 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: P A > max[ 0,X-S 0 ]

Example n n 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?

  P A > max[ 0 , X - S 0 ] P A > 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 n 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, C A > C E > max [ 0 , S 0 - Xe -rT ]

American Call Options n n n n For stocks that do not pay dividends, C A = C E .

The

exercise value

of an American call option is S 0 -X.

The

unexercised value

of an American call option is at least: C A > max [ 0 , S 0 - 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 n n The

Black-Scholes

option pricing formula prices

European options

on

non-dividend-paying

stocks.

Black-Scholes Call Option Formula: C E = S N(d 1 ) - Xe -rT N(d 2 ) N(d 1 ) = cumulative normal probability distribution, or NORMSDIST(.) in EXCEL .

d

1  ) 

T

.

5  2 )

T d

2 

d

1  

T

Call Option Sensitivities

Increase In:

S  T r X

Effect on Call Price

Intuition for Black-Scholes Payoff  R T

T

 0

X E C T

if

S T

if

S T

 

X X



E S

Pr

T S T

 

X

|

T



X

Pr

S T



X C

0  b g

T



e



E C T C

0 

e



e

 

T

|

X

Pr

S T T



X



X

Pr

S T



X

Intuition for Black-Scholes

Intuition for Black-Scholes

T

|

T



X

 Pr

S T



X

 

C

0 

e

 .



e

 .

( . ) ( 40 70 ) ( . ) ( )( . )  

Intuition for Black-Scholes

Intuition for Black-Scholes

C

0 

e



e

 

T

|

X

Pr

S T T



X



X

Pr

S T



X e



C

0 

e



e

  *

T

|

T



X X

Pr *

S T



X

Pr *

S T



X T

|

T



X

Pr *

S T



X



S

N 1 Pr *

S T



X

 N 2

Black-Scholes Put Option Formula n We can use the put-call parity relationship to derive the Black-Scholes put option formula: P E = C E - S + Xe -rT P E = -SN(-d 1 ) + Xe -rT N(-d 2 ) n We have used the fact that 1-N(d 1 ) = N(-d 1 ) and 1-N(d 2 ) = N(-d 2 ).

Put Option Sensitivities

Increase In:

S  T r X

Effect on Put Price

Example n 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 n The inputs for the Black-Scholes formula are:   S = $93.00  X = $100.00  r = 4.82%  = 32%  T = 77/365 n n n This gives d 1 = -0.351 and d 2 = -0.498.

The cumulative normal density for these values are N(d 1 ) = 0.3628 and N(d 2 ) = 0.3103.

Plugging these values into the Black-Scholes formula gives: c = $3.02 and p = $9.02.

Example n Microsoft Put and Call Options

Option

Apr. call 100 Apr. put 100

B-S Prices

$3.02

$9.02

Actual Prices

$3.25

$9.125

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

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

n This can be computed using SOLVER in EXCEL.

Hedging with Options n Initial investment (option premium) is required n You eliminate downside risks, while retaining upside potential

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

n At this point, we will

sell

DM, converting them back into dollars.

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

n We can lock in a minimum sale price by

buying put options

.

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

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

Deutschemark Falls to $0.30

n n 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.

n Our cash inflow is therefore $660,000

Deutschemark Rises to $0.90

n n 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.

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

n Our cash inflow is therefore $900,000

Debt and Equity n Consider a firm with zero coupon debt outstanding with a face value of F. The debt will come due in exactly one year.

n 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 n Similarly, the payoff to the firm’s bondholders one year from now will be: Payoff to Bondholders = V - max[0,V-F] n

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

.

Payoffs Debt and Equity Equityholders Bondholders 0 F Firm Value

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