WEMBA 2000
Real Options
1
What is an Option?
Definitions: An option is an agreement between two parties that gives the purchaser of the option
the right, but not the obligation, to buy or sell a specific quantity of an asset at a specified
price during a designated time period.
A Call is the right to purchase the underlying asset
A Put is the right to sell the asset
The Strike Price is the pre-specified price at which the option holder can buy (in the
case of a call) or sell (in the case of a put) the asset to the option seller
The Expiration date is the date (and time) after which the option expires
The Notional Amount is the quantity of the underlying asset that the option buyer has the
right to buy or sell under the terms of the option contract
The Premium is the price of the option contract (the amount paid by the buyer to the seller).
The option buyer’s maximum “downside” (possible loss) is the amount of the premium. Option
premia are quoted as percentage of notional amount.
To Exercise means to invoke the right to buy or sell the underlying asset under the terms
of the option contract
The Seller (or Writer) of the option receives a payment (the Option Premium) that then
obligates him to sell (in the case of a call) or buy (in the case of a put) the asset.
WEMBA 2000
Real Options
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Payoff Diagram for a Long Call Option
Strike Price = K
Price of Underlying Asset = S
Profit/Loss Analysis
Option
payoff
At expiration, there are two possible outcomes:
(i) S >=K. Exercise the call and purchase the
asset for K. Asset has market value S
Payoff = S - K
(ii) S <K. Option expires worthless.
Payoff = 0
K
S: Price of
Underlying Asset
at expiration
General Formula for call payoff
Long call payoff = Max (0, S - K)
WEMBA 2000
Real Options
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Payoff Diagram for a Long Put Option
Option
payoff
Strike Price = K
Price of Underlying Asset = S
Profit/Loss Analysis
At expiration, there are two possible outcomes:
(i) S <= K Exercise the put and sell the asset
for K. Asset has market value S
Payoff = K - S
(ii) S >K. Option expires worthless.
Payoff = 0
K
S: Price of
Underlying Asset
at expiration
General Formula put payoff
Long put payoff = Max (0, K - S)
WEMBA 2000
Real Options
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Payoff Diagrams for Short Options Positions
Short Call Position
Strike Price = K
Short Put Position
Strike Price = K
Option
payoff
Option
payoff
S: Price of
Underlying Asset
at expiration
K
Short call payoff = Min (0, K-S)
K
S: Price of
Underlying Asset
at expiration
Short put payoff = Min (0, S-K)
Notes:
(i) The short position payoff diagrams are mirror images of the long positions.
(ii) The above payoff charts do not include the cost of buying (or income from selling) the
option.
Question:
Which is potentially riskier, a long option position
or a short option position?
WEMBA 2000
Real Options
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Payoff Diagrams for some Option Combinations
"Covered Call" or "Buy-Write"
"Call Spread"
Position
Profit/Loss
Position
Profit/Loss
Profit/Loss
from stock
K
Profit/loss from
short call
Profit/loss from
short call
net profit/loss
net profit/loss
S: Price of
Underlying Asset
at expiration
S: Price of
Underlying Asset
at expiration
Profit/loss from
long call
Note: The above profit/loss charts include the cost of buying (or income from selling) the option
WEMBA 2000
Real Options
Factors that Influence Option Prices
The six variablesthat affect option prices:
1. Current (spot) price on the underlying security
2. Strike price
3. Time to expiration
4. Implied (expected) volatility on the underlying security
5. The riskfree rate over the time period of the option
6. Any dividends or other cashflows that will be paid or received on the underlying asset during
the life of the option
6
WEMBA 2000
Real Options
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Valuation of Options: Put-Call Parity
We construct two portfolios and show they always have the same payoffs, hence they must cost the same amount.
Portfolio 1: Buy 1 share of the stock today for price S0 and borrow an amount PV(X) = X e-rT
How much will this portfolio be worth at time T ?
Cashflow
Time = 0
Position
Cashflow
Time = T
Buy Stock
-S0
ST
Borrow
PV(K)
-K
Net: Portfolio 1
Portfolio
payoff
at time T
PV(K) - S0
ST - K
S
Payoff from
stock
net payoff
K
-K
Payoff from
borrowing
ST
WEMBA 2000
Real Options
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Valuation of Options: Put-Call Parity
Portfolio 2: Buy 1 call option and sell 1 put option with the same maturity date T and the same strike price K.
How much will this portfolio be worth at time T ?
Position
Cashflow
Time = 0
Buy Call
-c
Sell Put
p
Net: Portfolio 2
Portfolio
payoff
at time T
Cashflow: Time = T
ST < K
ST > K
0
ST - K
- (K - ST )
0
ST - K
ST - K
p-c
Payoff on
long call
net payoff
K
-K
Payoff on
short put
ST
WEMBA 2000
Real Options
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Valuation of Options: Put-Call Parity
Payoff from Portfolio 1 and Portfolio 2 is the same, regardless of level of ST , hence cost
of both portfolios (cashflows at time T = 0 ) must be the same.
Hence:
S0 - PV(K) = c - p
Put-Call Parity
Rearranging:
c = p + S0 - PV(K)
(1)
Put-Call parity: a worked example
Stock is selling for $100. A call option with strike price $90 and maturity 3 months has
a price of $12. A put option with strike price $90 and maturity 3 months has a price of $2.
The risk-free rate is 5%.
Question: Is there an arbitrage? Test Put-Call parity:
Right-hand side of (1): p + S0 - PV(K) = 2 + 100 - 90 e -0.05*0.25
= 13.12
Left-hand side of (1): c = 12
 13.12 !
Market Price of c is too low relative to the other three.
Buy the call, and Sell the "replicating portfolio".
WEMBA 2000
Real Options
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Valuation of Options: Put-Call Parity Example
Position
Buy Call
Sell Put
Sell stock
Cashflow
Time = 0
- 12
2
100
Cashflow: Time = T
ST < 90
ST > 90
0
ST - 90
ST - 90
0
- ST
- ST
Lend money
-90 e 0.05*0.25
90
90
Net Payoff
1.12
0
0
Result: arbitrage profit of 1.12 today, regardless of the value of the stock price!
WEMBA 2000
Real Options
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Valuation of Options: Black-Scholes Formula for Calls and Puts
S = Current stock price
K = Strike price on the option
T = Time to maturity of the option in years (e.g. 5 months = 5/12 = 0.417)
r = Riskfree rate of interest
 = Expected ("Implied") volatility (standard deviation) of the underlying stock over the
life of the option
Black-Scholes Call Price c = S N( d1 ) - X e -rT N( d2 )
(2)
where: d1 = ln (S/k) + (r +  2 / 2) T
 T
d2 = d1 -  T
N(d ) = cumulative standard normal probability of value less than d
Black-Scholes Put Price p = X e -rT N( - d2 ) - S N( - d1 )
(3)
WEMBA 2000
Real Options
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Valuation of Options: Black-Scholes Formula for Calls and Puts
Example: Options on Compaq stock
On Dec 20, Compaq stock closed at $76.75
3 month riskfree rate: 5.5% (e.g. yield on 3 month T-bill)
Estimated volatility: 41%
What are the values of 3 month call and put options with Strike = $75 ?
Black-Scholes formula inputs and calculations:
Observed inputs:
Option contract inputs:
S = 76.75
r = 5.5%
K = 75
T = 0.25
Estimated input (the future level of
volatility is not observable)
 = 41%
d1 = [ ln (76.75/75) + (0.055 + 0.412 / 2) 0.25 ]
0.41 0.25
= 0.2821
d2 = d1 -  T
= 0.0771
N(d1) = 0.6111
N(d2) = 0.5307
[obtained from Excel "normsdist" function]
[obtained from Excel "normsdist" function]
c = 7.638
p = 4.864
[from equation (2) ]
[from equation (3) ]
WEMBA 2000
Real Options
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Binomial Pricing Method 1: Creating a replicating portfolio
Bluejay Corp share price is $20. Possible price at the end of three months: either $22 or $18.
Value a call option on Bluejay with strike 21, expiration 3 months.
Riskfree rate = 2% over 3 months.
(i)
Share Price
(ii)
Option Value
22
20
22-21 = 1
c
18
{Reminder: the value
of the call
at expiration is
Max[0, S - K]}
0
(a) Create a portfolio: purchase one share of the stock, and borrow money at the riskfree rate
HINT: Choose amount to borrow so that the portfolio outcome is zero in one scenario
(iii)
Portfolio: Buy 1 share &
borrow PV(18)
22-18=4
20PV(18)
=2.35
18-18=0
Compare the payoff between
the call option and the portfolio.
How many call options do we
need to buy to make the payoffs
identical?
WEMBA 2000
Real Options
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Binomial Pricing Method 1: Creating a replicating portfolio
(iia)
=(ii)*4
(iii)
Option Value (4 calls)
4
Portfolio: Buy 1 share &
borrow PV(18)
4
equal
4*c
2.35
0
0
(b) Calculate number of call options to buy so that the payoff from the calls matches the portfolio
payoff in all scenarios. Hence the call price must equal the value of the portfolio (Law of One Price).
4 * c = 2.35
c = 0.59
Call premium (price)
How many shares of stock to buy to replicate the payoff from one call?
4 calls replicate payoffs from 1 share, hence 1 call is replicated by 0.25 shares.
The fraction of shares needed to replicate 1 call is called the delta () or hedge ratio.
delta ()
= 0.25
How do we create a
replicating portfolio
for puts?
WEMBA 2000
Real Options
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Binomial Pricing Method 1: Extending to two time-steps
Call Option Tree
Share Price Tree
24.2
[24.2 - 21] = 3.2
22
20
cu
19.8
18
c
0
cd
16.2
0
Bluejay Corp share price is currently $20. Possible price moves in each period: either up by 10%
or down by 10%. Period length: 3months.
Value a call option on Bluejay with strike 21, expiration 6 months.
Riskfree rate = 2% over each 3 month period.
Methodology:
Step 1: Calculate u and cu , the delta and call value at the upper intermediate node
Step 2: Calculate d and cd , the delta and call value at the lower intermediate node
(note: u and d will be different)
Step 3: Calculate  and c, the delta and the call price today
WEMBA 2000
Real Options
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Binomial Pricing Method 1: Extending to two time-steps
Step 1: Calculating cu and  u
Share Price Tree
Call Option Tree
3.2
24.2
cu
22
20
c
19.8
0
cd
18
0
16.2
Replicating Portfolio to calculate cu
22PV(19.8)
=2.59
24.2 - 19.8
= 4.4
19.8 - 19.8
=0
(a) Purchase 1 share and borrow
money so that the portfolio payoff
is zero in one scenario
Match replicating portfolio payoffs at ending nodes
equal
(1/u)cu
3.2*(1/u)
= 4.4
0
(b) Purchase the appropriate number of calls
so that the payoff at each terminal node
matches the payoffs from the portfolio.
u = 3.2/4.4 = 0.727
cu = u * 2.59 = 1.88
WEMBA 2000
Real Options
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Binomial Pricing Method 1: Extending to two time-steps
Step 2: Calculating cd and  d
Call payoff in either scenario is zero.
Hence cd = 0, replicating portfolio = 0.
By implication, d = 0
0
cd
0
Step 3: Calculating c and 
Replicating Portfolio to calculate c
20PV(18)
=2.35
22 - 18
=4
18 - 18
=0
(a) Purchase 1 share and borrow
money so that the portfolio payoff
is zero in one scenario
(note: this is identical to the 1-step tree)
Match replicating portfolio payoffs at ending nodes
equal
cu = 1.88*(1/)=4
(1/)*c
0
(b) Purchase the number of calls necessary
so that the payoff at each terminal node
matches the payoffs from the portfolio.
 = 1.88/4 = 0.47
c =  * 2.35 = 1.10
WEMBA 2000
Real Options
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Binomial Pricing Method 2: Creating a riskless portfolio
Bluejay Corp share price is currently $20. Possible price at the end of three months: either $22 or $18.
Value a call option on Bluejay with strike 21, expiration 3 months.
Riskfree rate = 2% over 3 months.
Share Price
Option Value
22
20
22-21 = 1
{Reminder: the value of the call
at expiration is Max[0, S - K]}
c
18
0
Create a riskless portfolio: sell 1 call, buy d shares (where d is a fraction of a share)
Riskless Portfolio
22 - 1
- c + 20
Question: how can we make this portfolio riskless?
18
WEMBA 2000
Real Options
Binomial Pricing Method 2: Creating a riskless portfolio
Riskless Portfolio
22 - 1
For the portfolio to be riskless, the two
outcomes must have identical values.
-c + 20
18
HINT: Choose  so that: 22 - 1 = 18
Portfolio "delta"
 = 0.25
Portfolio Terminal value = 4.5 (in either scenario)
Portfolio Present value = 4.5/(1.02)
= 4.41
(discount at riskfree rate)
Hence: 4.41 = -c + 20
c = 0.59
Call premium (price)
Note: this is the same call price and delta that we obtained using method 1.
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WEMBA 2000
Real Options
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Binomial Pricing Method 2: Creating a riskless portfolio
Stock Price
Option Value
q
1
q
22
20
0.59
1-q
1-q
18
0
Call price
q
= 0.59
= [1 * q + 0 * (1 - q)]/1.02
= 0.6
Stock price
q
= 20
= [22 * q + 18 * (1 - q)]/1.02
= 0.6
What does the value q represent?
It does not represent the probability that the stock price will move up or down!
It is sometimes referred to as the “risk-neutral” probability that the stock price will
move up or down.
WEMBA 2000
Real Options
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Binomial Pricing Method 2: Generalization
*Share Price
Option Value
Su
Su - cu
cu
+
S
Portfolio
=
c
Sd
S-c
Sd - cd
cd
For portfolio to be riskless, choose  so that Su - cu = Sd - cd
hence  = cu - cd
Su - Sd
(1)
Now the riskless terminal value, discounted at the riskless rate rf , should equal the portfolio cost:
Su - cu = S - c
(1 + rf )
Substitute for  from (1):
where
c = q cu + (1-q)cd
(1 + rf )
(2)
q = (1 + rf) - d
(u - d)
(3)
WEMBA 2000
Real Options
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Binomial Pricing Method 2: Generalization over two time-steps
Su
cu
Su2
cuu
Sud
cud
S
c
Sd
cd
Sd2
cdd
cuu = max[0, Su2 - K]
cud = max[0, Sud - K]
cdd = max[0, Sd2 - K]
q = (1+rf) - d
(u - d)
cu = [p cuu + (1-p)cud ]
(1+rf )
cd = …..
c = …..
S = Stock price today
u = proportional change in S on an up-move
d = proportional change in S on a down-move
rf = riskfree rate
c = call price today
cu = call value after one up-move
cd = call value after one down-move
cuu , cud , cdd = terminal call values
K = strike on the call
Example: compare with 2-step example using Method 1
S = 20, u=1.1, d = 0.9, rf = 2%
cuu = 3.2; cud = cdd = 0
q = [(1.02)-0.9]/(1.02) = 0.6
cu = [0.6 * 3.2 + 0.4 * 0]/1.02 = 1.88
cd = 0
c = [0.6 * 1.88 + 0.4 * 0]/1.02 = 1.10
compare these results with those from Method 1
WEMBA 2000
Real Options
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Valuation of Options: Binomial Pricing Method
What have we shown?
We can evaluate a call option either by creating a replicating portfolio of the underlying
stock and borrowing, or by creating a riskless portfolio of the call and the underlying stock
The two methods yield identical results
What other information do we obtain from these methods?
The delta or hedge ratio: the fraction of the underlying stock that we need to purchase relative
to selling a single call option to obtain a riskless portfolio
The risk-neutral probability of an upmove or downmove in the underlying stock
What are the underlying assumptions of these methods?
That we can freely buy and sell the underlying stock without transactions costs
That we can borrow or lend money at the riskless rate of interest
What are the limitations of these methods?
They become very complex over a large number of steps (although computers can help)
What is the connection between these methods and the Black-Scholes formula?
The Black-Scholes formula effectively represents the binomial tree model over many
hundreds or thousands of periods
Binomial Tree methodology:
Option price = delta * share price - bank loan
Black Scholes formula:
Option price = N(d1 )* S
- N(d2)* PV(K)
WEMBA 2000
Real Options
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Valuation of Options: Call and Put Price Sensitivities
As each input to the option pricing model varies, the call and put prices respond by increasing or decreasing
as follows:
Increase In:
S
X
T
r

Call Price
Put Price
Why?
[to be discussed in class]
WEMBA 2000
Real Options
25
Debt and Equity as Options
Suppose a firm has debt with a face value of $1MM outstanding that matures at the end
of the year. What is the value of debt and equity at the end of the year?
Firm Value (V)
0.3 MM
0.6 MM
0.9 MM
1.2 MM
1.5 MM
Payoff to shareholders
0
0
0
0.2 MM
0.5 MM
Payoffs
Equityholders
Bondholders
Payoff to debtholders
0.3 MM
0.6 MM
0.9 MM
1.0 MM
1.0 MM
Payoff to Equityholders
= max [0, V - $1MM]
equivalent to a call option, K=$1MM
Payoff to Bondholders
= V - max [0, V - $1MM]
equivalent to the total value of the firm
less a call option, K=$1MM
0
$1 MM
Firm Value V