Chapter 12: Basic option theory
Investment Science
D.G. Luenberger
Before we talk about options
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This course so far has dealt with deterministic cash
flows and single-period random cash flows.
Now we’d like to deal with random flows at each of
several time points, i.e., multiple random cash flows.
Multiple random cash flow theories are generally
very difficult. Here, we’d like to focus on a special
case: derivatives, i.e., those assets whose cash
flows are functionally related to other assets whose
price characteristics are assumed to be known.
Options are an important category of derivatives.
Option, I
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An option is the right, but not the obligation, to buy (or sell) an
underlying asset under specified terms. Usually there are a
specified price, called strike price or exercise price (K), and a
specified period of time, called maturity (T) or expiration date,
over which the option is valid.
An option is a derivative because whose cash flows are related
to the cash flows of the underlying asset.
If the option holder actually does buy and sell the underlying
asset, the option holder is said to exercise the option.
The market price of an option is called premium.
Option, II
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An option that gives the right to purchase (sell) the
underlying asset is called a call (put) option.
An American option allows exercise at any time
before and including the expiration date.
An European option allows exercise only on the
expiration date. In this course, we will focus on
European options because their pricing is easier.
The underlying assets of options can be financial
securities, such as IBM shares or S&P 100 Index, or
physical assets, such as wheat or corn.
Option, III
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Many options are traded on open markets. Thus,
their premiums are established in the market and
observable.
Options are wonderful instruments for managing
business and investment risk, i.e., hedging.
Options can be particularly speculative because of
their built-in leverages. For example, if you are very
sure that IBM’s share price will go up, betting $1 on
IBM call options may generate a much higher return
than betting $1 on IBM shares.
Notation
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S: the price of the underlying asset.
C: the value of the call option.
P: the value of the put option.
Value of call option at expiration
Value of put option at expiration
Value of option at expiration
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The value of the call at expiration: C = max (0, S –
K).
The call option is in the money, at the money, or out
of money, depending on whether S > K, S = K, or S
< K, respectively.
The value of the put at expiration: P = max (0, K –
S).
The put option is in the money, at the money, or out
of money, depending on whether S < K, S = K, or S
> K, respectively.
11.1 Binomial model, PP. 297-299
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A binomial model can be a single-period model or a
multiple-period model.
A basic period length can be a week, a month, a
year, etc.
If the price of an asset is known at the beginning of a
period, say S, the price of the asst at the end of the
period is one of only two possible values, S*u and
S*d, where u > 1 > d > 0. S*u (S*d) is expected to
happen with probability p (1 – p).
That is, we have uncertainty, but in the form of two
possible values.
Binomial model
Binomial lattice
S*u*u*u
S*u*u
S*u
S
S*u*u*d
S*u*d
S*d
S*u*d*d
S*d*d
S*d*d*d
Time 0
Time 1
Time 2
Time 3
Calibration, I
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Binomial models provide an uncertain structure for us to model
the underlying asset’s price dynamics.
This modeling is necessary because option value is a function
of the underlying asset’s price dynamics.
Thus, to obtain accurate valuation for an option, we need to do
a good job on modeling the underlying asset’s price dynamics.
That is, we need to choose binomial parameters, i.e., p, u, and
d, carefully such that the binomial-based price dynamics is
consistent with the observable (historical) price characteristics,
e.g., average return and standard deviation, of the underlying
asset.
Calibration, II
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Let v be the expected (average) annual return of the
underlying asset, v = E [ln(ST /S0 )], say 12%.
Let  be the yearly standard deviation of the
underlying asset, 2 = var [ln(ST /S0 )], say 15%.
Note that 12% return and 15% standard deviation
are about the kind of numbers that you would expect
from a typical S&P 500 stock.
Now we need to define the period length relative to a
year. If we define a period as a week, a period
length of t is 1/52.
Calibration, III
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Then, the binomial parameters can be selected as:
p = ½ + ½ *(v / )* (t)1/2.
u = e * (t)1/2.
d = 1/u.
e is exponential and has a value of 2.7183.
With these choices, the binomial model will closely
match the values of v and  (see pp. 313-315 for the
proof).
Calibration, IV
v
0.12
std
0.15
p
u
0.5555 1.021
d
del-t
0.97941 0.01923
After 4 weeks
108.677
106.4
104.25
102.1
100
104.248
102.1
100
97.941
100
97.94
95.925
95.9251
93.95
92.0162
1-period binomial option theory, I
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We assume that it is possible to borrow or
lend at the risk-free rate, r.
Let R = 1 + r, and u > R > d.
Suppose that there is a call option on the
underlying asset with strike price K and
expiration at the end of the single period.
Let Cu (Cd) be the value of the call at
expiration.
3 related lattices
If up
If down
S*u
S
S*d
Cu
C
R
Cd
1
R
Time 0
Time 1
Time 0
Time 1
No-arbitrage
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The key to price the call option at time 0 is to form a portfolio at
time 0: (1) the portfolio consists of the underlying asset and the
risk-free asset, (2) the portfolio’s value at time 1 is equal to the
value of the call at time 1, regardless whether it is up or down.
This portfolio is called a replicating portfolio: x dollar worth of
the underlying asset and b dollar worth of the risk-free asset.
No-arbitrage: because the replicating portfolio and the call yield
the value at time 1 regardless what might happen, the value of
the replicating portfolio and the call at time 0 must be the same.
That is, x + b = C.
Outcome matching
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The value of the replicating portfolio equals
to the value of the call at time 1 when it is up:
u * x + R * b = Cu.
The value of the replicating portfolio equals
to the value of the call at time 1 when it is
down: d * x + R * b = Cd.
Solve for x and b from the two equations.
1-period binomial Solution
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x = (Cu – Cd) / (u – d).
b = (u * Cd – d * Cu) / (R * (u – d)).
Based on no-arbitrage, we know that C = x + b.
C = (Cu – Cd) / (u – d) + (u * Cd – d * Cu) / (R * (u –
d)).
After some algebras, we have C = (1/R) * [q * Cu +
(1 – q) * Cd], where q = (R – d) / (u – d).
Note that p is not in the pricing equation because no
trade-off among probabilistic events is made.
1-month IBM call option, I
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Consider IBM with a volatility of its logarithm
of  = 20%. The current price of IBM is $62.
A call option on IBM has an expiration date 1
month from now and a strike price of $60.
The current interest rate is 10%,
compounded monthly. Suppose that IBM will
not pay dividends.
1-month IBM call option, II
std
0.2
u
d
del-t
R
q
interest rate
1.0594 0.9439 0.0833 1.0083 0.5577
0.1
stock
65.685
58.522
call
5.6849
3.1443
0
Time 0 Time 1
Time 1 Time 0
62
K
60
When no information about v and 
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If we have a primitive binomial problem in which we
have no information on expected return and
standard deviation, we then must know the two
possible outcomes.
Example: suppose the market price of a stock is $50.
The two possible outcomes for the stock price is
either $60 or $40 in a year.
A call option with a one-year expiration and a $50
exercise price.
Interest rate is 5%.
Duplicating portfolio
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We need to duplicate the call with the
strategy of buying stocks and borrowing
monies.
The duplicating strategy is to buy ½ share of
the stock and borrow $19.05. Why this
particular combination? We will talk about
this later.
When the stock price is up
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When the stock price is up, the payoff of
buying a call is $10 ($60 - $50).
When the stock price is up, the payoff the
duplicating strategy is also $10. The sum of
the following two positions is $10 ($30 - $20):
(1) buying ½ share: ½ * $60 = $30, and (2)
borrowing $19.05 at 5%: -$19.05 * 1.05 = $20.
When the stock price is down
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When the stock price is down, the payoff of
buying a call is $0.
When the stock price is down, the payoff the
duplicating strategy is also $0. The sum of
the following two positions is $0 ($20 - $20):
(1) buying ½ share: ½ * $40 = $20, and (2)
borrowing $19.05 at 5%: -$19.05 * 1.05 = $20.
No arbitrage
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The call and the duplicating strategy generate
identical payoffs at the end of the year.
No arbitrage principle implies that the current market
price of the call equals to the current market price of
the duplicating position.
The market price of the duplicating position is $5.95
($25 - $19.05). Buying ½ share costs $25 (1/2 *
$50).
The call price is $5.95.
Why ½ share?
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The duplicating portfolio was given to be
buying ½ share and borrowing $19.05
earlier.
Why ½ share? This amount is called the
delta of the call.
Delta = swing of the call / swing of the stock
= ($10 - $0) / ($60 - $40) = ½.
Why borrowing $19.05?
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Buying ½ share gives us either $30 or $20 at
expiration, which is exactly $20 higher than the
payoffs of the call, $10 and $0, respectively.
To duplicate the position, we thus need to borrow a
dollar amount such that we will need to pay back
exactly $20.
Given the future value is $20, the interest rate is 5%,
and the number of the time period is 1, we have the
present value to be $19.05 (use your financial
calculator).
Multiple-period pricing
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The usefulness of single-period binomial
pricing is that it can be applied to multipleperiod problems in a straightforward manner.
That is, the single period pricing, C = (1/R) *
[q * Cu + (1 – q) * Cd], is repeated at every
node of the lattice, starting from the final time
period and working backward toward the
initial time.
5-month IBM call option, I
std
0.2
u
1.05943
d
0.9439
del-t
R
0.08333 1.00833
Stock
q
0.5577
82.749
78.1067
73.7249
69.5889
65.6849
62
73.7249
69.5889
65.6849
62
58.5218
65.6849
62
58.5218
55.2387
58.5218
55.2387
52.1398
52.1398
49.2147
46.4538
Time 0
Time 1
Time 2
Time 3
Time 4
Time 5
K
60
5-month IBM call option, II
std
0.2
Stock
82.749
u
1.05943
d
0.9439
Call
22.749
del-t
R
0.08333 1.00833
q
0.5577
K
60
18.6026
73.7249
13.7249
14.7125
10.0848
65.6849
5.68495
11.189
6.95701
3.14428
58.5218
0
4.61068
1.73907
0
52.1398
8.210984
0
5.84508
2.972037
0.96186
0
0
46.4538
0
Time 5
Time5
Time4
Time3
Time2
Time1
Time0
1-month vs. 5-month
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The call price for 1-month IBM call is $3.14. The call
price for 5-month IBM call is $5.85.
Holding other factors constant, the longer the
maturity, the higher the call premium.
The reason for this is that additional time allows for a
greater chance for the stock to rise in value,
increasing the final payoff of the call option.
See Figure 12.3, p. 324.
Along the same line of seasoning, the higher the
standard deviation, the higher the call premium. You
should verify this numerically.
How about put option pricing?
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So far, we have focused on call pricing.
The reason for this is that for European options one
can calculate the value of a put option, P, based on
value the call option, C, when they have the same
strike price and maturity.
P = C – S + df * K, where df is risk-free discount
factor.
This relationship is called the put-call parity.
Put option pricing
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Consider a GM call option and a GM put
option, both have 3 months to expiration and
the same strike price, $35. The current price
of GM shares is $37.78. The call premium is
$4.25. The interest rate is 5.5%, so over 3
months, the discount factor is 0.986 (= 1 / (1
+ 0.055/4).
P = C – S + df * K = 4.25 – 37.78 + 0.986 *
35 = $1.00.
Options are interesting and important
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A combination of options can lead to a
unique payoff structure that otherwise would
not be possible.
Options make it happen!
Example: a butterfly spread, p. 325.
Question: who would hold a butterfly spread?