Lsh Options

Options

Finance 100

Prof. Michael R. Roberts

Copyright © Michael R. Roberts 1

Topic Overview

Options:
» Uses, definitions, types

Put-Call Parity

Valuation
» Black Scholes

Applications
» Portfolio Insurance
» Hedging
» Speculation and arbitrage


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1

Definitions

Call Option
A standard call option is an option giving the buyer the right to buy from
the seller an underlying asset for a fixed price (strike/exercise price) at any
time on or before a fixed date (expiration date)
Put Option
A standard put option is an option giving the buyer the right to sell to the
seller an underlying asset for a fixed price (strike/exercise price) at any
time on or before a fixed date (expiration date)
Exercise Styles:
» European: can be exercised at maturity only.
» American: can be exercised at any time before maturity
» Bermudan: can be exercised only at some predefined times (e.g. employee
stock options)
» Atlantic: can be exercised at times dependent on underlying asset (e.g. “cap”
and “barrier” options)


Options Markets

Exchange traded options
» Stock options (CBOE, Philli, AMEX, NYSE), ForEx options (Philli), Index options (CBOE), Futures
options (CBOE)
OTC options
Specification of an Option Contract
» Expiration Date (typically third Friday of the month)
» Strike Price
» Class & Series
» OTM, ATM, ITM (Deep)
» Splits & Dividends
Trading
» No margin investing!
Options Clearing Corporation (OCC)
» Similar to clearinghouse for futures
Regulation
» OCC
» SEC (options on stocks, stock indices, currencies & bonds)
» CFTC (options on futures)
Warrants & ESOPs


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2

Options Quotes for Amazon.com Stock


Values of Options at Expiration
Buying a Call

50

40
Net Payoff at Maturity

30

20

10

0
0 20 40 60 80 100
-10
Stock Price at Maturity

This is the payoff (at maturity) to the buyer of a call option:
» PayoffT = max(0,ST – K) - C


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3

Writing a Call

10

0
0 20 40 60 80 100

-10

-20

-30

-40

-50

This is the payoff (at maturity) to the seller (or writer) of a call option:
» PayoffT = -[max(0,ST – K) – C] = min(0,K – ST) + P


Buying a Put

50

40

30

20

10

0
0 20 40 60 80 100
-10

This is the payoff (at maturity) to the buyer of a put option:
» PayoffT = max(0,K - ST) – P


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4

Selling a Put

10

0
0 20 40 60 80 100

-10

-20

-30

-40

-50

This is the payoff (at maturity) to the seller (or writer) of a put option:
» PayoffT = -[max(0,K - ST) – P] = min(0,ST – K) + P


Sample Payoffs

What are the gross payoffs (ignoring the price of the
contract) to the buyer of a call option and a put option
if the exercise price is K=$50?
Stock Buy Write Buy Write
Price Call Call Put Put
max(0,ST - K) min(0,K - ST) max(0,K - ST) min(0,ST - K)
20 0 0 30 -30
40 0 0 10 -10
60 10 -10 0 0
80 30 -30 0 0


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5

Put-Call Parity
Example

What is the relationship between a put and a call
option that are otherwise identical. Consider:
» stock whose current price is $90 and pays no dividends
» a risk-free rate of 5%,
» One call and an otherwise identical put option each with
one year to expiration and strike price of 100.
Can we replicate the call option & what does no
arbitrage imply?


Put-Call Parity
Example (Cont.)

No arbitrage implies:


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6

Put-Call Parity Example

200
Call Price 10 Strike 100
Put Price 15.24 Spot Price 90 IR 5%
Combined Position
150 Future
Stock Buy Buy Buy Borrow Combined
Own Put
100 Price Call Put Asset PV(Strike) Position
Payoff at Expiration

0 -10 84.76 -90 -4.76 -10
50 20 -10 64.76 -70 -4.76 -10
40 -10 44.76 -50 -4.76 -10
0
60 -10 24.76 -30 -4.76 -10
0 50 100 150 200
80 -10 4.76 -10 -4.76 -10
-50
100 -10 -15.24 10 -4.76 -10
-100 120 10 -15.24 30 -4.76 10
Own Asset 140 30 -15.24 50 -4.76 30
-150 Borrowing 160 50 -15.24 70 -4.76 50
Future Stock Price 180 70 -15.24 90 -4.76 70
200 90 -15.24 110 -4.76 90

We see the combined position in the put, asset and
cash exactly replicates the payoffs to the call.


Put-Call Parity
General Expression
More generally, consider:
» An underlying asset with current price = S0, price at expiry = ST
» A put and a call option with T years to maturity and identical strike price = K
» An annualized risk-free return and dividend yield equal to r and d, respectively.
Today Expiration Date
ST < K ST > K
Buy Call -C 0 ST-K
Buy Put -P K-ST 0
-T
Buy 1 unit of Asset -S0(1+d) ST ST
-T
Borrow PV(Strike Price) K(1+r) -K -K
-T -T
Total -P-S(1+d) +K(1+r) 0 ST-K
C = P + S 0 (1 + d ) − T − K (1 + r )
−T


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7

Put-Call Parity General Expression
Continuous Compounding
More generally, consider:
» An underlying asset with current price = S0, price at expiry = ST
» A put and a call option with T years to maturity and identical strike price = X
» Continuously compounded risk-free return and dividend yield equal to r and d,
respectively.
Today Expiration Date
ST < K ST > K
Buy Call -C 0 ST-K
Buy Put -P K-ST 0
-dT
Buy 1 unit of Asset -S0e ST ST
-rT
Borrow PV(Strike Price) Ke -K -K
-dT -rT
Total -P-Se +Ke 0 ST-K

C = P + S 0 e − dT − Ke − rT

Put-Call Parity Implications

By rearranging the put-call parity relation, we find numerous
implications
» How can we replicate borrowing with options and the underlying asset ?

Ke − rT = − C + P + S 0 e − dT
» How can we replicate shorting the underlying asset with options ?

S 0 e − dT = C − P + Ke − rT
» What is the implies risk-free return ?
⎛ − C + P + S 0 e − dT ⎞1
r = − ln ⎜
⎜ ⎟
⎟T
⎝ K ⎠
» Protective Put = Fiduciary Call ?

S 0 e − dT + P = C + Ke − rT

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Put-Call Parity and Arbitrage
Example

A non-dividend paying 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%.
What does PCP imply?

What is the first step of your arbitrage strategy?


Put-Call Parity and Arbitrage
Example (Cont.)


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9

Call Option Valuation
Black-Scholes
Black Scholes price of a call option on a non-dividend-paying
stock, C
C = S × N (d1 ) − PV (K ) × N (d 2 )
S is current price of the stock
K is the exercise price
N(x) is the Standard Normal CDF, Pr(X<x)
ln[S / PV (K )] σ T
d1 = + and d 2 = d1 − σ T
σ T 2
σ is the annual volatility
T is the number of years left to expiration


Valuing a Call Option with Black-Scholes


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10

Valuing a Call Option with Black-Scholes

The BS parameters:
» S = (12.58+12.59)/2 = $12.585
» T = 45/365
» rf = 4.38%
» σ = 25%
» K = $12.50
PV ( K ) = 12.50 / (1.0438 )
45/365
= $12.434
ln[S / PV (K )] σ T ln (12.585 / 12.434 ) 0.25 45 / 365
d1 = + = + = 0.181
σ T 2 0.25 45 / 365 2
d 2 = d1 − σ T = 0.181 − 0.25 45 / 365 = 0.094

C = S × N (d1 ) − PV (K ) × N (d 2 )
= 12.585 ( 0.572 ) − 12.434 ( 0.537 ) = $0.52

Black-Scholes Call Option Values
Future Payoff

Value of Call before
Maturity

Value of Call at
Maturity

Future Stock Price


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Put Option Valuation
Black-Scholes
Black Scholes price of a call option on a non-dividend-paying
stock, P
P = PV (K )[1 − N (d 2 )] − S[1 − N (d1 )]
S is current price of the stock
K is the exercise price
N(x) is the Standard Normal CDF, Pr(X<x)
ln[S / PV (K )] σ T
d1 = + and d 2 = d1 − σ T
σ T 2
σ is the annual volatility
T is the number of years left to expiration


Stock Option Valuation with Dividends
Black-Scholes

Very simple adjustment to the previous formulas:
Define the ex-dividend stock price:
S x = S − PV (Div)
Substitute Sx for S in the formulas
» A special case is when the stock will pay a dividend that is
proportional to its stock price at the time the dividend is
paid. If q is the stock’s (compounded) dividend yield until
the expiration date, then:

S x = S / (1 + q)


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12

Call Option Sensitivities
The Option Pricing formula gives the following
sensitivities for a call option:

Effect on
Increase in… Call Price Intuition
S up More likely to finish ITM
σ up Asymmetric Payoff
T up Asymmetric Payoff
r up Time Value of $
K down Less likely to finish ITM


Compaq Options:
Using Black-Scholes (cont.)

Compaq (Ok…)
Black -Scholes prices and quoted prices:
Calls Puts
K BS Price Quoted Price BS Price Quoted Price
60 18.854 20.250 1.029 NA
65 14.942 15.500 2.027 2.500
70 11.539 12.875 3.534 3.000
75 8.691 8.750 5.597 6.125
80 6.394 6.000 8.211 8.250
85 4.604 4.375 11.331 NA


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Compaq Call Options
Market prices and Black-Scholes Prices

30.000

BS Price
25.000
Quoted Price

20.000
Option valu

15.000

10.000

5.000

0.000
50 55 60 65 70 75 80 85 90 95
Strike price


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?
Payoff to Payoff to
Asset Value Shareholders Debtholders
0.3 0.0 0.3
0.6 0.0 0.6
0.9 0.0 1.0
1.2 0.2 1.0
1.5 0.5 1.0
(Amounts are in millions of dollars)


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An Illustration

Net
Payoffs
Firm

Bondholders

Equityholders

Face Value
0 Future Firm Value
of Debt


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


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15

Junior vs. Senior Debt

150

125
Senior Debt

100
Payoff

75 Junior Debt

50

25
Equity

0
0 50 100 150 200 250 300
Asset Value


Hedging with Foreign Currency Options

Initial investment (option premium) is required
You eliminate downside risks, while retaining upside
potential
Example: Recall the American firm selling 20 machines
to a German company at 50,000ECU per machine.
» What’s our exposure ?

» What options position should we take to hedge the risk ?

(Assume that the puts are struck at $1.57/ECU)


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Hedging with Foreign Currency Options
(Cont.)
Scenario 1: Exchange rate falls to $1.00/ECU
» Profits from options position = ?

» Profits from sale of machines = ?

» Total profit in $US = ?
Scenario 2: Exchange rate rises to $2.00/ECU
» Profits from futures position = ?
» Profits from sale of machines = ?

» Total profit in $US =?
Punch line:


Hedging with Options vs. Futures

Payoffs
Payoff with Options

Payoff with
Futures

0 Exchange Rate

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Summary

Options are derivative securities

Put-Call Parity

Valuation: use Black-Scholes

Value of option increases with volatility of underlying assets

Use options for
» Volatility bets
» Portfolio Insurance
» Hedging


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