• Options Pricing, Arbitrage,
Hedging and Speculation
• Bodie et al. (2021) Ch. 21
• Hull (2017) Ch.9; Ch.13, Ch.15
15-1
21.1 Option Valuation: prior expiration: Introduction
• Option premium = Intrinsic value + time value
• Intrinsic value 内在値: payoff could be obtained by
immediate exercise
– : Call: max(S – X, 0); Put: max(X – S, 0)
− Because the option holder can choose not to
exercise, the payoff cannot be worse than zero.
• Time value 時間值 (volatility value波動值) – the
difference between the option premium and the intrinsic
value
• Time value is the part of the option’s value that may be
attributed to the fact that it still has positive time to
expiration.
• So, time value is positive only before expiration. 15-2
15-3
Option Valuation: Introduction
• Even if a call option is out of the money now, it
still will sell for a positive price because there is
always a chance that the stock price will
increase, while imposing no risk of additional
loss if the stock price falls, and then the option
will expire with zero value.
• As the stock price increases substantially, it
becomes likely that the call option will be
exercised by expiration. Ultimately, with
exercise all but assured 幾 乎 可 以 肯 定 , the
volatility value becomes minimal.
15-4
15-5
Option Valuation: Introduction
• As the stock gets ever larger, the option value at
present approaches the adjusted intrinsic value,
i.e. So – PV(X).
• It is because you are virtually certain that the
option will be exercised and the stock purchased
for X dollars, it is as if you own the stock already.
• The adjusted intrinsic value is So– PV(D) – PV(X)
if the stock pays dividends before expiration.
• Figure 21.1 illustrates the call option valuation
function.
• The option value curve shows that call value is
near zero when the stock price is very low
because there is almost no chance of exercising it.
15-6
Option Valuation: Introduction
• When the stock price is very high, the call option
value approaches adjusted intrinsic value. The
volatility value becomes minimal.
• In the midrange case, where the option is
approximately at the money, the volatility value
of the option is quite high although exercise
today would have a negligible or negative payoff.
• The slope is greatest when the option is deep in
the money. In this case, exercise is all but
assured, and the call option increases one-forone with the stock price. The volatility value
becomes minimal.
• The option premium can be estimated using
Binomial option pricing model or Black-Scholes
model.
15-7
Figure 21.1 Call Option Value
before Expiration
15-8
Class exercise
• Before expiration, the time value of an in
the money call option is always
•
A. equal to zero.
B. positive.
C. negative.
D. equal to the stock price minus the
exercise price.
E. None of these is correct.
15-9
Class exercise
• Before expiration, the time value of an atthe-money put option is always
•
A. equal to zero.
B. equal to the stock price minus the
exercise price.
C. negative.
D. positive.
E. none of the above
15-10
Class exercise
• A call option has an intrinsic value of zero
if the option is
•
A. at the money.
B. out of the money.
C. in the money.
D. A and C.
E. A and B.
15-11
Class exercise
• Prior to expiration
•
A. the intrinsic value of a call option is
greater than its actual value.
B. the intrinsic value of a call option is
always positive.
C. the actual value of a call option is
greater than the intrinsic value.
D. the intrinsic value of a call option is
always greater than its time value.
E. None of these is correct.
15-12
Notation
co (Co) = current European (American) call option value
p0 (P0) = current European (American) put option value
So = Current stock price
ST = Stock price at option maturity
X = Exercise price
r = Risk-free interest rate (annualizes continuously
compounded with the same maturity T as the option)
T = time to maturity of the option in years
Standard deviation of annualized cont. compounded
rate of return on the stock
D = PV of dividends during the option’s life
15-13
Effect of Variables on Option
Pricing
Variable
S0
X
T

r
D
c0
+
–
+
+
+
–
p0
–
+
++
–
+
C0
+
–
+
+
+
–
P0
–
+
+
+
–
+
15-14
Effect of stock price and strike price
on Option Pricing
• When stock price increases and strike
price decreases, intrinsic value of call
option increases and put option decreases.
• The option premiums of the corresponding
options will change accordingly.
15-15
Effect of stock price and strike price on
Option Pricing
• Profits for speculation prior expiration:
– If you expect stock price to rise, the payoff and
intrinsic value of a call option will rise, and the
payoff and intrinsic value of a put option will
fall.
– Buy a call option now and sell it out when its
price of the call will have risen
– Or, you write a put option now and buy it back
when its price of the put will have fallen.
15-16
Effect of stock price and strike price on
Option Pricing
• Profits for speculation prior expiration:
– If you expect stock price to fall, the payoff and
intrinsic value of a call option will fall, and the
payoff and intrinsic value of a put option will
rise.
– Buy a put option now and sell it out when its
price of the put will have risen
– Or, you write a call now and buy it back when
its price of the call will have fallen.
15-17
Effect of time to expiration on Option
Pricing
• The longer an option has until expiration,
the greater the chance it will end up in the
money.
• As expiration approaches, the option's time
value decreases.
15-18
Effect of volatility on Option Pricing
• Volatility of a stock price is a measure of how
uncertain we are about future stock price
movements.
• As volatility increases, the chance that the stock
will do very well or very poorly increases. For the
owner of a stock, these two outcomes tend to
offset each other.
• However, the owner of a call benefits from price
increases but has limited downside risk in the
event of price decreases because the most the
owner can lose is the price of the option only.
15-19
Effect of volatility on Option Pricing
• Likewise, the owner of a put benefits from
price decreases, but has limited downside
risk in the event of price increases.
• The values of both calls and puts therefore
increase as volatility increases.
15-20
Effect of risk-free interest rate on Option Pricing
• Call options
• Interest rate is the opportunity cost of buying
stocks. If interest rate becomes higher, that makes
buying call options instead of the stocks more
attractive, thus leading to higher call option
premium.
• Call option writers need to either have the same
amount of stocks in inventory or have cash locked
up in their account as margin. Either way, the
options writer is denied the right to sell the stocks
or reallocate the cash into those higher interest Tbills. This loss of interest by the writers is
compensated by a higher call option premium.15-21
15-22
Effect of risk-free interest rate on Option Pricing
• Put options
• Put options are substitutes for shorting shares.
Interest rate is the opportunity cost of buying put
options because you can earn interests on the cash
received from shorting shares but buying put options
cannot earn interest. This makes buying put options
when interest rate rises less attractive than shorting
the shares, thus lower put option premium.
• Put option writers usually have a short position in the
underlying stock for hedging, which gives the writers
cash earning interest in the account. When interest
rate rises, put options premium can be lower to
neutralize additional gains by the writers so that it
remains a fair trade on both sides from the start.15-23
15-24
Effect of dividend on Option Pricing
• Dividends have the effect of reducing the
stock price on the ex-dividend date.
• Dividends have negative effect on call and
positive effect on put.
15-25
Class exercise
• Buyers of put options anticipate the value
of the underlying asset will __________ and
sellers of call options anticipate the value of
the underlying asset will _______.
•
A. increase; increase
B. decrease; increase
C. increase; decrease
D. decrease; decrease
E. Cannot tell without further information.
Fundamentals of Futures and Options Markets, 8th Ed, Ch 9, Copyright © John C. Hull 2013
26
15-26
Class exercise
• Which one of the following variables influence
the value of call options?
I) Level of interest rates.
II) Time to expiration of the option.
III) Dividend yield of underlying stock.
IV) Stock price volatility.
•
A. I and IV only.
B. II and III only.
C. I, II, and IV only.
D. I, II, III, and IV.
E. I, II and III only.
15-27
Class exercise
• Other things equal, the price of a stock call
option is positively correlated with the
following factors except
•
A. the stock price.
B. the time to expiration.
C. the stock volatility.
D. the exercise price.
E. None of these is correct.
15-28
Class exercise
• Other things equal, the price of a stock put
option is positively correlated with the
following factors
•
A. the stock price.
B. the time to expiration.
C. the stock volatility.
D. the exercise price.
E. the time to expiration, the stock volatility,
and the exercise price.
15-29
Class exercise
• The price of a stock put option is
__________ correlated with the stock price
and __________ correlated with the striking
price.
•
A. positively; positively
B. negatively; positively
C. negatively; negatively
D. positively; negatively
E. not; not
15-30
Class exercise
Other things equal, the price of a stock put
option is negatively correlated with the
following factors
A. the stock price.
B. the time to expiration.
C. the stock volatility.
D. the exercise price.
E. the time to expiration, the stock volatility,
and the exercise price.
15-31
Class exercise
• Lower dividend payout policies have a
__________ impact on the value of the
call and a __________ impact on the
value of the put compared to higher
dividend payout policies.
A. negative, negative
B. positive, positive
C. positive, negative
D. negative, positive
E. zero, zero
15-32
21.2 Restrictions on Option Values
• Value of a call cannot be negative, so C > 0
• Payoff of a call at expiration (T):
CT = (ST – X) > 0
• Value of leveraged equity at T:
Stock Value
ST +D
Payback of loan –(X+D)
ST – X > = < 0
• Value of the call must be greater than the
value of levered equity
C0 > S0 - ( X + D ) / ( 1 + rf )T ; or more generally,
C0 > S0 - PV ( X ) - PV ( D ) (lower bound)
15-33
15-34
Restrictions on Option Values
• We interpret PV(D) as PV of any and all
dividends to be paid prior to the option
expiration.
• Value of a call cannot exceed the stock
value, so C0 ≦ S0 (Upper bound)
15-35
Figure 21.2 Range of Possible Call
Option Values
15-36
Early exercise and dividends
• A call option holder wants to close out position:
—Exercise it at time t and get St – X > 0;
—Sell it at Ct and get at least St - PV(X) - PV(D).
 For an option on a non-dividend paying stock, we
never exercise before expiration since
Ct > St - PV(X) > St – X
 The values of otherwise identical American and
European call options on stocks paying no
dividends before expiration are equal.
15-37
Early exercise and dividends
•
St – X
St - PV(X)
St - PV(X) - PV(D)
15-38
Figure 21.3 Call Option Value as a
Function of the Current Stock Price
15-39
Early exercise of American puts
• The payoff for a put option at time t: X - St > 0
• Suppose St falls to zero (nearly zero) if the firm
goes bankrupt (nearly bankrupt).
• For American put options, immediate exercise
gives investors immediate receipt of the
exercise price which can be invested to start
generating income. Delay in exercise means a
time-value-of-money cost.
• American put must be worth more than the
European put
15-40
Early exercise of American puts
• From figure 21.4, once the stock price drops
below a critical value, S*, exercise becomes
optimal.
• At that point the option-pricing curve is
tangent to the straight line depicting the
intrinsic value of the option.
• If and when the stock price reaches S*, the
put option is exercised and its payoff equals
its intrinsic value.
15-41
Figure 21.4 Put Option Values as a
Function of the Current Stock Price
15-42
Class Exercise
• An American call option buyer on a nondividend paying stock will
A. always exercise the call as soon as it is in
the money.
B. only exercise the call when the stock price
exceeds the previous high.
C. never exercise the call early.
D. buy an offsetting put whenever the stock
price drops below the strike price.
E. None of these is correct.
15-43
21.4 Black-Scholes Option Valuation
• Option premium can be estimated using
the Black-Scholes pricing formula before
expiration.
• The Black-Scholes pricing formula was
developed to calculate the value of a
European call option on non-dividend
paying stocks.
• But it can be modified to calculate
dividend-paying call, European put and
American call options.
15-44
21.4 Black-Scholes Option Valuation
Co = SoN(d1) – Xe-rTN(d2)
(21.1)
d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)
d2 = d1 – (T1/2)
where
Co = Current European call option value
So = Current stock price
N(d) = probability that a random draw from a
normal distribution will be less than d
15-45
Black-Scholes Option Valuation
Continued
X = Exercise price
e = 2.71828, the base of the natural log
r = Risk-free interest rate (annualizes continuously
compounded with the same maturity as the
option)
T = time to maturity of the option in years
ln = Natural log function
Standard deviation of annualized continuously
compounded rate of return on the stock
15-46
Figure 21.6 A Standard Normal Curve
15-47
Example 21.2 Black-Scholes Valuation
So = $100
X = $95
r = 0.10
T = 0.25 (3 months)
 = 0.50
d1 = [ln(100/95) + (0.10+(052/2))0.25] / (05
(0.251/2))
= 0.43
d2 = 0.43 – ((050.251/2)
= 0.18
15-48
Probabilities from Normal Distribution
N (0.43) = 0.6664
d
N(d)
0.42
0.6628
0.43
0.6664
0.44
0.6700
15-49
Probabilities from Normal Distribution
Continued
N (0.18) = 0.5714
d
0.16
0.18
0 .20
N(d)
0.5636
0.5714
0.5793
15-50
Cumulative Normal Distribution
15-51
15-52
European Call Option Value
Co = SoN(d1) – Xe-rTN(d2)
Co = $100 x 0.6664 – $95 e -0.10 X 0.25 x 0.5714
Co = $13.70
15-53
The N(d) Function
• N(d) is the probability that a normally
distributed variable with a mean of zero
and a standard deviation of 1 is less than d.
• In other words, N(d) refers to a cumulative
probability distribution for a standardized
normal distribution of less than d.
15-54
The N(d) Function
• In Excel, the command is:
=NORMSDIST(d)
• In our case,
For example:
=NORMSDIST(0.43)
We obtain 0.666402.
15-55
Class exercise
Which of the inputs in the Black-Scholes
Option Pricing Model are directly
observable
A. the price of the underlying security.
B. the risk-free rate of interest.
C. the time to expiration.
D. the variance of returns of the underlying
asset return.
E. A, B, and C.
15-56
Implied Volatility 引伸波幅
• Using Black-Scholes and the actual price of the
option, solve for volatility.
• Implied volatility of an option is the standard
deviation of stock returns consistent with an
option’s market price.
• If your true volatility is larger than the implied
volatility, the option is underpriced and you
should buy or take a long position.
• If your true volatility is smaller than the implied
volatility, the option is overpriced and you should
sell or write or take a short position.
15-57
Call Option Value
• It can be backed out of an option-pricing
model by finding the stock volatility that
makes the option’s value equal to its
observed price.
15-58
Spreadsheet 21.1 Spreadsheet to
Calculate Black-Scholes call Option
Values
15-59
Figure 21.8 Implied Volatility of the S&P
500 (VIX Index)
15-60
European Put Option Valuation
• European put option:
• P = Xe-rT [1-N(d2)] – S0 [1-N(d1)]
(21.3)
• Example: Using the sample call data
S = 100 r = 0.10 X = 95  = 0.5 T =0 .25
$95e-0.1x0.25(1–0.5714) – $100(1–0.6664) =
$6.35
15-61
Dividends
• Dividends cause stock prices to reduce on the
ex-dividend date by the amount of the dividend
payment.
• European options on dividend-paying stocks
are valued by substituting the stock price less
the present value of all the dividends
anticipated during the life of the option,
discounted from the ex-dividend dates to the
present at the risk-free rate, into Black-Scholes.
• Only dividends with ex-dividend dates during
life of option should be included
• The “dividend” should be the expected
62
15-62
reduction in the stock price
Dividends
• Consider a European call option on a stock
with ex-dividend dates in two months and
five months. The dividends on each exdividend dates is expected to be $0.50.
The So is $40, the X is $40, the σ is 30%
per annum, the continuously compounded r
is 9% per annum, and T is 6 months.
• PV of Dividends is:
$0.5 e-0.09 x 2/12 + $0.5 e-0.09 x 5/12 = $0.9741
So – PV(D) = $40 – $0.9741 = $39.0259
15-63
Black-Scholes Valuation with dividends
So –PV(D) = 39.0259
X = 40
r = 0.09
T = 0.50 (6 months)
 = 0.30
d1 = [ln(39.0259/40) + (0.09+(032/2))0.50] /
(03 (0.51/2))
= 0.2020
d2 = 0.2020 – (03)0.51/2) = -0.01013
N(0.20) = 0.5793 N(-0.01) = 0.4960
15-64
Black-Scholes Valuation with dividends
Co = (So – PV(D)) N(d1) – Xe-rT N(d2)
Co = $39.0259 x 0.5793 – $40 e -0.09 X 0.50 x
0.4960
Co = $3.6407
15-65
Dividend and Put Option Valuation
•
•
•
(21.3) is valid for European puts on nondividend paying stocks.
If the underlying stocks pay a dividend
before expiration, find European put values
by substituting S0 – PV (Dividends) for S0
(21.3) describe only the lower bound on the
true value of the American put.
15-66
European Options on Stocks
Providing a known Dividend Yield
• Dividends cause stock prices to reduce on the exdividend date by the amount of the dividend
payment. The payment of a known dividend yield
at rate q therefore causes the growth rate in the
stock price to be less than it would otherwise be by
an amount of q.
• If with a dividend yield of q, the stock price grows
from So today to ST at time T, then in the absence
of dividends, it would grow from So today to ST eqT
at time T. Alternatively, in the absence of dividends,
it would grow from Soe-qT today to ST at time T.
67
15-67
European Options on Stocks
Providing a known Dividend Yield
We get the same probability distribution for
the stock price ST at time T in each of the
following cases:
1. The stock starts at price S0 and provides
a known dividend yield = q
2. The stock starts at price S0e–q T and
provides no income
15-68
European Options on Stocks
Providing known Dividend Yield
continued
We can value European options by
reducing the stock price to S0e–q T and
then behaving as though there is no
dividend
15-69
Pricing formula (Equations 15.4 and 15.5)
c  S0e qT N (d1)  Xe  rT N (d 2 )
p  Xe
 rT
N( d 2 )  S0e
 qT
N(d1)
2
ln( S0 / X)  (r  q   / 2)T
where d1 
 T
d 2  d1   T
15-70
Valuing European Index Options
We can use the formula for an option on a
stock paying a dividend yield
Set S0 = current index level
Set q = average annualized dividend yield
expected on the index during the life of the
option
15-71
Valuing European stock index
option
• Consider a European call option on a stock
index with a known dividend yield of 3% per
year. The So is 930 points, the X is 900
points, the σ is 20% per annum, the
continuously compounded r is 8% per
annum, and T is 2 months.
15-72
Black-Scholes Valuation with dividend
yield
So = 930 X = 900
r = 0.08
T = 2/12 (i.e. 2 months)
 = 0.20
d1 = [ln(930/900) + (0.08-0.03+(022/2)) 2/12] /
(02 (2/12)1/2)) = 0.5444
d2 = 0.5444 – (02 (2/12)1/2) = 0.4628
N(0.54) = 0.7054; N(0.46) = 0.6772
15-73
Black-Scholes Valuation with dividend
yield
c  S0e qT N(d1)  Xe  rT N(d 2 )
Co = 930 e -0.03 X 2/12 x 0.7054
– 900 e -0.08 X 2/12 x 0.6772
Co = 51.3425
The contract cost ($) is Co x multiplier.
If multiplier = 100 for S&P500; 50 for Hang
Seng index
15-74
Currency Options
• Currency options trade on the Philadelphia
Exchange (PHLX)
• There also exists a very active over-thecounter (OTC) market
• Currency options are used by corporations
to buy insurance when they have an FX
exposure
15-75
The Foreign Interest Rate
• We denote the foreign interest rate by rf
• When a U.S. company buys one unit of the
foreign currency it has an investment of S0
dollars
• The return from investing at the foreign rate is
rf S0 dollars
• This shows that the foreign currency provides
a “dividend yield” at rate rf
15-76
Valuing European Currency
Options
• A foreign currency is an asset that provides
a “dividend yield” equal to rf
• We can use the formula for an option on a
stock paying a dividend yield:
Set S0 = current exchange rate
Set q = rƒ
15-77
Formulas for European Currency Options
(Equations 15.11 and 15.12)
c  S0e  rf T N (d1)  Xe  rT N (d 2 )
p  Xe  rT N (d 2 )  S0e  rf T N (d1)
where d1 
ln( S0 / X )  (r  rf  2 / 2)T
 T
d 2  d1   T
15-78
Currency options
• Consider a four-month European call option
on the British pound. Suppose that the
current exchange rate So is $1.6000 per
pound, X = 1.6000, the σ is 14.10% per
annum, the continuously compounded US r
is 8% per annum, UK rf is 11% per annum,
and T is 4 months.
15-79
Call Currency Option Value
So = $1.6 X = $1.6
r = 0.08
rf = 0.11 T = 4/12 (4 months)
 = 0.141
d1 = [ln(1.6/1.6) + (0.08-0.11+(01412/2)) 4/12] /
(0141 (4/12)1/2)) = –0.0821
d2 = -0.0821 – (0141 (4/12)1/2) = -0.1635
N(-0.08) = 0.4681 N(-0.16) = 0.4364
15-80
Call Currency Option Value
c  S0e
 rf T
N(d1)  Xe  rT N(d 2 )
Co = $1.6 e -0.11 X 4/12 x 0.4681
– 1.6 e -0.08 X 4/12 x 0.4364
Co = $0.0421
The contract cost ($) = Co x contract size.
For example: GBP: 62,500; NZD: 10,000
Euros: 10,000; AUD: 10,000; JPY: 1,000,000
CAD: 10,000
15-81
Dividends and American Call Option Valuation
• American call options should never be
exercised early when the underlying stock
pays no dividends.
• When dividends are paid, it is sometimes
optimal to exercise at a time immediately
before the stock goes ex-dividend because the
dividend will make both the stock and the call
option less valuable.
• If the dividend is sufficiently large and the call
option is sufficiently in the money, it may be
worth exercising the call option early in order
to avoid the adverse effects of the dividend on
15-82
the stock price.
Black’s approximation
• Black’s approximation is an approximate method
for computing the value of an American call
option on a dividend-paying stock.
• Black’s approximation involves calculating the
prices of two European call options:
• 1. Assuming early exercise, a European call that
matures just before the latest ex-dividend date
during the life of the option, and
• 2. Assuming no early exercise, a European call
that matures at the same time as the American
call being valued, but with the stock price
reduced by the PV(D)
15-83
Black’s approximation
• Consider an American call option with exdividend dates in 3 months and 5 months
and has an expiration date (T) of 6 months.
The dividend (D) on each ex-dividend date
is expected to pay $0.70. Also, S0 = $40,
X=$40,  = 0.30 per annum, continuously
compounded r = 10% per annum.
15-84
Black’s approximation
1. T is 5 months or 0.4167 year
PV(First D) = $0.7 e -0.1 x 3/12 = $0.6827
So – PV(First D) = $40 – $0.6827 = $39.3173
d1 = [ln(39.3173/40) + (0.10+(032/2))0.4167] /
(0.3 (0.41671/2)) = 0.2231
d2 = 0.2231 – (03)0.41671/2) = 0.0294
N(0.22) = 0.5871; N(0.03) = 0.5120
15-85
Black’s approximation
Co = (So – PV(First D)) N(d1) – Xe-rT N(d2)
Co = $39.3173 x 0.5871 – $40 e -0.10 X 0.4167 x
0.5120
Co = $3.4391
15-86
Black’s approximation
2. PV(D) = $0.7 e -0.1 x 3/12 + $0.7 e -0.1 x 5/12
= $1.3541.
T = 6 months (i.e. 6/12 year)
So –PV(D) = $40 – $1.3541 = $38.6459
d1 = [ln(38.6459/40) + (0.10+(032/2))0.50] /
(03 (0.501/2)) = 0.1794
d2 = 0.1794 – (03)0.501/2) = -0.0327
N(0.18) = 0.5714 N(-0.03) = 0.4880
15-87
Black’s approximation
Co = (So – PV(D)) N(d1) – Xe-rT N(d2)
Co = $38.6459 x 0.5714 – $40 e -0.10 X 0.50 x
0.4880
Co = $3.5142
$3.5142 > $3.4391, the price of the American
call option is $3.5142.
15-88
Black’s approximation
 The pseudo-American call option value is
the maximum of the value derived by
assuming no early exercise and the value
derived by assuming early exercise.
15-89
https://www.hkex.com.hk/eng/sorc/tools/calcula
tor_stock_option.aspx
15-90
15-91
15-92
15-93
The Put–Call Parity Relationship
• Strategy 1: Buy a European call and write a
European put with X
• Strategy 2: Borrow money (PV(X)) and buy the
underlying stock on the call and put
• Each strategy produces the same cash flows at
expiration, so each costs the same.
𝐶 − 𝑃 = 𝑆0 − 𝑋𝑒 −𝑟𝑇
15-94
15-95
The Put–Call Parity
Alternatively:
•Strategy 3: Buy a European call and buy a T-bill
with par value = X, at PV(X)
•Strategy 4: Buy a European put and a stock
•Each strategy produces the same cash flows at
expiration, so each costs the same.
𝐶 + 𝑋𝑒
−𝑟𝑇
= 𝑆0 + 𝑃
15-96
Put–Call Parity - Disequilibrium Example
Stock Price = 110 Call Price = 17
Put Price = 5 Maturity = 1 yr X = 105
Continuously compounded Risk Free = 5%
𝐶 + 𝑋𝑒
−𝑟𝑇
= 𝑆0 + 𝑃
116.879 > 115
Since the leveraged equity is less
expensive, acquire the low-cost alternative
and sell the high cost alternative
15-97
15-98
Put–Call Parity
If the stock pays dividends, then:
𝐶 − 𝑃 = 𝑆0 − 𝑃𝑉 𝐷 − 𝑃𝑉(𝑋)
𝐶 + 𝑃𝑉(𝑋 + 𝐷) = 𝑆0 + 𝑃
15-99
Class exercise
• According to the put-call parity theorem, the value
of a European put option on a non-dividend
paying stock is equal to:
•
A. the call value plus the present value of the
exercise price plus the stock price.
B. the call value plus the present value of the
exercise price minus the stock price.
C. the present value of the stock price minus the
exercise price minus the call price.
D. the present value of the stock price plus the
exercise price minus the call price.
E. None of these is correct.
15-100
Arbitrage Opportunities
• Suppose that
c0 = 3
S0 = 31
T = 0.25
r = 10%
X =30
D = $1.0
• What are the arbitrage possibilities when
p0 = 2.8?
• c0 + (X+D) e-rT < p0 + S0
• 3 + 30.23 = 33.23 < 2.8 + 31 = 33.80
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John101
C. Hull 2013
15-101
The Put–Call Parity
• An arbitrageur can short sell the stock ($31) and
write the European put ($2.8), buy the European
call ($3) and buy the zero-coupon bonds
($31+2.8-3=$30.8) at 10% for 3 months. Initial
cash flow is zero.
• If ST > X ($30), e.g. $32, we exercise the
European call by paying X = $30, receiving the
stock, and close out the European call . We repay
the stock and $1 dividend to close out the short
sale positions. The bonds grows to $30.8 e0.1x3/12
= $31.58. We earn $31.58 - $30 - $1.0 = $0.58
15-102
The Put–Call Parity
• If ST < X ($30), e.g. $28, the European put is
exercised by receiving the stock and paying X =
$30 to close out the European put. We repay
stock and $1 dividend to close short sale
positions. The bonds grows to $30.8 e0.1x3/12 =
$31.58. We earn $31.58 - $30 -$1.0 = $0.58.
15-103
15-104
Arbitrage Opportunities
• Suppose that
c0 = 3
S0 = 31
T = 0.25
r = 10%
X =30
D = $1.0
• What are the arbitrage possibilities when
p0 = 2.0?
• c0 + (X+D) e-rT > p0 + S0
• 3 + 30.23 = 33.23 > 2.0 + 31 = 33
Fundamentals of Futures and Options Markets, 8th Ed, Ch 10, Copyright © John105
C. Hull 2013
15-105
The Put–Call Parity
• An arbitrageur can write the European call ($3),
buy the European put ($2.0), buy the stock ($31)
and borrow ($3 – 2 – 31 = -$30) at 10% for 3
months.
• If ST > X ($30), e.g. $32, the European call is
exercised by receiving $30 and paying the stock
to close out the European call position. We
receive $1 dividend. We repay the debt at $30
e0.1 x 3/12 = $30.76. We earn $30 + $1.0 - $30.76 =
$0.24.
15-106
The Put–Call Parity
• If ST < X ($30), e.g. $28, we exercise the
European put by selling the stock and
receiving X = $30 to close out the European
put position. We receive $1 dividend. We
repay the debt $30 e0.1 x 3/12 = $30.76. We earn
$30 + $1.0 - $30.76 = $0.24.
15-107
15-108
21.5 Using the Black-Scholes Formula
Hedging: Hedge ratio or delta
– The number of stocks required to hedge
against the price risk of holding a short
position on one option: Call = N (d1)
Put = N (d1) - 1
• Hedge ratio is also equal to the slope of option
curve, i.e. △C/△S for the call and △P/△S for
the put.
• Hedge ratios are near zero for deep out-of-the
money options, approach 1.0 for deep in-themoney calls and approach -1.0 for deep in-themoney puts.
15-109
Class exercise
• A hedge ratio of 0.85 implies that a
hedged portfolio should consist of
A. long 0.85 calls for each short stock.
B. short 0.85 calls for each long stock.
C. long 0.85 shares for each short call.
D. long 0.85 shares for each long call.
E. none of the above.
15-110
Class exercise
• A portfolio consists of 225 shares of stock and
300 calls on that stock. If the hedge ratio for the
call is 0.4, what would be the dollar change in
the value of the portfolio in response to a one
dollar decline in the stock price?
A. -$345
B. +$500
C. -$580
D. -$520
E. none of the above
15-111
Using the Black-Scholes Formula
• Option Elasticity (similar to effective gearing
ratio)
Percentage change in the option’s value
given a 1% change in the value of the
underlying stock
• Although hedge ratios are less than 1.0, call
options have elasticities greater than 1.0.
The rate of return on a call responds more
than
one-for-one
with
stock
price
movements.
15-112
Figure 21.9 Call Option Value and
Hedge Ratio
15-113
Class exercise
• The elasticity of a stock call option is
always
A. greater than one.
B. smaller than one.
C. negative.
D. infinite.
E. none of the above.
15-114
Class exercise
• The elasticity of an option is
•
A. the volatility level for the stock that the option
price implies.
B. the continued updating of the hedge ratio as
time passes.
C. the percentage change in the stock call option
price divided by the percentage change in the
stock price.
D. the sensitivity of the delta to the stock price.
E. volatility level for the stock that the option price
implies and the percentage change in the stock
call option price divided by the percentage change
15-115
in the stock price.
Class exercise
• A put option is currently selling for $6 with an
exercise price of $50. If the hedge ratio for the
put is -0.30 and the stock is currently selling for
$46, what is the elasticity of the put?
A. 2.76
B. 2.30
C. -7.67
D. -2.76
E. -2.30
15-116
Class Exercise
• An American-style call option with six months to
maturity has a strike price of $35. The underlying
stock now sells for $43. The call premium is $12.
• What is the intrinsic value of the call?
A. $12
B. $8
C. $0
D. $23
E. None of these is correct.
•
15-117
Class Exercise
• What is the time value of the call?
A. $8
B. $12
C. $0
D. $4
E. cannot be determined without more
information.
•
15-118
Class Exercise
• If the option has delta of 0.5, what is its
elasticity?
A. 4.17
B. 2.32
C. 1.79
D. 0.5
E. 1.5
15-119
Portfolio Insurance
• Buying Puts - results in downside protection
with unlimited upside potential
• Limitations
– Tracking errors if indexes are used for the
puts
– Maturity of puts may be too short
– Hedge ratios or deltas change as stock
values change
15-120
Figure 21.10 Profit on a Protective Put
Strategy
15-121
15-122
Figure 18.12 S&P 500 Cash-to-Futures
Spread in Points at 15 Minute Intervals
15-123
Hedging On Mispriced Options
Option value is positively related to volatility:
• If an investor believes that the volatility
that is implied in an option’s price is too
low, a profitable trade is possible
• Profit must be hedged against a decline in
the value of the stock
• Performance depends on option price
relative to the implied volatility
15-124
Hedging and Delta
The appropriate hedge will depend on the
delta
Recall the delta is the change in the value of
the option relative to the change in the
value of the stock
Delta =
Change in the value of the option
Change of the value of the stock
15-125
Mispriced Option: Text Example
Implied volatility
= 33%
Investor believes volatility should = 35%
Option maturity
= 60 days
Put price P
= $4.495
Exercise price and stock price
= $90
Risk-free rate r
= 4%
Delta
= -.453
15-126
15-127
15-128
Figure 18.13 Implied Volatility of the S&P
500 Index as a Function of Exercise
Price
15-129
15-130
15-131