# Option Valuation

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```					Option Valuation
Lecture XXI
   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
cost \$2.
– 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:
p

Strike Price

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
Option.
– 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(.)
 0.
– Proposition 2: F(S,T;T,x)=f(S,T;T,x)=Max(S-
x,0), G(S,T;T,x)=g(S,T;T,x)=Max(S-x,0).
– Proposition 3: F(S,t;T,x)  S-x, G(S,t;T,x) 
x-S.
– Proposition 4: For T2>T1 F(.;T2,x)  F(.;T1,x),
G(.;T2,x)  G(.;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)
 g(.,x2)
– 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.
Valuing Options
   Intuitive Determinants of European
Option Prices.
• 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
price” option:
Distribution

Payoff

x
S
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
Bernoulli gambles.
• A Bernoulli event is the probability
distribution function used for a coin toss.

Px  p 1  p             x  0,1
x         1 x
• Assume a very simple payoff structure


y  100  10 x  1
2

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
option worth?

f 100;1, p   5 p  01  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  51  p 

Again, if p=.5, the put option is worth
\$2.5.
   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:
• 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  10z  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 p1  p   01  p 
2                            2

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 p1  p   01  p 
2                                2

which equals 6.25 if p=.5.
Binomial Distribution
3z

2z
p3

z          p2              2z

p            z            p2(1-p)

0     p(1-p)           z

0
1-p                        p(1-p)2
0
(1-p)2
(1-p)3
• Mathematically, the probability of r “heads”
out of n draws becomes

 n r
Pr n, p     p 1  p                p 1  p 
nr     n!       r        nr
r               r!n  r !
 
Black-Scholes
 The Black-Scholes pricing model
extends the binomial distribution to
continuos time.
 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 
rf T

d1 
 X  r T  1 
ln S      f
T
 T                2
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
density function:

d
 z    2

N d  
1

   2p
exp 
 2
 dz

   Example:
• 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
as:
d1 

ln 50
45
 .06  14  1  .2    1  .65
2           4
.2 1
4
d 2  d1     .2    1  .4264
4
• 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
\$7.62.
Option Value of Investments
   Moss, Pagano, and Boggess. “Ex Ante
Modeling of the Effect of Irreversibility
and Uncertainty on Citrus Investments.”
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
attractive.
– In most investments, investors can be
construed to have limited liability with the
distribution being truncated at the loss the the
entire investment.
– 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
process:

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
process

dx   x dt   x dz
  r   vm
• 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-
FVV).

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
2
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 
V   *
 I
                *
V

V 
*

   1
I
1

1  r  d    r  d  1  2
                         r 
   2
                                2 2
2          2
 

2
2    
 then simplifies to

               1

1       8r    2   
  1  1  2        
2                
                   
   Estimating 
• 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
investment as
N t
CFi
Vt  
i 1 (1  r ) i
• Converting this value to an infinite
streamed investment then involves:
Vt
APVt 
         1         
 1                
    (1  r )  N   
       r           
                   
                   

APVt
Vt 
*
r
• The parameters of the stochastic process
can then be estimated by

dV
 d lnV 
V
 lnVt 1   lnVt 
   Application to Citrus
• The simulated results indicate that the
present value of orange production was
\$852.99/acre with a standard deviation of
\$179.88/acre.
• 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.

```
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 views: 5 posted: 12/1/2011 language: English pages: 45