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Example 17 Consider a European call op- tion with strike price 18

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Example 17 Consider a European call op- tion with strike price 18 Powered By Docstoc
					Example 1.7 Consider a European call op-
tion with strike price 18 at time 2 on a stock
whose share price will follow the following bi-
nomial tree:                      share option
                                  price value

                                   $$$
                                      $X
                                       $   30   12
                                 $$
                              $$$
                 ¨¨
                   ¨
                   B   24   $$
                            ˆˆ
                              ˆˆˆ
                ¨                ˆˆ
               ¨                   ˆˆˆ

           ¨¨
             ¨¨
                                      ˆˆ
                                       z
                                           21        3
         ¨¨
   20   ¨
        r
         rr
           rr
             rr
               rr                   $$
                                      $X
                                       $   20        2
                                 $$$
                 rr           $$$
                   j
                   r
                       15   $$
                            ˆˆˆ
                               ˆˆˆ
                                  ˆˆ
                                    ˆˆˆ
   time      E                         ˆ
                                       z
                                           12        0

The interest rate is 5% per unit time (so
r = 0.05).

We can price the option with this stock price
model by finding the risk neutral probabilities
at each time 0 or 1 node, finding (for each
possible price) the option values for an in-
vestor who arrives at time 1 and finally using
these time 1 prices to find the time 0 price.
                                                12
Example 1.7 (continued) To find risk neu-
tral probabilities (if they exist) at any node
of the tree we must try to solve Rp = 0 so
we should write out R. Remember Rij is the
return from wager i when outcome j occurs.
The only possible wagers are on the stock or
the option, the only outcomes are that the
stock price goes from S up to uS or down to
dS. We have
          stock price change
             up       down        wager
                             
            uS − S    dS − S
          1+r       1+r           stock
    R   =
                             
            Cu − C    Cd     
                                   option
           1+r       1+r − C

The first term of Rp = 0 gives us
           1+r−d
       ¯
       p=            (for the up jump)
             u−d
while the second term gives us
                  ¯           ¯
                  pCu + (1 − p)Cd
              C=
                        1+r
We get the exactly the same result using
self-replicating portfolios (which will be use-
ful next term).
                                           13

				
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Description: Example 17 Consider a European call op- tion with strike price 18