What is an option?
• In a general sense, an option is exactly
what its name implies - An option is the
opportunity to buy or sell one share of
stock or lot of commodity at some point in
the future at some state price.
– For example, a call option entitles the
purchaser to purchase a stock or commodity in
the future at some state price.
– Assume that a call contract stated that the
holder of the contract was entitled to purchase
one share of IBM at $120 at some point in the
future. Suppose further that the right cost $2
per share. At the time the contract matured,
suppose that the price of IBM was $130. The
gain to the holder of the instrument would be
$8. In other words, the investor would exercise
the option to purchase one share of IBM at
$120 and sell the share for $130. Hence, the
investor would gross $10. Of this $10, the call
– On the flip side, the instrument that gives the
bearer the right to sell a stock at a fixed price in
the future is called a put.
• Both of these contracts are called
contingent claims. Specifically, the claim
only has value contingent on certain
outcomes of the economy.
– In the call security, suppose that the market
price for IBM was $110. The rational investor
would choose not to exercise their right.
– Exercising their right would mean purchasing a
stock for $120 and selling it for $110 for a gross
loss of $10 and a total loss of $12.
– Graphically (ignoring the time value of money)
the payoff on buying a call is:
Price of the call 45 o Stock Price
Factors affecting the price of options
• Technically, there are two types of options:
a European option and an American
– A European option can only be exercised on
the expiration date.
– The American option can be exercised on any
date up until the expiration date.
Given these differences let: F(S,t;T,x)
denote the value of an American call
option with stock price S on date t and
an expiration data T for an exercise
price of X. Given this notation f(S,t;T,x)
is the price of a European call option,
G(S,t;T,x) is the price of an American
put option, and g(S,t;T,x) is the price of
the European put option.
• Risk Neutral Propositions - Simply assume
that investors prefer more to less.
– Proposition 1: F(.) 0, G(.) 0, f(.) 0, g(.)
– Proposition 2: F(S,T;T,x)=f(S,T;T,x)=Max(S-
– Proposition 3: F(S,t;T,x) S-x, G(S,t;T,x)
– Proposition 4: For T2>T1 F(.;T2,x) F(.;T1,x),
– Proposition 5: F(.) f(.) and G(.) g(.).
– Proposition 6: For x1 > x2 F(.,x1) F(.,x2) and
f(.,x1) f(.,x2), and G(.,x1) G(.,x2) and g(.,x1)
– Proposition 7: S = F(S,t;,0) F(S,t;T,x)
f(S,t;T,x). The first equality involves the
definition of a stock in a limited liability
economy. If you purchase a stock, you
purchase the right to sell the stock between
now and infinity. Further, given limited liability,
you will not sell the stock for less than zero.
– Proposition 8: f(0,.)=F(0,.)=0.
Intuitive Determinants of European
• Three of the previous results bear restating
– The value of a call option is an increasing
function of the spot stock price (S).
– The value of a call option is a decreasing
function of the strike price (x).
– The value of a call option is an increasing
function of the time to maturity (T).
• The value of an option is an increasing
function of the variability of the underlying
asset. To see this, think about imposing
the probability density function over a “zero
Binomial Pricing Model
The simplest form of option pricing
model is referred to as a binomial
pricing model. It is based on a series of
• A Bernoulli event is the probability
distribution function used for a coin toss.
Px p 1 p x 0,1
x 1 x
• Assume a very simple payoff structure
y 100 10 x 1
Under the Bernoulli structure, the value of
the payoff is y=95 with probability (1-p) and
y=105 with probability p.
• Assume a strike price for a call option of
$100. If the event is x=1, implying that
y=$105, the value of the call option is $5.
However, if the event is x=0 implying that
y=$95, the value of the call option is $0.
• The question is then: How much is the call
f 100;1, p 5 p 01 p
If p=.5, then the call option is worth
$2.5. How much is the put option worth?
g 100;1, p 0 p 51 p
Again, if p=.5, the put option is worth
The binomial probability function is the
sum of a sequence of Bernoulli events.
• For example, if we link to coin tosses
together we have three possible outcomes:
2 heads, 2 tails or one head and one tail.
• Let z be the sum of two Bernoulli events. z
could take on the value of zero, one or two:
z 0 x1 0 x2 0
z 1 x1 1 x2 0 or x1 0 x2 1
z 2 x1 1 x2 1
• Extending the payoff formulation
y $100 10z 1
• In this case, y=$110 if z=2 which occurs
with probability p2, y=$100 if z=1 which
occurs with probability 2p(1-p), and y=$90
if z=0 which occurs with probability (1-p)2.
• Now the call option is worth
f 100;2, p 10 p 0 2 p1 p 01 p
which again equals 2.5 if p=.5. The call
option for a strike price of $95 is now
f 95;2, p 15 p 5 2 p1 p 01 p
which equals 6.25 if p=.5.
z p2 2z
p z p2(1-p)
0 p(1-p) z
• Mathematically, the probability of r “heads”
out of n draws becomes
Pr n, p p 1 p p 1 p
nr n! r nr
r r!n r !
The Black-Scholes pricing model
extends the binomial distribution to
The derivation of the Black-Scholes
model is beyond this course. However,
the formula for pricing a call option is
c S N d1 Xe N d 2
X r T 1
ln S f
d 2 d1 T
where S is the price of the asset (stock
price), X is the exercise price, rf is the
riskless interest rate, and T is the time to
expiration. N(.) is the integral of the normal
• Assume that the current stock price is $50,
• the exercise price of the American call
option is $45,
• the riskless interest rate is 6 percent,
• and the option matures in 3 months.
• Given that the interest rate is specified as
an annual interest rate, T is implicitly in
years. 3 months is then ¼ of a year.
• In addition, we need an estimate of
consistent with this increment in time.
Assume it to be .2.
• The two constants can then be computed
.06 14 1 .2 1 .65
d 2 d1 .2 1 .4264
• The two N(.) can be derived from a
standard normal table as N(d1)=.742 and
N(d2)=.6651. Plugging these values back
into the option formula yields a call price of
Option Value of Investments
Moss, Pagano, and Boggess. “Ex Ante
Modeling of the Effect of Irreversibility
and Uncertainty on Citrus Investments.”
• Traditional courses in financial
management state that an investment
should be undertaken if the Net Present
Value of the investment is positive.
• However, firms routinely fail to make
investments that appear profitable
considering the time value of money.
• Several alternative explanation for this
phenomenon have been proposed.
However, the most fruitful involves risk.
– Integrating risk into the decision model may
take several forms from the Capital Asset
Pricing Model to stochastic net present value.
– However, one avenue which has gained
increased attention during the past decade is
the notion of an investment as an option.
• Several characteristics of investments
make the use of option pricing models
– In most investments, investors can be
construed to have limited liability with the
distribution being truncated at the loss the the
– Alternatively, Dixit and Pindyck have pointed
out that the investment decision is very
seldomly a now or never decision. The
decision maker may simply postpone
exercising the option to invest.
Derivation of the value of waiting
• As a first step in the derivation of the value
of waiting, we consider an asset whose
value changes over time according to a
geometric Brownian motion stochastic
d V V dt V dt
• Given the stochastic process depicting the
evolution of asset values over time, we
assume that there exists a perfectly
correlated asset that obeys a similar
dx x dt x dz
• Comparing the two stochastic processes
leads to a comparison of and .
– The relationship between these two values
gives rise to the execution of the option.
– Defining d= to the the dividend associated
with owning the asset. is the capital gain
while “operating” return.
– If d is less than or equal to zero, the option will
never be exercised. Thus, d >0 implies that the
operating return is greater than the capital gain
on a similar asset.
• Next, we construct a riskless portfolio
containing one unit of the option to some
level of short sale of the original asset
P F (V ) FV (V ) V
P is the value of the riskless portfolio, F(V)
is the value of the option, and FV(V) is the
derivative of the option price with respect
to value of the original asset.
• Dropping the Vs and differentiating the
riskfree portfolio we obtain the rate of
return on the portfolio. To this
differentiation, we append two assumption:
– The rate of return on the short sale over time
must be -d V (the short sale must pay at least
the expected dividend on holding the asset).
– The rate of return on the riskfree portfolio must
be equal to the riskfree return on capital r(F-
dF FV dV d V FV dt r F FVV dt
• Combining this expression with the original
geometric process and applying Ito’s
Lemma we derive the combined zero-profit
and zero-risk condition
1 2 2
V FVV r d V FV rF 0
In addition to this differential equation we
have three boundary conditions
F (0) 0, F (V ) V I , FV (V ) 1
* * *
The solution of the differential equation
with the stated boundary conditions is:
F (V ) V
1 r d r d 1 2
then simplifies to
1 8r 2
1 1 2
• In order to incorporate risk into an
investment decision using the Dixit and
Pindyck approach we must estimate .
• This one approach to estimating is
through simulation. Specifically, simulating
the stochastic Net Present Value of an
i 1 (1 r ) i
• Converting this value to an infinite
streamed investment then involves:
(1 r ) N
• The parameters of the stochastic process
can then be estimated by
lnVt 1 lnVt
Application to Citrus
• The simulated results indicate that the
present value of orange production was
$852.99/acre with a standard deviation of
• Clearly, this investment is not profitable
given an initial investment of $3,950/acre.
• The average log change based on 7500
draws was .0084693 with a standard
deviation of .0099294.
• Assuming a mean of the log change of
zero, the computed value of is 25.17
implying a /(-1) of 1.0414.
• Hence, the risk adjustment raises the
hurdle rate to $4113.40. Alternatively, the
value of the option to invest given the
current scenario is $163.40.