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Options

Global Financial Management

Campbell R. HarveyFuqua School of Business

Duke Universitycharvey@mail.duke.edu

http://www.duke.edu/~charvey

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Overview

Options:» Uses, definitions, types

Put-Call Parity» Futures and Forwards

Valuation» Binomial» Black Scholes

Applications» Portfolio Insurance» Hedging

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Definitions

Call Optionis a right (but not an obligation) to buy an asset at a pre-arranged price (=exercise price) on or until a pre-arranged date (=maturity).

Put Optionis a right (but not an obligation) to sell an asset at a pre-arranged price (=exercise price) on or until a pre-arranged date (=maturity).

European Optionscan be exercised at maturity only.

American Optionscan be exercised at any time before maturity

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Examples of Options

Securities Equity options

WarrantsUnderwritingCall provisionsConvertible bondsCapsInterest rate optionsInsuranceLoan guarantees

Risky bonds Equity

Real Options

Options to expandAbandonment optionsOptions to delay investmentModel sequences

Options are everywhere!

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Values of Options at ExpiryBuying a Call

Payoff

Stock Price X0

Payoff = max[0, ST - X]

Buy Call Option

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Payoff

Stock Price

X0

Sell Call Option

Payoff = - max[0, ST - X]

Values of Options at ExpiryWriting a Call

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Payoff

Stock Price X0

Payoff = max[0, X - ST ]

Buy Put Option

X

Values of Options at ExpiryBuying a Put

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Payoff

Stock Price

X0

Sell Put Option

Payoff = - max[0, X - ST]

-X

Values of Options at ExpirySelling a Put

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Example

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

StockPrice

Buy Call Write Call Buy Put Write Put

20 0 0 30 -30

40 0 0 10 -10

60 10 -10 0 0

80 30 -30 0 0

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Valuation of Options: Put-Call Parity

Principle: » Construct two portfolios» Show they have the same payoffs» Conclude they must cost the same

Portfolio I: Buy a share of stock today for a price of S0 and simultaneously borrowed an amount of PV(X)=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 PV(X) -X

Portfolio I PV(X) - S0 ST - X

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Payoff of Portfolio I

Payoff

Stock Price

ST

-X

ST - X

0

Payoff on Stock

Payoff on Borrowing

Net Payoff

X

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Put-Call Parity

Portfolio II: 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?

Since the two portfolios have the same payoffs at date T, they must have the same price today.

The put-call parity relationship is:

This implies: Call - Put = Stock - Bond

Position 0 T

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

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

Net Position PE-CE ST - X

CE - PE = S0 - PV(X)

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Put-Call Parity

Payoff

Stock Price

-X

X

ST - X

0

Payoff on short put

Payoff on long call

Net Payoff

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Put-Call Parity and Arbitrage

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?

From Put-Call Parity, the price of the call option should be equal to:» CE = PE+ S0 - Xe-rT=$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 PV($90)= 88.88 for 3 months.

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Put-Call Parity and Arbitrage

The cash flows for this investment are outlined below:

Hence, realize an arbitrage profit of 1.12» This is independent of the value of the stock price!

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

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Options and Futures

Compare this with a futures contract that specicifies that you buy a stock at X at time T. The futures contract trades today at F0.

» What is the price of the futures if there is no arbitrage?– Construct zero-payoff portfolio: Buy a Put, Write a Call,

and buy the futures contract

» Hence, the relationship between futures and options is:

Position 0 T

Write Call CE -Max[0,ST - X]

Buy Put -PE Max[0,ST - X]

Buy Futures -F0 ST - X

Net Position CE-PE-F0 0

F C PE E0

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Options and Futures

Payoff

Stock Price

-X

X

ST - X

0

Payoff on short put

Payoff on long call

Payoff on Future

Call is right to purchase

Short Put is obligation to sell

Future combines both When is F0=0?

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Debt and Equity as Options

Suppose a firm has debt with a face value of $1m outstanding that matures at the end of the year. What is the value of debt and equity at the end of the year?

Asset Value Payoff toShareholders

Payoff toDebtholders

0.3m 0 0.3m

0.6m 0 0.6m

0.9m 0 0.9m

1.2m 0.2m 1.0m

1.5m 0.5m 1.0m

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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. 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.

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Debt and Equity

Payoffs

Firm Value0

Equityholders

Bondholders

F

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Valuing OptionsEstablish bounds for Options

Upper bound on European call:» Compare to following portfolio: buy one share, borrow PV of

exercise price» Consider value at maturity:

Hence, since the call is worth more at maturity, CE>S-PV(X) before maturity

S<X S>X

Call 0 S-X

Share S S

Borrow X X

Portfolio S-X<0 S-X

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Bounds on Option Values

CE>S-PV(X); dominates portfolio of stock and borrowing X.

CE<S, otherwise buy stock straightaway

S, C

Stock Price

C=S C=S-PV(X)

PV(X)

PV(X)

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Example on Option Bounds I

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.

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

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Example on Option Bounds II

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

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

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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.

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

Stock Price = 100

125

80

High State

Low Stater=10%

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Pricing Contingent ClaimsStep 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:

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.

80

11072 73

..

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Consider the following option:

Maturity: 1 year

Exercise price: 110

Type: European

How does the option value develop?

The present value of $1 in the good state is $0.606, hence the option value is:

Option value = $0.606*15=$9.09

Pricing Contingent ClaimsStep 2: Value an Option

Option Value = ?

15

0

High State

Low State

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Why does this work?Contingent Claim Pricing and Arbitrage

Compare two portfolios:

Portfolio 1: 1 Call option

Portfolio 2: 1/3 share; 1 loan which pays off 80/3 at the end

Value of call option

= Share price - Loan value = You can make an option through buying a share and

borrowing.

Asset/State Stock Price = 80 Stock Price = 125

Loan -80/3 -80/3

Call Option 0 15

Portfolio 1 0 15

Portfolio 2 0 15

$100 $80 /

.$9.

3

3

11009

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Arbitrage: The General Idea

General Rule:

Use arbitrage principle by constructing portfolio with same payoffs as option (this is called replication).

Portfolio has delta shares and loan which pays exactly the 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:

Options can be valued by replicating their payoffs through forming portfolios of other assets.

Having an option is similar to buying stock and borrowing.

Spread of Option Values

Spread of Stock Prices

15 0

125 80

1

3

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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

periods with two-states

in each.

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.

73

100100

137

117

85

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The Black-Scholes FormulaAlternative Solution:

Repeat the above process until infinity;

Continuum of different states.

Use mathematical theory to determine result of this process.

Black-Scholes Formula:

Option value= [delta x share price] - [bank loan]

N(d1) x P - N(d2) x PV(X)

where:

d

S PV X

T

T

d d T

N d

1

2 1

2

ln /

( )

Cumulative Normal Density

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Call Option Sensitivities

Increase In: Effect on Call Price

S

T

r

X

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

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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

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Black-Scholes Put Option Formula

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

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

PE = CE - S + Xe-rT

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

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Put Option Sensitivities

Increase In: Effect on Put Price

S

T

r

X

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

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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?

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How to Use Black-Scholes

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

d2 = -0.498. The cumulative normal density for these values are

N(d1) = 0.3628

N(d2) = 0.3103. Plugging these values into the Black-Scholes formula gives:

c = $3.02

p = $9.02.

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How to Use Black-Scholes

Microsoft Put and Call Options

Option B-S Prices Actual Prices

Apr. call 100 $3.02 $3.25

Apr. put 100 $9.02 $9.125

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Implied Volatility

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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.

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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..

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Option Trading Strategies:The Straddle

Payoff

Stock Price X0

X

PutPayoff

CallPayoff

StraddlePayoff

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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.

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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

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Portfolio Insurance

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

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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