What is an Option?

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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 buyers
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.
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WEMBA 2000 Real Options 2
Payoff Diagram for a Long Call Option
Strike Price K Price of Underlying Asset S
Profit/Loss Analysis At expiration, there are
two possible outcomes (i) S gtK. Exercise the
call and purchase the asset for
K. Asset has market value S
Payoff S - K (ii) S ltK. Option expires
worthless. Payoff 0
Option payoff
S Price of Underlying Asset at expiration
K
General Formula for call payoff Long call payoff
Max (0, S - K)
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WEMBA 2000 Real Options 3
Payoff Diagram for a Long Put Option
Strike Price K Price of Underlying Asset S
Profit/Loss Analysis At expiration, there are
two possible outcomes (i) S lt K Exercise the
put and sell the asset for K.
Asset has market value S Payoff
K - S (ii) S gtK. Option expires worthless.
Payoff 0
Option payoff
K
S Price of Underlying Asset at expiration
General Formula put payoff Long put payoff Max
(0, K - S)
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WEMBA 2000 Real Options
WEMBA 2000 Real Options 4
Payoff Diagrams for Short Options Positions
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
S Price of Underlying Asset at expiration
K
K
Short put payoff Min (0, S-K)
Short call payoff Min (0, K-S)
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?
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WEMBA 2000 Real Options 5
Payoff Diagrams for some Option Combinations
"Covered Call" or "Buy-Write"
"Call Spread"
Position Profit/Loss
Position Profit/Loss
Profit/loss from short call
Profit/Loss from stock
net profit/loss
net profit/loss
S Price of Underlying Asset at expiration
S Price of Underlying Asset at expiration
K
Profit/loss from long call
Profit/loss from short call
Note The above profit/loss charts include the
cost of buying (or income from selling) the option
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WEMBA 2000 Real Options 6
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
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WEMBA 2000 Real Options 7
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 Cashflow Position Time 0 Time
T Buy Stock -S0 ST Borrow PV(K)
-K Net Portfolio 1 PV(K) - S0 ST - K
Portfolio payoff at time T
S
Payoff from stock
net payoff
K
ST
Payoff from borrowing
Payoff from borrowing
-K
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WEMBA 2000 Real Options 8
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 ?
Cashflow Cashflow Time
T Position Time 0 ST lt K ST
gt K Buy Call - c 0 ST -
K Sell Put p - (K - ST )
0 Net Portfolio 2 p - c ST - K
ST - K
Portfolio payoff at time T
Payoff on long call
net payoff
K
ST
Payoff on short put
-K
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WEMBA 2000 Real Options 9
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
-0.050.25
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".
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WEMBA 2000 Real Options 10
Valuation of Options Put-Call Parity Example
Cashflow Cashflow Time
T Position Time 0 ST lt 90 ST
gt 90 Buy Call - 12 0 ST
- 90 Sell Put 2 ST - 90
0 Sell stock 100 - ST
- ST Lend money -90 e 0.050.25 90
90 Net Payoff 1.12 0
0
Result arbitrage profit of 1.12 today,
regardless of the value of the stock price!
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WEMBA 2000 Real Options 11
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)
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WEMBA 2000 Real Options 12
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 Estimated input (the future level
of volatility is not observable) S
76.75 K 75 ? 41 r 5.5 T 0.25 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 obtained from
Excel "normsdist" function N(d2)
0.5307 obtained from Excel "normsdist"
function c 7.638 from equation (2) p
4.864 from equation (3)
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WEMBA 2000 Real Options 13
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.
(ii)
(i)
Share Price Option Value
Reminder the value of the call at
expiration is Max0, S - K
22
22-21 1
20
c
18
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
Portfolio Buy 1 share borrow PV(18)
(iii)
Compare the payoff between the call option and
the portfolio. How many call options do we need
to buy to make the payoffs identical?
22-184
20- PV(18) 2.35
18-180
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WEMBA 2000 Real Options 14
Binomial Pricing Method 1 Creating a replicating
portfolio
(iia) (ii)4
(iii)
Portfolio Buy 1 share borrow PV(18)
Option Value (4 calls)
4
4
equal
4c
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.
How do we create a replicating portfolio for
puts?
delta (?)
? 0.25
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WEMBA 2000 Real Options 15
Binomial Pricing Method 1 Extending to two
time-steps
Call Option Tree
Share Price Tree
24.2
24.2 - 21 3.2
22
cu
20
19.8
c
0
18
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
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WEMBA 2000 Real Options 16
Binomial Pricing Method 1 Extending to two
time-steps
Step 1 Calculating cu and ? u
Share Price Tree
Call Option Tree
3.2
cu
c
0
cd
0
Match replicating portfolio payoffs at ending
nodes
Replicating Portfolio to calculate cu
equal
3.2(1/?u) 4.4
24.2 - 19.8 4.4
22- PV(19.8) 2.59
(1/?u)cu
0
19.8 - 19.8 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
(a) Purchase 1 share and borrow money so that the
portfolio payoff is zero in one scenario
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WEMBA 2000 Real Options 17
Binomial Pricing Method 1 Extending to two
time-steps
Step 2 Calculating cd and ? d
0
Call payoff in either scenario is zero. Hence cd
0, replicating portfolio 0. By implication,
?d 0
cd
0
Step 3 Calculating c and ?
Match replicating portfolio payoffs at ending
nodes
Replicating Portfolio to calculate c
equal
cu 1.88(1/?)4
22 - 18 4
20- PV(18) 2.35
(1/?)c
0
18 - 18 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
(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)
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WEMBA 2000 Real Options 18
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
22-21 1
Reminder the value of the call at expiration is
Max0, S - K
20
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?
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WEMBA 2000 Real Options 19
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? ?
0.25 Portfolio Terminal value 4.5 (in
either scenario) Portfolio Present value
4.5/(1.02) (discount at riskfree rate)
4.41 Hence 4.41 -c 20? c 0.59

Portfolio "delta"
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 20
Binomial Pricing Method 2 Creating a riskless
portfolio
Stock Price
Option Value
22
1
q
q
0.59
20
1-q
1-q
0
18
Call price 0.59 1 q 0 (1 -
q)/1.02 q 0.6
Stock price 20 22 q 18 (1 -
q)/1.02 q 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.

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WEMBA 2000 Real Options 21
Binomial Pricing Method 2 Generalization
?Share Price Option Value Portfolio
?Su
?Su - cu
cu


?S-c
?S
c
?Sd - cd
?Sd
cd
For portfolio to be riskless, choose ? so that
?Su - cu ?Sd - cd hence ? cu - cd
Su - Sd 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) c
q cu (1-q)cd (1 rf )
where q (1 rf) - d (u - d)
(1)
(2)
(3)
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WEMBA 2000 Real Options 22
Binomial Pricing Method 2 Generalization over
two time-steps
Su2
cuu
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
Su
cu
Sud
S
c
cud
Sd
cd
Sd2
cdd
cuu max0, Su2 - K cud max0, Sud - K cdd
max0, Sd2 - K q (1rf) - d (u -
d) cu p cuu (1-p)cud (1rf
) cd .. c ..
Example compare with 2-step example using Method
1
S 20, u1.1, 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
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WEMBA 2000 Real Options 23
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)
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WEMBA 2000 Real Options 24
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 Call
Price Put Price Why? to
be discussed in class S X T r ?

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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) Payoff to shareholders Payoff to
debtholders 0.3 MM 0 0.3 MM
0.6 MM 0 0.6 MM 0.9 MM
0 0.9 MM 1.2 MM 0.2
MM 1.0 MM 1.5 MM 0.5 MM 1.0 MM
Payoffs
Equityholders
Payoff to Equityholders max 0, V -
1MM equivalent to a call option,
K1MM Payoff to Bondholders V -
max 0, V - 1MM equivalent to the
total value of the firm less a call
option, K1MM
Bondholders
0
1 MM
Firm Value V