FINANCE 384: Corporate Valuation, Investment
Decisions and Risk Management
STUDENT LECTURE NOTE 8 Sp14CR
I. PUT/CALL PARITY THEOREM OPTION PRICING1
A. Two Portfolio Insurance Strategies
1. Long Stock/Protective Put:
a. Suppose that you wish to invest in a particular share,
but because it will not be part of your long-term
portfolio you are trying to determine a way to protect
it against a short-term loss. One strategy is to
purchase a put as a _____ against the long stock
position.
“Watching for riches consumeth the flesh, and the cares thereof driveth away
sleep.”
– The Holy Bible, Apocrypha—Ecclesiasticus 31:1
“Whosoever will save his life shall lose it: and whosever will lose his life for my
sake shall find it. For what is a man profited, if he shall gain the whole world,
and lose his soul?”
– Ibid., Matthew 16:25-26
“The love of money is the root of all evil.”
– Ibid., Timothy I, 1:8
b. Assume for simplicity that the put employed is _____-_____ so that S0 = K and expires at time T. The
initial value of the portfolio equals S0 + P. If the share
price falls, the put is exercised and the share can be
1
This discussion draws upon that in Bodie, Kane and Marcus, Investments, 4th
Ed. Chapter 20, (1999).
1
sold for K. If ST > S0, then the put finishes out-of-themoney and is not exercised. Thus, the portfolio is
insured against ________ risk, and allows receiving
upside price appreciation at the cost of a put.
FIGURE 8.1
Value of Protective Put Portfolio at Expiration
K > ST
K < ST
Stock
ST
ST
+ Put
K - ST
0
= Total
2. Calls plus Bills:
a. An alternative strategy that provides downside
protection with unlimited ______ potential is to buy
call options and treasury bills. For example, suppose
one call (for the purchase of 100 shares) with an
exercise price of K=100 is purchased. If exercised the
cost of the shares would be $10,000. At the same time
a T-Bill with a maturity value (at time T) equaling
$10,000 is also purchased. More generally, you
would purchase a risk-free zero-coupon bond with a
face value of K, for each call held with exercise price
K.
b. Consider the value of this portfolio at time T when
the call expires and the bill matures. The bill will be
worth K and this provides a floor for the value of the
portfolio. If the share price falls below K, then the
call expires _________. Conversely, if the share price
goes up, then the call will be exercised and worth
2
ST – K. Thus, the cost of the call provides the
opportunity to realise upside potential.
FIGURE 8.2
Value of Calls + Bills Portfolio at Expiration
K > ST
K < ST
Call Value
0
ST – K
+ Riskless Bills
K
K
= Total
3. No Arbitrage Relationship:
a. It may be noted that the terminal values to the two
portfolios shown in Figure 8.1 and 8.2 are the ____
in both the situation where K > ST or where K < ST.
Thus, since no riskless arbitrage profits should be
possible in equilibrium, the cost of the two portfolios
should be the same. It can be concluded that the
following relationship will hold:
S0 + P = C + KR-t,
(8.1)
where: R = 1 + Rf (risk-free rate), and
t = time-to-maturity as a fraction of a year.
b. Equation (8.1) may be rearranged into the form of
the put-call parity theorem that is most typically
depicted.
C – P = S0 - KR-t.
(8.2)
c. The version of the theorem shown in (8.2) is used
because it highlights the fact that the “no arbitrage”
3
difference between a call and an otherwise similar
puts’ prices should reflect the __________ between
the stock price and the discounted exercise price.
d. Students should note that assets with positive signs
are assumed to be held long, whereas a negative sign
denotes a short position. Equation (8.2) can also be
solved directly to find the “synthetic” call and put
prices.
e. The “synthetic” call price is shown in (8.3). This
demonstrates that a long call is equivalent to a ____
stock purchase, _________ the discounted exercise
price and a ____ put purchase.
C = S0 - KR-t + P.
(8.3)
f. The “synthetic” put price is shown in (8.4) below.
Thus, a long put is equivalent to the stock sold _____,
_______ the discounted exercise price and a ____ call
purchase.
P = -S0 + KR-t + C.
(8.4)
Ex. 8.1
You are given the following quotation for August 29,
2002 from the Australian Financial Review (AFR).
Assume that the Dec ’02 options expire on December 24,
2002, a 365-day year and that the 120-day Bank Bill
Swap Reference Rate equals 5.08%. Use the put/call
parity theorem to determine the price of the Dec’02 call
(K=12.25) price assuming the call expires in 117 days.
4
DERIVATIVES – SHARE OPTIONS
Exer
Series
Price
PUT OPTIONS
Fair
Value
Last
Sale
Woolworth’s Ltd Last Sale Price $12.00
Dec 02
12.25
0.76
1.01
Vol
000’s
Open
Int
133
513
Implied
Volatility
Buyer
Seller
36.78
36.67
Delta
Annual
%
Return
-.49
13.16
A8.1. n = 2 + 30 + 31 + 30 + 24 = ___ days.
C = 12 – 12.25*1/((1.0508)117/365) + 1.01 = ________.
F.Y.I. Call price using Black-Scholes = 0.957818.
II. THE BLACK-SCHOLES OPTION PRICING MODEL2
A. Originally presented in: Black, F., and M. Scholes.
(1973), “The Pricing of Options and Corporate
Liabilities,” Journal of Political Economy 81, pp. 637654.
1. Assumptions of the Model
a. ______ Capital Markets, i.e., no transaction costs or
taxes. There are no short-sale constraints. Investors
are allowed the full use of short-sale proceeds.
b. All investors can borrow or lend at the risk-free rate
which is ________ over the asset's life.
c. The stock pays no dividends, thus the model can be
used to value either American or European calls—
since neither will be exercised before expiration.
d. Trading is continuous and markets are always open.
2
This section is based on my tOSU 920b lecture notes written by Professor Rene Stulz.
5
e. Stock price changes follow a specific stochastic
process called a diffusion process, as given below:
dS
 μdt  σdz.
S
This formula basically says that the (instanttaneous) %Change in the stock price is equal to the
expected stock return per unit of time plus the
instantaneous return's standard deviation times the
change in a random standard-normally distributed
variable, z.
2. The Black-Scholes Formulas:
C(S,T,K) = S*N(d1) - K*e-rT*N(d2),
(8.5)
P(S,T,K) = S*[N(d1)-1] - K*e-rT*[N(d2)-1],
(8.6)
where:
 ln( S / K )  (r  0.5 * 2 ) * T 
d1 = 
,

*
T


(8.7)
 ln( S / K )  (r  0.5 * 2 ) * T 
d2 = 
,

*
T


(8.8)
= d1 - (σ * T ),
and where:
6
e
T
r

= natural log of 1,
= time remaining to expiration in years,
= continuously compounded riskless rate,
= standard deviation of the continuously
compounded annual rate of return on the
stock
N(d) = the probability that a deviation less than d
will occur in a normal distribution with a
mean of zero and a standard deviation
equal to one.
We need to find T, r, K, S, and (S).
a. To find T, _____ number of days between today
and expiration date (calendar days), then divide
by 365.
b. To find S read WSJ.
c. To find K read the option contract (or check the
WSJ listing).
d. To find r, need to find the effective annual yield
of a T-bill that matures as close to expiration as
possible, then use Ed Kane Annualised DiscountBasis Formula to find Ytm.
 360 * d 
EK Ann Ytm: rt,T = 
,
360

(d
*
n)


(8.9)
where: d = annualised discount asked yield, and
n = days to maturity, from t to T.
7
e. (S) standard deviation of stock.
By assumption, the continuously-compounded
returns will follow a ______ distribution. All daily,
(or weekly) continuously-compounded returns also
follow the same distribution, i.e., they have same
mean and standard deviation. If S(j) is the stock
price at date j, then for “n” days an estimate of the
mean of the continuously compounded returns is:
 1  n 1  S ( j  1) 
.
̂ =   *  ln 

 n  j 0  S ( j) 
(8.10)
Furthermore an unbiased estimate of the standard
deviation is:
2
 1
n

1




S
(
j

1
)

 =  
*  ln 
 ˆ  


  n  1 j  0   S ( j ) 
 
1/ 2
^
. (8.11)
Past stock prices can be found in WSJ. How should
n be chosen? Not too long and not too short. (Prof.
René Stulz, Finance 410 Lecture Note #8, University
of Rochester)
If daily data we should probably use about 50 to 200
observations. If using weekly data try 40-60 weeks.
If monthly data should probably use at least 30-50
price observations.
8
3. Now, an example.
Ex. 8.2
Use the Black-Scholes Option Pricing model in
F384_LN08_SS_Sp14CR spreadsheet and the following
information to determine the theoretical prices for the
Yum! Brands (YUM) Dec’07 Call (K=30, 35, 40 & 45)
and the corresponding Dec’07 Puts. Assume a 365-day
year, the options expire on 21 December, 2007 (i.e., 50
days), the ASK Yld for a T-Bill which matures closest to
the expiration is 3.84% and that the relevant variance for
the stock, 2(RYUM), has been calculated as 0.1134128
using the approach in equation (8.11) above. Determine the
percentage by which these model prices differ from the
CBOE option quotes below, from 1 November, 2007.
YUM (YUM BRANDS INC)
39.08 -0.01
Bid N/E Ask N/E Size N/ExN/E Vol 5732200
Nov 01, 2007 @ 16:01 ET
Last
Sale
Calls
Net
Vol
Open
Puts
Int
Last
Sale
Net
Vol
Open
Int
07 Dec 30.00 (YUMLF-E)
8.70
0.0
20
0 07 Dec 30.00 (YUMXF-E)
0.05
0.0
15
18
07 Dec 35.00 (YUMLG-E)
4.60
0.0
28
104 07 Dec 35.00 (YUMXG-E)
0.35
+0.05
2
607
07 Dec 40.00 (YUMLH-E)
1.75
-0.40 148
3123 07 Dec 40.00 (YUMXH-E)
1.71
+0.40
71
259
07 Dec 45.00 (YUMLI-E)
0.25
-0.05 280
231 07 Dec 45.00 (YUMXI-E)
5.17
0.0
2
2
A8.2.
Results for the Black-Scholes option calculations are
shown in the Spreadsheet 8 excerpt below.
First determine the effective riskless rate using the Ed
Kane Annualised Discount-Basis Ytm approach (given
in equation (8.9)) and then find d1 and d2.
9
 360 * 0.0384 
EK Ann Ytm: rt,T = 
= _______%.

 360  (0.0384 * 50) 
Excel Eqn.(I52): (=H4) =(360*B4)/(360-(B4*D4))
52
53
54
55
56
57
58
59
60
61
62
63
B
Stock (S)
d1
2.22611
1.58394
0.98938
0.43585
-0.08193
-0.56832
-1.02689
C
$39.08
N(d1)
0.9869966
0.9433958
0.8387603
0.6685287
0.4673508
0.2849105
0.1522361
D
Term(t)
d2
2.10146
1.45929
0.86473
0.31121
-0.20657
-0.69296
-1.15153
E
0.136986
F.Y.I.
F
2(R )
VARP(fn)
G
0.11341280
0.11581568
Black-Scholes Option Prices
N(d2)
Strike(K)
Call
0.9822000
30.00
$9.2612
0.9277575
32.50
$6.8748
0.8064072
35.00
$4.7034
0.6221797
37.50
$2.9174
0.4181712
40.00
$1.6254
0.2441677
42.50
$0.8119
0.1247564
45.00
$0.3650
H
Rf Rate
Call Quotes
$8.70
$4.60
$1.75
$0.25
I
3.86059%
Put
$0.0230
$0.1234
$0.4388
$1.1396
$2.3345
$4.0077
$6.0476
The detailed calculations shown below are for the call
and put with the $35.00 strike price.
 ln(39.08 / 35)  (.0386059  (0.5 * 0.1134128)) * (50 / 365) 
d1 = 
,
0.1134128 * (50 / 365)


 0.110262766  0.013056479 
=
= 0._____5727.

0.124643492


Excel Eqn.(B59):
=(LN($C$52/L59)+($I$52+($G$52/2))*$E$52)/(($G$52^0.5)*($E$52^0.5))
d2 = d1 - (σ * T )
10
= 0.989375727 –
0.1134128 * (50/365) )
= 0._____2235.
Excel Eqn.(D59): =B59-($G$52^0.5)*($E$52^0.5)
Next determine N(d1) and N(d2) using a Cumulative
Standard Normal Table (or using the appropriate Excel
function—which is much easier).
N(d1) = NORMSDIST(d1) = 0.____603.
Excel Eqn.(C59): =NORMSDIST(B59)
N(d2) = NORMSDIST(d2) = 0.____072.
Excel Eqn.(E59): =NORMSDIST(D59)
Find the discounted exercise price as:
K*e-(r*T) = (35)*(2.71828 -(.0386059)(0.136986))
= 35 * 0.994725492 = __.____922.
Then the option prices are found as:
Dec’07 Call(K=35) = (39.08 * 0.8387603)
- (34.8153922 * 0.8064072)
= $______ vs. quote(=$4.60), is _____% higher.
Excel Eqn.(G59):
=($C$52*$C59)-(($F59*EXP(-$I$52*$E$52))*$E59)
11
Dec’07 Put (K=35) = [(39.08)*(0.8387603-1)]
- [(34.8153922)*(0.8064072-1)]
= $_______ vs. Quote(=$0.35), is _____% higher.
Excel Eqn.(I59):
=($C$52*($C59-1))-($F59*EXP(-$I$52*$E$52))*($E59-1)
The results of comparing the Black-Scholes calculated
option prices to the CBOE quotes are shown in the
following figure. Although some options appear to be
underpriced by the model, most appear to be
overpriced. The most likely reason is that the stock
price standard deviation has been estimated on the
“____” side.
B/S Call Call Quote
$9.2612
$8.70
$4.7034
$4.60
$1.6254
$1.75
$0.3650
$0.25
%Diff
+6.45
+2.25
-7.12
+46.0
B/S Put
$0.0230
$0.4388
$2.3345
$6.0476
Put Quote
$0.05
$0.35
$1.71
$5.17
%Diff
-54.0
+25.4
+36.5
+17.0
B. Interpretation of the Model
In a risk-neutral economy the two terms in the B/S call
model (8.5) have natural interpretations.
The first term is the discounted expected value of the
terminal stock price, given that the terminal stock price
_______ the exercise price, times the probability that
the terminal stock price is higher than K.
12
The second term is the discounted exercise price times
the probability that the terminal stock price ______ the
exercise price.
C. Instructive Query
Why does the B-S model use a measure of total risk
(the standard deviation) rather than some measure of
market risk? Ans: Simply because the option's value is
highly correlated with—and responds to—the total risk
of the stock, not just the stock's market risk.
The question largely turns on what risky influences are
supposed to be relevant and thereby priced. Due to
diversification, in the CAPM only market factors are
supposed to cause any change in portfolio return.
Cannot diversify away non-market risk of an option
because its return depends directly on one individual
stock.
Q: Will holding a portfolio of options diversify away
nonmarket risk?
A: ___.
III. Comparative Statics of the Model (The Greeks)
13
A. The B/S Model may be differentiated with respect to
its different parameters, S, T, K, σ(R) and r. These
resulting relationships are a prediction of how the
option prices will react in response to an incremental
change in the option variable, ceteris paribus.
B. Results for Calls
C/S
= Abs Value of │N(d1)│ > 0
= Delta = 1/β.
C/K = -e-rT*N(d2) < 0.
(8.12)
(8.13)
C/T = Ke-rT[σZ(d2)/2*( T )+rN(d2)] > 0 (8.14)
= Theta().
C/r = T*Ke-rT*N(d2) > 0 = Rho (P).
(8.15)
C/σ = S* ( T )*Z(d1) > 0 = Lambda.
(8.16)
D/S = 2C/S2 = Z(d1)/[S*σ*( T )] > 0
(8.17)
= Gamma().
Z(d) is the standard normal density at d. This is the
incremental change in the standard normal distribution
at d. It is equal to the following expression:
( d ) / 2

 N(d)   e
Z(d) = 
=
.


d

  2 * 
2
C. Results For Puts
14
(8.18)
P/S = C/S - 1 = N(d1) – 1 < 0 = Delta. (8.19)
P/K = C/K + e-rT > 0.
(8.20)
P/T = C/T - K*r*e-rT > 0 = Theta().
(8.21)
P/σ = C/σ > 0.
(8.22)
P/r = C/r - K*T*e-rT < 0.
(8.23)
2P/S2 = d/S = Z(d1)/(S*σ* T ) > 0 =  (8.24)
D. The Importance of Delta
1. Delta describes the change in the value of the
option for a small change in the value of the
underlying asset (ceteris paribus).
2. Option deltas are often referred to as hedge ratios
because they can be used to determine the number
of calls that would be written or puts that would be
purchased to hedge a long stock position. This was
previously discussed in F384 LN #5 and the
Option Strategies Handout.
3. Call option deltas range from about ____ for
deeply out-of-the-money calls, to 0.5 for at-themoney calls, to 1.0 for deeply in-the-money calls.
4. Put option deltas decline from nearly ____ for
deeply out-of-the-money puts, to -0.5 for at-themoney puts, to -1.0 for deeply in-the-money puts.
15
5. These delta characterisations will be illustrated in
the next example using the results determined in
Ex. 8.2.
Ex. 8.3
Use the data inputs previously developed for the Yum!
Brands Inc. Dec’07 options to calculate the Black/Scholes
option prices and deltas for a call and put with a strike
price of $37.50. Do this for stock prices ranging from $30
to $45 in increments of $2.50.
A8.3. A facsimile from Spreadsheet 8 showing the results of
calculating the Yum! Brands Inc. Dec’07 (K=37.50)
Black/Scholes Call and Put option values as well as the
deltas is shown below. As the actual calculations for the
option values have been previously shown, they are not
repeated. However, the Excel equations used here are
shown for the case where S=30.
5
6
7
8
9
10
11
12
13
14
15
16
17
J
Ex. 8.3
K
L
M
Graphing Call and Put Deltas
N
Strike(K)
O
37.50
Term(t)
0.1369863
2(R )
0.1134128
Rf Rate
3.86059%
Stock (S)
30.00
32.50
35.00
37.50
40.00
42.50
45.00
d1
-1.6855037
-1.0433306
-0.4487711
0.1047506
0.6225355
1.1089197
1.5674949
Call Delta
N(d1)
0.045946
0.148398
0.326798
0.541713
0.733205
0.866268
0.941500
d2
-1.81015
-1.16797
-0.57341
-0.01989
0.49789
0.98428
1.44285
N(d2)
0.0351364
0.1214087
0.2831820
0.4920644
0.6907199
0.8375101
0.9254688
Call
$0.0677
$0.2941
$0.8746
$1.9592
$3.5628
$5.5754
$7.8455
Equation to find d1:
Excel Eqn.(K11):
16
P
Q
Put
$7.3699
$5.0963
$3.1768
$1.7614
$0.8650
$0.3776
$0.1477
Put Delta
-0.95405
-0.85160
-0.67320
-0.45829
-0.26679
-0.13373
-0.05850
=(LN($J11/$O$5)+($O$7+($M$7/2))*$K$7)/(($M$7^0.5)*($K$7^0.5))
Equation to find N(d1) and B/S Call Delta:
Excel Eqn.(L11): =NORMSDIST(K11)
Equation to find d2:
Excel Eqn.(M11): =$K11-($M$7^0.5)*($K$7^0.5)
Equation to find N(d2):
Excel Eqn.(N11): =NORMSDIST(M11)
Equation to find B/S Call Price:
Excel Eqn.(O11):
=($J11*$L11)-(($O$5*EXP(-$O$7*$K$7))*$N11)
Equation to find B/S Put Price:
Excel Eqn.(P11):
=($J11*($L11-1))-($O$5*EXP(-$O$7*$K$7))*($N11-1)
Equation to find B/S Put Delta:
Excel Eqn.(Q11): =L11-1
Below you will find the graphs created using an X-Y
type graph where Stock Price is the X-variable and
Delta is the Y-variable.
17
Call Option Deltas
1.000
Delta
0.800
0.600
0.400
0.200
0.000
$30.0
$32.5
$35.0
$37.5
$40.0
$42.5
Stock Price
$45.0
Call Delta
Put Option Deltas
Delta
0.000
$30.0
-0.200
$32.5
$35.0
$37.5
$40.0
$42.5
$45.0
-0.400
-0.600
-0.800
-1.000
Stock Price
P ut Delt a
18