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Unformatted text preview: = 20 z0 10z1 10z3 = 0 Subtracting the second equation from the ﬁrst and the fourth from the third
gives 25z2 = 25 and 40z3 = 20 so z2 = 1 and z3 = 1/2. Plugging in these
values, we have two equations in two unknowns:
z0 + 20z1 = 20 z0 10z1 = 5 Taking di↵erences, we conclude 30z1 = 15, so z1 = 1/2 and z0 = 10.
The reader may have already noticed that z0 = 10 is the option price. This
is no accident. What we have shown is that with $10 cash we can buy and
sell shares of stock to produce the outcome of the option in all cases. In the
terminology of Wall Street, z1 = 1/2, z2 = 1, z3 = 1/2 is a hedging strategy
that allows us to replicate the option. Once we can do this it follows that
the fair price must be $10. To do this note that if we could sell it for $12 then
we can take $10 of the cash to replicate the option and have a sure proﬁt of $2. 183 6.2. BINOMIAL MODEL 6.2 Binomial Model In this section we consider the Binomial model in which at each time the stock
price the stock is multiplied by u (for ‘up’). or multiplied by d (for ‘down’). As
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- Spring '10
- The Land