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Unformatted text preview: olio with two equations and two
unknowns 2 Find the stock ∆ to hedge the option position 3 Find the riskneutral probability using the stock price The last one is generally the easiest to do. It gives us some
intuition about the state pricing, but it doesn’t give us the
replicating portfolio. Generality of the Result
We can repeat the steps from the numerical examples to see these
results are general. We want to price an option:
Stock: S0 S0 u
S0 d Bond: B 1
1 To ﬁnd ∆ to have a hedged portfolio we have
S0 u ∆ − fu = S0 d ∆ − fd
or
∆= fu − fd
S0 u − S0 d f =??? fu
fd Generality of the Result: Continued
The ﬁnal payoﬀ (up or down) is S0 u ∆ − fu , so its PV is
(S0 u ∆ − fu )B = S0 ∆ − f
We can solve this for f to get
f = S0 ∆(1 − uB ) + fu B
Using ∆ = (fu − fd )/(S0 u − S0 d ) we get
f = B (qfu + (1 − q )fd )
where q = [1/B − d ]/(u − d )
This q works for any payoﬀ since q does not depend on fu or fd . Dividends
We can also modify this argument for dividends. Now the cash
ﬂows are:
Stock: S0 S0 u + Du
S0 d + Dd If we assume dividends are always a fraction of the stock price
(ﬁxed payout ratio), we have Du = αS0 u and Dd = αS0 d for some
α.
The discounted value of the future CFs using q and the risk free
rate must give the price:
S0 = 1
(q (1 +...
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This document was uploaded on 10/28/2013.
 Spring '13

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