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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:
      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  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.”
     • 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
  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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