55-S T S T -55

0 S T >X

0 S T -55

S T -55>0

24

Valuing Options as Contingent Claims

Idea:

  Investors attach different values to

states

in which assets pay off: $1 is worth more in bad times than in good times.

Values depend on preferences for insuring against bad times and discounting (time value of money).

 Value of $1 in good times or bad times (or a continuum of states) can be inferred from prices of stocks and bonds.

125 High State Stock Price = 100 r=10% 80 Low State

Procedure:

» Determine value of $1 in good and in bad state » Use the value to infer the value of the option 25

Pricing Contingent Claims

Step 1: Determine the value of states

Method

 Break up payment to shareholders into two components: » Shareholders receive at least 80 for sure (in good

and

bad state).

» Shareholders receive an additional 45 if the share price is high, otherwise nothing.

Steps:

1. The present value of a safe payment of 80 is simply: 80  2. The value shareholders attach to the uncertain 45=125-80 must be the difference between the current share price and the value of the safe payment: 100 - 72.73 = 27.27

3. The present value of $1 in the good state is 27.27/45=0.606.

26

Pricing Contingent Claims

Step 2: Value an Option  Consider the following option: Maturity: Exercise price: Type: Option Value = ?

1 year 110 European 15 How does the option value develop?

0 High State Low State 27

Why does this work?

Contingent Claim Pricing and Arbitrage  Compare two portfolios:

Portfolio 1:

1 Call option Asset/State Stock Price = 80 Stock Price = 125

Portfolio 2:

1/3 share; 1 loan which pays off 80/3 at the end Loan -80/3 -80/3 Call Option Portfolio 1 Portfolio 2 0 0 0 15 15 15 $100  3 $80 / 3  $9.

09 28

Arbitrage: The General Idea

General Rule:

 Use arbitrage principle by constructing portfolio with same payoffs as option (this is called

replication

).

  1 lowest value of the delta shares. delta is called the

option delta

:  If portfolio replicates option, then it must have the same value as the option.

Implications:

29

Options with Many States

 Suppose there are more than two possible states at the end of the period. Then: subdivide period.

Example:

3 states at the end of the period: Divide movement into two 117 100 periods with two-states in each.

85

Solution:

  Value the option for each of the mid-period nodes and then fold it backwards into the first node.

Repeat this for ever smaller intervals to cover larger numbers of states.

137 100 73 30

The Black-Scholes Formula

Alternative Solution:

 Repeat the above process until infinity; Continuum of different states.

 Use mathematical theory to determine result of this process.

d

1 

d

2  Option value= ln  

d

1  

T

[delta

x

     2

T

share price]

N(d 1 ) x P

-

-

[bank loan]

N(d 2 ) x PV(X)

31

Call Option Sensitivities

 The Option Pricing formula gives the following sensitivies for a call option:

Increase In:

S  T r X

Effect on Call Price

32

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 33

Black-Scholes Put Option Formula

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

P

E

= C

E

- S + Xe

-rT  Use Put-Call Parity and the fact that the normal distribution is symmetric around the mean:

P

E

= -SN(-d

1

) + Xe

-rT

N(-d

2

)

34

Put Option Sensitivities

 The Option Pricing formula gives the following sensitivies for a put option:

Increase In:

S  T r X

Effect on Put Price

35

Example

  On February 2, 1996, Microsoft stock closed at a price of $93 per share. » 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?

36

How to Use Black-Scholes

    The inputs for the Black-Scholes formula are: » S = $93.00 s » X = $100.00 s  r = 4.82% = 32% » T = 77/365 This gives: d 1 = -0.351

d 2 = -0.498.

The cumulative normal density for these values are N(d 1 ) = 0.3628

N(d 2 ) = 0.3103.

Plugging these values into the Black-Scholes formula gives: c = $3.02

p = $9.02.

37

How to Use Black-Scholes

 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

38

Implied Volatility

39

Implied Volatilities

   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.

This can be computed using SOLVER in EXCEL.

40

Applications of Options I:

Volatility Bets   Suppose you have no information about the return of the stock, but you believe that the market

underrates

the volatility of the stock: » Give an example!

– How can you trade?

Buy

Straddle:

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

41

Payoff

Option Trading Strategies: The Straddle

X 0 Put Payoff Straddle Payoff X Call Payoff Stock Price 42

Hedging with Options

  Initial investment (option premium) is required You eliminate downside risks, while retaining upside potential

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

.

– 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.

43

Heding with Currency Options

  

Scenario I:

Deutschmark 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    

Scenario II:

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 44

Portfolio Insurance

 Reconsider the case of a fund manager who wishes to insure his portfolio 45

Summary

     Options are derivative securities: » Replicate payoffs with combinations of underlying assets Put and Call prices are linked Valuation as contingent claims » Use Black-Scholes as approximation Value of option increases with volatility of underlying assets Use options for » Volatility bets » Portfolio Insurance » Hedging 46