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Chapter 25 - Option Valuation
CHAPTER 25
OPTION VALUATION
CHAPTER WEB SITES
Section Web Address
25.1 http://www.optionseducation.org/
25.2 www.numa.com
25.3 www.ivolatility.com
www.margrabe.com/optionpricing.html
25.4 finance.yahoo.com
CHAPTER ORGANIZATION
25.1 Put-Call Parity
Protective Puts
An Alternative Strategy
The Result
Continuous Compounding: A Refresher Course
25.2 The Black-Scholes Option Pricing Model
The Call Option Pricing Formula
Put Option Valuation
A Cautionary Note
25.3 More about Black-Scholes
Varying the Stock Price
Varying the Time to Expiration
Varying the Standard Deviation
Varying the Risk-Free Rate
Implied Standard Deviations
25.4 Valuation of Equity and Debt in a Leveraged Firm
Valuing the Equity in a Leveraged Firm
Options and the Valuation of Risky Bonds
25.5 Options and Corporate Decisions: Some Applications
Mergers and Diversification
Options and Capital Budgeting
25.6 Summary and Conclusions
25-1
Chapter 25 - Option Valuation
ANNOTATED CHAPTER OULTINE
25.1 Put-Call Parity
Terminology Review:
Call – right, but not the obligation, to buy the underlying asset at the specified
price on or before a specified date
Put – right, but not the obligation, to sell the underlying asset at the specified
price on or before a specified date
Exercise or strike price – price specified in the option contract
A. Protective Puts
The strategy:
Buy one share of stock at price, S.
Buy one put option with strike price, E, and put premium, P.
Example: Suppose you buy Citigroup stock for $45, and at the
same time, you purchase a put option with a strike price of $40.
You pay $1.80 for the option, and it expires in one year. You plan
to sell the stock in one year.
Consider the following possible payoffs:
Stock
Price
Put Value Combined Value Total Gain or Loss
25 15 40 -5.80
30 10 40 -5.80
35 5 40 -5.80
40 0 40 -5.80
45 0 45 -1.80
50 0 50 +3.20
55 0 55 +8.20
60 0 60 +13.20
65 0 65 +18.20
The maximum loss has been limited to $5.80.
B. An Alternative Strategy
Suppose, instead, you buy a call option with a strike price of E and
a call price of C. You invest the remainder in a Treasury Bill.
25-2
Chapter 25 - Option Valuation
Example: A $40 call option on Citigroup stock is selling for $7.78,
and the T-bill has an interest rate of 2.5%. We want to look at the
same investment as in part A, so you invest a total of $46.80. So,
you invest 46.80 – 7.78 = 39.02 in T-bills. Consider the payoffs.
Stock
Price
Call Value T-bill
39.02(1.025)
Combined Value Total Gain or Loss
25 0 40 40 -5.80
30 0 40 40 -5.80
35 0 40 40 -5.80
40 0 40 40 -5.80
45 5 40 45 -1.80
50 10 40 50 +3.20
55 15 40 55 +8.20
60 20 40 60 +13.20
65 25 40 65 +18.20
The payoffs are the same with both strategies.
C. The Result
If the combined value is the same at the end, under all situations,
then the cost today must be the same.
This leads to the famous put-call parity (PCP) condition:
S + P = C + PV(E)
where the present value is computed using the risk-free rate.
The PCP condition can be rearranged to solve for any of the
components.
D. Continuous Compounding: A Refresher Course
Effective annual rate with continuous compounding:
EAR = eq – 1
where q is the quoted rate.
Suppose you have a quoted rate of 5% per year with continuous
compounding:
EAR = e.05 – 1 = .05127 or 5.127%
Time value of money calculations with continuous compounding:
25-3
Chapter 25 - Option Valuation
FV = PVeRt
PV = FVe-Rt
where R = continuously compounded rate, and t = number of
periods in terms of years
Example: What is the present value of $1000 to be received in
three months if the annual continuously compounded rate is 8%?
PV = 1000e-.08(3/12) = 980.20
PCP with continuous compounding
S + P = C + Ee-Rt
Example: Given the following, what does the call have to sell for
to prevent arbitrage?
S = 80; P = 6; E = 85; R = 10% with continuous compounding; t =
9 months (9/12)
80 + 6 = C + 85e-.1(9/12)
C = 86 – 78.86 = 7.14
25.2 The Black-Scholes Option Pricing Model
A. The Call Option Pricing Formula
The Formula:
C = SN(d1) – Ee-RtN(d2)
where N(d1) and N(d2) are probabilities that we compute using the
following formulas and then look the numbers up in the standard
normal tables.
d1=
ln
(
S
E
)
+
(
R+σ2
2
)
t
σ
√
t
d2=d1−σ
√
t
where is the standard deviation (or volatility) of the underlying
asset returns.
25-4
Chapter 25 - Option Valuation
Example: Consider a stock that is currently selling for $35. You are
looking at a call option that has an exercise price of $30 and
expires in 6 months. The risk-free rate is 4%, compounded
continuously. The volatility of stock returns is .25. What is the call
price?
d1=
ln
(
35
30
)
+
(
. 04+. 252
2
)
(
6
12
)
.25
√
6
12
=1 .07353
d2=1. 07353−.25
√
6
12=. 89675
From Table 25.3
N(d1) = N(1.07) = (.8554 + .8599)/2 = .8577
N(d2) = N(.90) = .8159
C = 35(.8577) – 30e-.04(6/12)(.8159) = $6.03
B. Put Option Valuation
Lecture Tip: The Black-Scholes model can also be adjusted to
solve for the value of a put option directly.
P = Ee-RtN(-d2) – SN(-d1)
where d1 and d2 are computed as before. You just change the sign
before looking it up in the table.
As an example, consider the previous information but find the
value of a put instead of a call.
N(-d1) = N(-1.07) = (.1401 + .1446)/2 = .1424
N(-d2) = N(-.9) = .1841
P = 30e-.04(6/12)(.1841) – 35(.1424) = $0.43
A slightly different answer will be found below using the PCP. The
difference is due to rounding.
Example: Consider the previous call option example and use put-
call parity to find the value of the put.
S + P = C + PV(E)
35 + P = 6.03 + 30e-.04(6/12)
P = $0.44
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Chapter 25 - Option Valuation
C. A Cautionary Note
Both PCP and the Black-Scholes model are strictly for European
options that can be exercised only at expiration. There are times
when it would be optimal to exercise a put option early, but these
models will not capture that additional value, called the “early
exercise premium.”
25.3 More on Black-Scholes
Table 25.4 illustrates the relationship between option values and
the five major inputs.
A. Varying the Stock Price
Call prices have a direct relationship with the stock price, while put
prices have an inverse relationship. This relationship is called
delta.
For European options:
Call delta = N(d1)
Put delta = N(d1) – 1
You can use delta to estimate the new option value given a small
change in the stock price.
Lecture Tip: Delta is the first derivative of the OPM with respect
to S. Gamma is the second derivative with respect to S.
An option’s delta changes as S changes, and gamma measures the
rate of change. Delta is often used to determine how many options
are needed to hedge a portfolio. As S changes, the number of
options needed will change because delta depends on S.
The larger the gamma, the smaller the change in S required to
cause a significant change in delta. The larger the change in delta,
the greater the need to rebalance the portfolio and the higher the
trading costs. Therefore, portfolio managers will often look at both
the gamma and the delta when deciding which options to use for
hedging.
B. Varying the Time to Expiration
For American calls and puts, the value of the option increases as
the time to expiration increases.
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Chapter 25 - Option Valuation
It is never optimal to exercise call options on non-dividend paying
stocks early. Therefore, the value of a European call will also
increase as time increases.
However, it may be optimal to exercise a put option early, and a
European put prevents early exercise. Therefore, there are
situations in which a shorter time to expiration would actually be
more valuable, and the relationship between European put value
and time is ambiguous.
The relationship between option value and time to expiration is
called theta.
Intrinsic value
call: max[S – E, 0]
put: max[E – S, 0]
Option value = intrinsic value + time premium
Time premium – option value associated with the time left to
expiration, decreases as expiration approaches
Example: Consider the previous option valuation examples. What
is the intrinsic value and the time premium for each option?
Call: C = 6.03
intrinsic value = max[35 – 30, 0] = 5
time premium = 6.03 – 5 = 1.03
Put: P = .44
intrinsic value = max[30 – 35, 0] = 0
time premium = .44 – 0 = .44
C. Varying the Standard Deviation
The relationship between volatility and option value is called vega.
As volatility increases, the value of the option increases.
The potential loss is limited to your premium. However, the greater
the volatility, the larger the potential gain.
D. Varying the Risk-Free Rate
The value of a call increases as the risk-free rate increases. The
opposite is true for puts. However, the impact is very small,
especially for “realistic” rates.
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Chapter 25 - Option Valuation
The relationship between the risk-free rate and option value is
called rho.
E. Implied Standard Deviations
We can observe option values, underlying asset values, exercise
prices, and risk-free rates in the market. The one variable that is
not observable is the standard deviation, or volatility.
The OPM can be used to estimate the expected standard deviation
of returns – called the implied standard deviation or implied
volatility.
There is not a closed-form solution – the easiest way to find the
implied volatility is to use an options calculator.
25.4 Valuation of Equity and Debt in a Leveraged Firm
Equity can be viewed as a call option on the assets of a business.
When a debt payment is due, stockholders can choose not to
exercise the option and the assets pass to the bondholders.
Paying off the debt is the same as exercising the option.
A. Valuing the Equity in a Leveraged Firm
Example: For simplicity, assume a firm has a 5-year, zero coupon
bond with a face value of $20 million. The firm’s assets have a
market value of $30 million. The volatility of asset returns is .3,
and the continuously compounded risk-free rate is 5%. What is the
market value of equity? Of debt?
S = 30; E = 20; t = 5; = .3; R = .05
d1=
ln
(
30
20
)
+
(
. 05+
(
. 3
)
2
2
)
(
5
)
.3
√
5=1 . 31
d2=1. 31−. 3
√
5=. 64
N(1.31) = (.9032 + .9066)/2 = .9049
N(.64) = .7389
Equity = 30(.9049) – 20e-.05(5)(.7389) = 15.63788 million
Debt = 30 – 15.63788 = 14.36212 million
25-8
Chapter 25 - Option Valuation
What is the firm’s cost of debt?
14.36212 = 20e-R(5)
R = .06623 = 6.623%
B. Options and the Valuation of Risky Bonds
A protective put can be used to reduce the risk of bonds.
Buy a put with an exercise price of $20 million
Value of risky bond + put = value of risk-free bond
14.36212 + P = 20e-.05(5)
P = 1.2139 million
Increasing the value of the put decreases the value of the risky
bond.
Value of debt = Value of risk-free debt – put
Debt = Ee-Rt – P
PCP: S + P = C + Ee-Rt
S = C + (Ee-Rt – P)
Assets = Equity + Debt
PCP and the balance sheet identity are the same.
25.5 Options and Corporate Decisions: Some Applications
A. Mergers and Diversification
Use option valuation to investigate whether diversification is a
good reason for a merger – from a stockholder’s viewpoint.
If synergies do not exist, then a merger will reduce volatility
without increasing cash flow.
Decreasing volatility decreases the value of the call option (equity)
and the put option. Decreasing the value of the put increases the
value of the debt.
So, a merger for diversification reasons, transfers value from the
stockholders to the bondholders.
25-9
Chapter 25 - Option Valuation
B. Options and Capital Budgeting
If a firm has a substantial amount of debt, stockholders may prefer
riskier projects, even if they have a lower NPV.
The riskier project increases the volatility of the asset returns. The
increased volatility increases the value of the call (equity) and the
put. The increased put value decreases the value of the debt. This
transfers wealth from the bondholders to the stockholders.
Stockholders may even prefer a negative NPV project if it
increases volatility enough.
The wealth transfer from bondholders to stockholders may
outweigh the negative NPV.
Lecture Tip: Bondholders recognize the desire of stockholders to
take on riskier projects. Consequently, provisions are put into the
bond indentures to try to prevent this wealth transfer. These
provisions add to the firm’s cost either directly through a higher
interest rate or through additional monitoring costs. These costs
are all considered agency costs.
25.6 Summary and Conclusions
25-10
