It gives us some intuition about the state pricing

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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 risk-neutral 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.

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