CHAPTER. 22. Options and Corporate Finance: Basic Concepts. 22.1 Options 22.2 Call Options 22.3 Put Options 22.4 Selling Options 22.5 Reading The Wall Street Journal 22.6 Combinations of Options 22.7 Valuing Options 22.8 An Option‑Pricing Formula 22.9 Stocks and Bonds as Options
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22.2 Call Options
22.3 Put Options
22.4 Selling Options
22.5 Reading The Wall Street Journal
22.6 Combinations of Options
22.7 Valuing Options
22.8 An Option‑Pricing Formula
22.9 Stocks and Bonds as Options
22.10 CapitalStructure Policy and Options
22.11 Mergers and Options
22.12 Investment in Real Projects and Options
22.13 Summary and Conclusions
Chapter OutlineAlmost every issue of corporate stocks and bonds has option features.
In addition, capital structure and capital budgeting decisions can be viewed in terms of options.
22.1 OptionsIntrinsic Value
Speculative Value
+
=
Options Contracts: PreliminariesWhen exercising a call option, you “call in” the asset.
22.2 Call OptionsC= Max[ST –E, 0]
Where
ST is the value of the stock at expiry (time T)
E is the exercise price.
C is the value of the call option at expiry
Buy a call
60
40
Option payoffs ($)
20
80
120
20
40
60
100
50
Stock price ($)
–20
Exercise price = $50
–40
60
40
Option payoffs ($)
20
80
120
20
40
60
100
50
Stock price ($)
–20
Exercise price = $50
Sell a call
–40
40
Option payoffs ($)
20
80
120
20
40
60
100
Stock price ($)
–20
–40
Call Option ProfitsBuy a call
10
50
–10
Exercise price = $50; option premium = $10
Sell a call
When exercising a put, you “put” the asset to someone.
22.3 Put OptionsP= Max[E – ST, 0]
60
50
40
Option payoffs ($)
20
Buy a put
0
80
0
20
40
60
100
50
Stock price ($)
–20
Exercise price = $50
–40
40
Option payoffs ($)
20
Sell a put
0
80
0
20
40
60
100
50
Stock price ($)
–20
Exercise price = $50
–40
–50
60
40
Option payoffs ($)
20
Sell a put
10
Stock price ($)
80
50
20
40
60
100
–10
Buy a put
–20
Exercise price = $50; option premium = $10
–40
The seller (or writer) of an option has an obligation.
The purchaser of an option has an option.
Buy a call
40
Option payoffs ($)
Buy a put
Sell a call
Sell a put
10
Stock price ($)
50
40
60
100
Buy a call
–10
Buy a put
Sell a put
Exercise price = $50; option premium = $10
Sell a call
–40
This option has a strike price of $135;
a recent price for the stock is $138.25
July is the expiration month
This makes a call option with this exercise price inthemoney by $3.25 = $138¼ – $135.
Puts with this exercise price are outofthemoney.
On this day, 2,365 call options with thisexercise price were traded.
The CALL option with a strike priceof $135 is trading for $4.75.
Since the option is on 100 shares of stock, buying this option would cost $475 plus commissions.
On this day, 2,431 put options with thisexercise price were traded.
The PUT option with a strike price of $135 is trading for $.8125.
Since the option is on 100 shares of stock, buying this option would cost $81.25 plus commissions.
If you understand this, you can become a financial engineer, tailoring the riskreturn profile to meet your client’s needs.
22.6 Combinations of OptionsProtective Put payoffs
Value at expiry
$50
Buy the stock
Buy a put with an exercise price of $50
$0
Value of stock at expiry
$50
Value at expiry
Buy the stock at $40
$40
Protective Put strategy has downside protection and upside potential
$0
$10
$40
$50
Buy a put with exercise price of $50 for $10
Value of stock at expiry
$40
$30
Covered Call StrategyValue at expiry
Buy the stock at $40
Covered Call strategy
$0
Value of stock at expiry
$40
$50
Sell a call with exercise price of $50 for $10
$40
Buy a call with exercise price of $50 for $10
40
Option payoffs ($)
30
Stock price ($)
40
60
30
70
Buy a put with exercise price of $50 for $10
$50
A Long Straddle only makes money if the stock price moves $20 away from $50.
This Short Straddle only loses money if the stock price moves $20 away from $50.
Option payoffs ($)
Sell a put with exercise price of
$50 for $10
Stock price ($)
30
70
40
60
$50
–30
Sell a call with an
exercise price of $50 for $10
–40
Portfolio value today = c0 +
(1+ r)T
bond
PutCall Parity: p0 + S0 = c0 + E/(1+ r)TPortfolio payoff
Call
Option payoffs ($)
25
Stock price ($)
25
Consider the payoffs from holding a portfolio consisting of a call with a strike price of $25 and a bond with a future value of $25.
Portfolio payoff
Portfolio value today = p0 + S0
Option payoffs ($)
25
Stock price ($)
25
Consider the payoffs from holding a portfolio consisting of a share of stock and a put with a $25 strike.
Portfolio value today = p0 + S0
E
= c0 +
Option payoffs ($)
Option payoffs ($)
(1+ r)T
25
25
Stock price ($)
Stock price ($)
25
25
PutCall Parity: p0 + S0 = c0 + E/(1+ r)TSince these portfolios have identical payoffs, they must have the same value today: hence
PutCall Parity: c0 + E/(1+r)T = p0 + S0
This section considers the value of an option prior to the expiration date.
A much more interesting question.
22.7 Valuing OptionsCall Put
The value of a call option C0 must fall within
max (S0 – E, 0) <C0<S0.
The precise position will depend on these factors.
Profit
ST
Call
Option payoffs ($)
25
Time value
Intrinsic value
ST
E
Outofthemoney
Inthemoney
loss
The value of a call option C0 must fall within max (S0 – E, 0) <C0<S0.
Then we will graduate to the normal approximation to the binomial for some realworld option valuation.
22.8 An Option‑Pricing Formula$28.75 = $25×(1.15)
$21.25 = $25×(1 –.15)
Binomial Option Pricing ModelSuppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or $21.25. The riskfree rate is 5%. What is the value of an atthemoney call option?
S0
$25
S0
S1
C1
$28.75
$3.75
$25
$21.25
$0
Borrow the present value of $21.25 today and buy 1 share.
The net payoff for this levered equity portfolio in one period is either $7.50 or $0.
The levered equity portfolio has twice the option’s payoff so the portfolio is worth twice the call option value.
S0
S1
debt
portfolio
C1
( – ) =
– $21.25
$7.50
$28.75
=
$3.75
$25
– $21.25
$21.25
$0
=
$0
The value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt:
S0
S1
debt
portfolio
C1
( – ) =
– $21.25
$7.50
$28.75
=
$3.75
$25
– $21.25
$21.25
$0
=
$0
We can value the call option todayas half of the value of thelevered equity portfolio:
S0
S1
debt
portfolio
C1
( – ) =
– $21.25
$7.50
$28.75
=
$3.75
$25
– $21.25
$21.25
$0
=
$0
$2.38
The Binomial Option Pricing ModelIf the interest rate is 5%, the call is worth:
S0
S1
debt
portfolio
C1
( – ) =
– $21.25
$7.50
$28.75
=
$3.75
$25
– $21.25
$21.25
$0
=
$0
The most important lesson (so far) from the binomial option pricing model is:
the replicating portfolio intuition.
Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.
D =
Swing of stock
Delta and the Hedge RatioThe delta of a put option is negative.
Value of a call = Stock price ×Delta – Amount borrowed
$2.38 = $25 × ½ – Amount borrowed
Amount borrowed = $10.12
+

´
q
V
(
U
)
(
1
q
)
V
(
D
)
=
V
(
0
)
+
(
1
r
)
f
The RiskNeutral Approach to ValuationS(U), V(U)
We could value V(0) as the value of the replicating portfolio. An equivalent method is riskneutral valuation
q
S(0), V(0)
1 q
S(D), V(D)
S(U), V(U)
S(0) is the value of the underlying asset today.
q
q is the riskneutral probability of an “up” move.
S(0), V(0)
1 q
S(D), V(D)
S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.
V(U) and V(D) are the values of the asset in the next period following an up move and a down move, respectively.
q
´
+

´
q
V
(
U
)
(
1
q
)
V
(
D
)
=
V
(
0
)
S(0), V(0)
+
(
1
r
)
f
1 q
S(D), V(D)
´
+

´
q
S
(
U
)
(
1
q
)
S
(
D
)
=
S
(
0
)
+
(
1
r
)
f
The RiskNeutral Approach to ValuationA minor bit of algebra yields:
´
$
28
.
75
$
25
(
1
.
15
)
$28.75,C(D)
q
$25,C(0)
=
´

$
21
.
25
$
25
(
1
.
15
)
1 q
$21.25,C(D)
Example of the RiskNeutral Valuation of a Call:Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The riskfree rate is 5%. What is the value of an atthemoney call option?
The binomial tree would look like this:
´

(
1
r
)
S
(
0
)
S
(
D
)
f
=
q

S
(
U
)
S
(
D
)
´

(
1
.
05
)
$
25
$
21
.
25
$
5
=
=
=
q
2
3

$
28
.
75
$
21
.
25
$
7
.
50
Example of the RiskNeutral Valuation of a Call:The next step would be to compute the risk neutral probabilities
$28.75,C(D)
2/3
$25,C(0)
1/3
$21.25,C(D)

C
(
U
)
$
28
.
75
$
25
=

C
(
D
)
max[$
25
$
28
.
75
,
0
]
Example of the RiskNeutral Valuation of a Call:After that, find the value of the call in the up state and down state.
$28.75, $3.75
2/3
$25,C(0)
1/3
$21.25, $0
+

´
q
C
(
U
)
(
1
q
)
C
(
D
)
=
C
(
0
)
+
(
1
r
)
f
´
+
´
2
3
$
3
.
75
(
1
3
)
$
0
=
C
(
0
)
(
1
.
05
)
$28.75,$3.75
$
2
.
50
2/3
=
=
C
(
0
)
$
2
.
38
(
1
.
05
)
$25,C(0)
1/3
$21.25, $0
Example of the RiskNeutral Valuation of a Call:Finally, find the value of the call at time 0:
$25,$2.38
This riskneutral result is consistent with valuing the call using a replicating portfolio.
=
´

´
rT
C
S
N(
d
)
Ee
N(
d
)
0
1
2
2
σ
+
+
ln(
S
/
E
)
(
r
)
T
2
=
d
1
s
T
=

s
d
d
T
2
1
The BlackScholes ModelThe BlackScholes Model is
Where
C0 = the value of a European option at time t = 0
r = the riskfree interest rate.
N(d) = Probability that a standardized, normally distributed, random variable will be less than or equal to d.
The BlackScholes Model allows us to value options in the real world just as we have done in the 2state world.
Find the value of a sixmonth call option on the Microsoft with an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsicvalue of the option is $10—our answer must be at least that amount.
+
2
ln(
S
/
E
)
(
r
.
5
σ
)
T
=
d
1
s
T
+
+
2
ln(
160
/
150
)
(.
05
.
5
(
0
.
30
)
).
5
=
=
d
0
.
5282
1
0
.
30
.
5
=

s
=

=
d
d
T
0
.
52815
0
.
30
.
5
0
.
31602
2
1
The BlackScholes ModelLet’s try our hand at using the model. If you have a calculator handy, follow along.
First calculate d1 and d2
Then,
=
´

´
rT
C
S
N(
d
)
Ee
N(
d
)
0
1
2
=
d
0
.
5282
1
=
d
0
.
31602
2

´
=
´

´
.
05
.
5
C
$
160
0
.
7013
150
e
0
.
62401
0
=
C
$
20
.
92
0
The BlackScholes ModelN(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
The underlying asset comprise the assets of the firm.
The strike price is the payoff of the bond.
If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an inthemoney call, they will pay the bondholders and “call in” the assets of the firm.
If at the maturity of the debt the shareholders have an outofthemoney call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire.
22.9 Stocks and Bonds as OptionsThe underlying asset comprise the assets of the firm.
The strike price is the payoff of the bond.
If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an inthemoney put.
They will put the firm to the bondholders.
If at the maturity of the debt the shareholders have an outofthemoney put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire.
22.9 Stocks and Bonds as OptionsE
c0 = S0 + p0 –
(1+ r)T
Value of a riskfree bond
Value of a call on the firm
Value of a put on the firm
Value of the firm
=
–
+
22.9 Stocks and Bonds as OptionsStockholder’s position in terms of put options
Stockholder’s position in terms of call options
For example, recall the incentive shareholders in a levered firm have to take large risks.
22.10 CapitalStructure Policyand OptionsAssets BV MV Liabilities BV MV
Cash $200 $200 LT bonds $300
Fixed Asset $400 $0 Equity $300
Total $600 $200 Total $600 $200
What happens if the firm is liquidated today?
$200
$0
The bondholders get $200; the shareholders get nothing.
NPV = –$200 +
(1.10)
Selfish Strategy 1: Take Large RisksThe Gamble Probability Payoff
Win Big 10% $1,000
Lose Big 90% $0
Cost of investment is $200 (all the firm’s cash)
Required return is 50%
Expected CF from the Gamble = $1000 × 0.10 + $0 = $100
NPV = –$133
PV of Stocks With the Gamble:
$30
$70
$20 =
$47 =
(1.50)
(1.50)
Selfish Stockholders Accept Negative NPV Project with Large RisksThe stocks are worth more with the high risk project because the call option that the shareholders of the levered firm hold is worth more when the volatility of the firm is increased.
$42.55
22.11 Mergers and OptionsThecontingent value rights paidthe difference between $42.55 andGeneral Mills’ stock price in oneyear up to a maximum of $4.55.
Cash payment to newly issued shares
$4.55
$0
Value of General Mills in 1 year
Strike $42.55
$42.55
$42.55
– $38.00
$4.55
$42.55
Sell a put
Strike $38
–$38
22.11 Mergers and OptionsCash payment to newly issued shares
$0
Value of General Mills in 1 year
$38
Value of General Mills in 1 year
Value of a share
Value of a share plus cash payment
$42.55
$4.55
$0
Value of General Mills in 1 year
$38
$42.55
The next chapter will take up this point.
22.12 Investment in Real Projects & Optionsc0–
= S0 + p0
(1+ r)T
22.13 Summary and Conclusions1.Current price of underlying stock.
2. Dividend yield of the underlying stock.
3. Strike price specified in the option contract.
4. Riskfree interest rate over the life of the contract.
5. Time remaining until the option contract expires.
6. Price volatility of the underlying stock.