; 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

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

• pg 1
```									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   \$\$


¨                
¨                   

¨¨
¨¨

z
21        3
¨¨
20   ¨
r
rr
rr
rr
rr                   \$\$
\$X
\$   20        2
\$\$\$
rr           \$\$\$
j
r
15   \$\$




time      E                         
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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