Options

(Text reference: Chapters 17, 18)

� background

� option payoffs

� put-call parity

� option investment strategies

� exotic options

� factors affecting option values

� arbitrage bounds on option values

� Black-Scholes option valuation

� binomial option valuation

1

Background

� modern option trading began on the CBOE in 1973, although

financial option contracts can be traced back to ancient Greece

� a call option gives its holder the right to buy the underlying security

for a price X (known as the strike price or exercise price) on or before

an expiry date T

� a put option is similar except that the holder has the right to sell

� European options can only be exercised at T , American options can

be exercised at any time up to and including T

� many options trade on financial exchanges (options on individual

stocks or stock indexes, FX, options on futures), others are OTC

(interest rate options, exotic options) or packaged as part of other

securities

2

Option payoffs

ST

$

X

payoff at T

profit

Call Writer

ST

$

X

payoff at T

profit

Call Owner

ST

$

X

payoff at T

profit

Put Writer

ST

$

X

payoff at T

profit

Put Owner

3

� the payoff of an option if it were to be exercised is known as the

intrinsic value of the option

� prior to maturity, the option will have some additional value which is

called the time value of the option

$

SX

Call $

SX

Put

4

� options can be combined to create any desired payoff pattern

� e.g. suppose you buy a call with a strike of $95, sell 2 calls with a

strike of $100, and buy a call with a strike of $105. Your payoff is

max�ST � 95 � 0 � � 2 � max

�ST � 100 � 0 ��� max

�ST � 105 � 0 � , which

looks like:

$

ST

5

Put-call parity

� consider a portfolio A with a long position in the underlying asset (e.g. stock)

and a put option

� consider another portfolio B with a call option (same strike and maturity as

the put in A) and a risk free pure discount bond maturing at the same time as

the call with a face value equal to the strike

� the payoffs at maturity T for these portfolios are:

ST � X ST � X

A Stock ST ST

Put X ST 0

X ST

B Call 0 ST X

Bond X X

X ST

6

� since the two portfolios have the same future payoffs in every possible future

state, by no-arbitrage they must sell for the same value today:

St � Pt � Ct � PV � X �� an implication of this relationship is that given any three of stock, bond, call,

and put, we can always create the fourth synthetically

– example: suppose there is no risk free bond. We can construct one by

purchasing the stock, selling a call, and buying a put.

– example: suppose there is no put option. We can create one by buying a

bond and a call and shorting the stock.

� the expression above assumes that no dividends are paid by the stock before

the options expire. If there are dividends, then the parity relationship

becomes:

St � Pt � Ct � PV � X � � PV � dividends �� even this assumes that the options are European; more complicated arbitrage

bounds are available for American options

7

Options and investments

� options (and other derivatives such as futures or forward contracts) are highly

leveraged

– example: suppose you buy a one year call option on a stock for $4. The

stock is currently trading at $50, and the option is “at-the-money” (i.e.

strike equals current stock price). If the stock price rises 20% to $60 after

one year, your rate of return is � 10 4 ��� 4 � 150%; if the stock drops

20% to $40, your rate of return is 100%.

� there is some evidence that the options markets react faster to new

information than the stock markets

� many types of securities or contracts feature embedded options (convertibles,

savings bonds, retractables/extendibles, mortgages, mortgage-backed

securities, futures contracts, some swaps, insurance, segregated funds,

index-linked GICs, etc.) Although it may be quite difficult to value these

options, keep in mind that no-one gives away free options!

8

� investment strategies using traded options

1. bullish strategies:

– if you are optimistic that the price will rise, buy a call:

– if you are confident of a price increase, and quite sure the price won’t

fall, sell a put:

9

– if you are confident of a price increase but somewhat worried about a

price decrease, use a bull spread:

– if you are bearish in the short-term and bullish in the longer term, use

a diagonal spread:

10

2. bearish strategies

– if you are very sure the price will fall, buy a put:

– if you are quite certain the price will not rise, sell a call:

11

– if you believe the price will fall, and are fairly sure it will not rise, use

a bear spread:

– if you are bullish in the short term and bearish in the long term, use a

diagonal spread:

12

– if you own a stock but you are worried about a price fall, you can use a

put hedge (a.k.a. “protective put”):

3. Neutral strategies

– if you are confident that prices will fluctuate in a narrow range, sell a

straddle:

13

– if you expect a broader range of fluctuation, sell a strangle:

– if you are less certain about the range of fluctuation, use a long

butterfly:

14

– if your view is pessimistic in the short term but optimistic in the longer

term, use a calendar spread:

– if you own the stock but expect no movement in its price, use a

covered call:

15

4. volatile strategies

– if you expect very high volatility, buy a straddle:

– if you expect high volatility, buy a strangle:

16

– if you expect moderate volatility, use a short butterfly:

� many other combinations are possible

� important note: any strategy which involves selling American options

faces the risk of early exercise!

17

Exotic options� so far we have discussed plain vanilla puts and calls, but there is an

increasing variety of exotic options, with various different features:

– Bermudan options can be exercised at some points before expiry

– digital (a.k.a. binary) options are pure bets: if the underlying asset ends

up above the strike, the holder of a digital call receives $1, otherwise the

payoff is zero

– Asian options have payoffs which depend on the average price of the

underlying asset over time

– lookback options have payoffs which are functions of either the

maximum or minimum value of the underlying asset over time

– barrier options are like vanilla puts and calls that also depend on whether

the underlying asset ever reaches certain levels during the life of the

option

– quanto options have payoffs defined by variables in one currency but are

paid in a different currency

18

Factors affecting option values

� we will deal once again with plain vanilla contracts, though most of

these also affect other kinds of options

� in general, we have the following:

Factor Effect on call value Effect on put value

Stock price + -

Strike price - +

Stock price volatility + +

Time to expiration + +

Interest rate + -

Dividend - +

19

Arbitrage bounds on option values

� call options:

– since it is a limited liability instrument, a call option cannot have

a negative value

– at most, a call option can be worth St

– we can also show that a call must be worth at least

St � PV�dividends � � PV

�X �

20

– thus we have:

� important implication: on a non-dividend paying stock, it is never

optimal to exercise an American call option prior to maturity

21

� put options:

– cannot have a negative value

– worth at most X

– we can show that a put must be worth at least

PV � dividends � � PV � X � St :

– note that it may be optimal to exercise an American put option prior to

maturity, even if the underlying stock pays no dividends

22

Black-Scholes option valuation

� to get precise option values (instead of just arbitrage bounds), we must make

assumptions about the underlying stock’s distribution of returns; the

Black-Scholes model assumes that this distribution is lognormal

� in this case, the value of a European call option on a non-dividend paying

asset is:

Ct � St N � d1 � Xe � r�T � t � N � d2 ���

where

d1 � ln � St � X � � � r � σ2 � 2 � � T t �σ � T t

d2 � d1 σ � T t

r � continously compounded risk free interest rate

σ � standard deviation of stock’s return (continuously compounded)

N ��� � � cumulative normal distribution function

23

� the value of a European put option on a non-dividend paying asset is:

Pt � Xe � r�T � t � N � d2 � St N � d1 �

� example: St � $50, X � $50, r �� 05, T t � 0 � 5 years, σ �� 40

– then:

d1 � ln � 50 � 50 � � � � 05 �� 402 � 2 � � � 5 �� 40 � � 5 � 0 � 2298

d2 � 0 � 2298 � 40 � � 5 � 0 � 0530

– from text Table 18.2, we have:

N���

22 �� � 5871 N� � � 06 �� � 4761

N���

24 �� � 5948 N� � � 04 �� � 4841

� N���

2298 ��� � 24 � � 2298�24 � � 22 � � 5871 � N

� � � 0530 ��� � � 04 � � � � 0530 �� � 04 � � � � 06 ��� � 4761

� � 2298 � � 22�24 � � 22 � � 5948

� � � 0530 � � � � 06 �� � 04 � � � 0

�06 � � � 4841

� 590873 0�478900

24

– note that we can calculate these values directly on a spreadsheet: using

the NORMSDIST function in Excel gives N � d1 � � 0 � 590880 and

N � d2 � � 0 � 478853.

– then

Ct � 50 � � 590873 � 50e � � 05� �

5 � � � 478900 � � 6 � 1899

– using the accurate values from Excel gives Ct � 6 � 1925

– to calculate the value of a put option instead of a call, we have (using the

NORMSDIST function):

N � � 2298 � �� 409120 N � � 0530 � � � 521147

Pt � 50e � � 05� �

5 � � � 521147 � 50 � � 409120 � � 4 � 9580

– we can also calculate this using put-call parity:

Pt � Ct � Xe � r�T � t � St

� 6 � 1925 � 48 � 7655 50 � 4 � 9580

25

Binomial option valuation

� the fundamental idea underlying option pricing is replication, i.e.

finding a combination of other assets which gives the same payoff as

the option and then noting that absence of arbitrage � option price

equals cost of replicating portfolio

� the easiest context to analyze is the binomial model

� e.g. one period version:

100

130

76.92

Stock price tree Call price tree

What is the price of a call with X � 98 if r � 10% per year?

26

� form a portfolio with h shares of stock and B invested in a risk free asset. The

current value of the portfolio V � 100h � B. Pick h and B to replicate the

option:

130h � 1 � 1B � 32

76 � 92h � 1 � 1B � 0� h �� 6029 � B � 42 � 16

V � 100 � � 6029 � 42 � 16 � 18 � 13 � C0

� suppose we divide the year into two periods and the stock price tree is:

100

114.02

87.71

130

100

76.92

27

� consider the same call option:

C0

Cu

Cd

Cuu � 32

Cud � 2

Cdd � 0� note that dividing the year into n intervals produces n � 1 terminal stock

prices and n � 1 terminal option values, assuming:

1. u � d � d � u

2. option is not path-dependent

28

� Cu:

130h � 1 � 0488B � 32

100h � 1 � 0488B � 2

� h � 1, B � 93 � 44, Cu � 114 � 02 � 1 � 93 � 44 � 20 � 58

� Cd :

100h � 1 � 0488B � 2

76 � 92h � 1 � 0488B � 0

� h �� 0867, B � 6 � 36, Cd � 87 � 71 � � 0867 � 6 � 36 � 1 � 24

� C0:

114 � 02h � 1 � 0488B � 20 � 58

87 � 71h � 1 � 0488B � 1 � 24

� h �� 7348, B � 60 � 26, C0 � 100 � � 7348 � 60 � 26 � 13 � 22

29

� note that replication involves a dynamic portfolio strategy, i.e. we

have to rebalance the portfolio after six months. Is the call value

really $13.22? Only if we don’t have to add any further investment to

the replicating portfolio over time, i.e. if the replicating portfolio is

self-financing.

� e.g. suppose the stock price rises to $114.02 after six months. The

initial portfolio with 0.7348 shares and -60.26 is now worth

114 � 02�

� 7348 � � 60 � 26�1 � 0488 � � 20 � 58. The portfolio is then

rebalanced to 1 share of stock and -93.44 in the risk free asset, which

is worth 114 � 02 � 93 � 44 � 20 � 58.

� in this binomial framework, it is always possible to replicate using a

self-financing strategy

30

� more formally, let the current stock price be S, the up factor be u, the

down factor be d, and define R � 1 � r f (for the appropriate periods)

S0

Su

Sd

Suu

Sud

Sdd

� note that Su � S0 � u, Sd � S0 � d, Suu � S0 � u2, Sud � S0 � u � d,

Sdd � S0 � d2

� absence of arbitrage � u � R � d

31

� for the option:

C0

Cu

Cd

Cuu

Cud

Cdd

� Cu:

hS0u2 � RB � Cuu

hS0ud � RB � Cud

� hS0�u2 � ud � � Cuu � Cud

� h �Cuu � Cud

S0u�u � d �

32

� substituting in the replicating portfolio for h gives:

Cuu � Cud

S0u�u � d ��� S0u2 � RB � Cuu

�

�Cuu � Cud � u

u � d� RB � Cuu

� then:

B �

�Cuu �

�Cuu � Cud � u

u � d ��� 1R

�

�Cuu

�u � d � �

�Cuu � Cud � u

u � d ��� 1R

�uCud � dCuu

R�u � d �

33

� solving for Cu � hSu � B gives:

Cu �

�Cuu � Cud

S0u�u � d ����� S0u � uCud � dCuu

R�u � d �

�Cuu � Cud

u � d� uCud � dCuu

R�u � d �

�

�R � d � Cuu �

�u � R � Cud

R�u � d �

�1R � ��� R � d

u � d � Cuu ��

u � Ru � d � Cud �

� check previous example for Cu (R � 1 � 0488, u �� 1 � 3 � 1 � 1402,

d � 1 � 1 � 3 � � 8771):

h �Cuu � Cud

Su�u � d � �

32 � 2130 � 100

� 1

34

B �uCud � dCuu

R�u � d � �

1 � 1402�2 � � � 8771

�32 �

1 � 0488�1 � 1402 � � 8771 � � � 93 � 44

Cu �1R � � � R � d

u � d � Cuu ��

u � Ru � d � Cud �

�1

1 � 0488 � ��� 1 � 0488 � � 87711 � 1402 � � 8771 � 32 �

�1 � 1402 � 1 � 04881 � 1402 � � 8771 � 2 �

� 20 � 58

� note that there is nothing unique about this particular node—these

formulas apply throughout the tree

35

� we have:

S

S�

S �C

C�

C �

h �C

�

� C �

S� � S �

B �uC � � dC

�

R�u � d �

C �1R

� �R � du � d � C � �

�u � Ru � d � C � �

36

� check Cd in previous example:

87 � 71

100

76 � 92

Cd

2

0

h � 2 0100 76 � 92 � � 0867

B � 1 � 1402 � 0 � � 8771 � 2 �1 � 0488 � 1 � 1402 � 8771 � � 6 � 36

C � 11 � 0488

���1 � 0488 � 87711 � 1402 � 8771 � � 2 �

�1 � 1402 1 � 04881 � 1402 � 8771 � � 0 �

� 1 � 24

37

� continue back to the initial value:

100

114 � 02

87 � 71

Cd

20 � 58

1 � 24

h � 20 � 58 1 � 24114 � 02 87 � 71 �� 7350

B � 1 � 1402 � 1 � 24 � � 8771 � 20 � 58 �1 � 0488 � 1 � 1402 � 8771 � � 60 � 29

C � 11 � 0488

���1 � 0488 � 87711 � 1402 � 8771 � � 20 � 58 �

�1 � 1402 1 � 04881 � 1402 � 8771 � � 1 � 24 �

� 13 � 22

38

� of course, given h and B, it seems easier to simply calculate C � hS � B, but

there are some significant benefits to using the apparently more complicated

option pricing expression

� recall:

C � 1R

� �R du d � � C � �

�u Ru d � � C � �

� note that since u � R � d:

0 � R du d � 1 and 0 � u R

u d � 1

� furthermore:R du d � u R

u d � 1

� these terms can be interpreted as “probabilities”, even though they have

nothing to do with the actual probabilities of the stock price moving up or

down

39

� if these were the actual probabilities, then the expected gross return

on the stock would be:�R � du � d � u � �

u � Ru � d � d � R

i.e. 1 � r f , which would be the expected return on the stock if

everyone was risk-neutral. Hence these “probabilities” are called

“risk-neutral probabilities”.

� let:R � du � d

� π

� note that π is the risk-neutral probability that the stock price will rise

� then:

Ct �1R

�πC

�

t�

1 ��1 � π � C �t

�1 �

40

� extending to two periods:

C0

π

1 � π

Cu

π

1 � π

Cd

π

1 � π

Cuu

Cud

Cdd

Cu � 1R � πCuu � � 1 π � Cud �

Cd � 1R � πCud � � 1 π � Cdd �

C0 � 1R � πCu � � 1 π � Cd �

� 1R2

�π2Cuu � 2π � 1 π � Cud � � 1 π � 2Cdd �

41

� the expression at the bottom of slide 41 is the discounted expected value of

the terminal payoff of the option using the risk-neutral probabilities, i.e.

C0 � 1R2 Eπ � CT �

� note that nothing in the argument is specific to call options, so exactly the

same expression applies to put options or in fact any type of claim contingent

on the stock (though we may have to check for early exercise in the case of

American options)

� for the n period binomial model, we have

C0 � 1Rn Eπ � CT �

so for a call option

C0 � 1Rn

n

∑j 0

�nj � π j � 1 π � n � j max

�S0u jdn � j X � 0 �

42

where �nj � � n!

j! � n j � ! ��nj � π j � 1 π � n � j

is the risk-neutral probability of a path with j up moves and n j down

moves, and max � S0u jdn � j X � 0 � is the option payoff for such a path.

� e.g. n � 3:

S0

S0u

S0d

S0u2

S0ud

S0d2

S0u3

S0u2d

S0ud2

S0d3

43

j Description # of paths Total Probability

0 3 down moves � 30 � � 3!

0! � 1! � 1 1 � � 1 π � 3

1 2 down moves, 1 up � 31 � � 3!

1! � 2! � 3 3 � π � 1 π � 2

2 1 down move, 2 up � 32 � � 3!

2! � 1! � 3 3 � π2 � 1 π �3 3 up moves � 3

3 � � 3!3! � 0! � 1 1 � π3

� complications with path-dependency since there are a total of 2n paths

� see handout for an example with a standard European put and an Asian call

44

� returning to the two-period call option example with S0 � 100, r f � � 0488,

u � � 1 � 3, d � 1 � u, and X � 98:

S0 � 100

C0 � 13 � 22

h �� 7348

B � 60 � 26

Su � 114 � 02

Cu � 20 � 58

h � 1

B � 93 � 44

Sd � 87 � 71

Cd � 1 � 24

h �� 0867

B � 6 � 36

Sud � 100

Cud � 2

Suu � 130

Cuu � 32

Sdd � 76 � 92

Cdd � 0

45

� we also have

C0 � 1R2

�π2Cuu � 2π � 1 π � Cud � � 1 π � 2Cdd �

where π � � R d ��� � u d � � � 1 � 0488 � 8771 ��� � 1 � 1402 � 8771 � � � 6528 is

the risk-neutral probability that the stock rises. Thus

C0 � � 1 � 1 � 1 ��� 65282 � 32 � � 2 � � 6528 � � 1 � 6528 � � 2 � � � 1 � 6528 � 2 � 0 � �

� 13 � 22

� some concluding comments about option valuation

– it can be cheaper to replicate using futures rather than the underlying

stock because transactions costs are lower in futures markets

– the same type of results holds in other contexts: the condition that

risk-neutral probabilities exist and that option prices may be computed as

discounted expected values is equivalent to the absence of arbitrage. In

fact, this provides a relatively simple way to derive the Black-Scholes

formula.

46

– since we only have to compute an expected value, we can use Monte

Carlo methods to price options (these can be very efficient for

high-dimensional and/or path-dependent options, though early exercise is

difficult to handle)

– remember the underlying hedging argument. In some cases people

simply value an option using Black-Scholes without thinking about

whether it makes any sense (e.g. executive stock options, American).

– if we let r be the continuously compounded annual risk free rate, σ be the

annual standard deviation of the stock, let ∆t be the time interval (e.g.

T � 1 year, with n � 50 steps, ∆t �� 02). Then set u � eσ � ∆t , d � 1 � u,

π � � er∆t d ��� � u d � to converge to the Black-Scholes price as ∆t � 0.

– example: S0 � X � 100, r �� 08, T � 1, σ � � 3. The Black-Scholes price

of a European put is $8.02. For the binomial model:

n 10 20 40 80 160 320 640

European put 7.73 7.88 7.95 7.99 8.00 8.01 8.02

American put 8.77 8.84 8.87 8.89 8.90 8.90 8.90

47