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section4 - Derivative Securities – Fall 2007– Section 4...

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Unformatted text preview: Derivative Securities – Fall 2007– Section 4 Notes by Robert V. Kohn, extended and improved by Steve Allen. Courant Institute of Mathematical Sciences. Lognormal price dynamics and passage to the continuum limit. After a brief recap of our pricing formula, this section introduces the lognormal model of stock price dynamics, and explains how it can be approximated using binomial trees. Then we use these binomial trees to price contingent claims. The Black-Scholes analysis is obtained in the limit δt → 0. As usual, Baxter–Rennie captures the central ideas concisely yet completely (Section 2.4). Hull has a lot of information about the lognormal model scattered through Chapter 13. As usual, we’ll focus initially on options on a (non-dividend-paying) stock. Then, at the end of these notes, we ask what’s different for options on a forward price. ********************** Recap of the multiperiod option pricing formula . Recall what we achieved at the end of the last section: if the risk-free rate is constant and the risky asset price evolution is described by a multiplicative binomial tree with s up = us now and s down = ds now then the value at time 0 of a contingent claim with maturity T = Nδt and payoff f ( s T ) is V ( f ) = e- rT · E RN [ f ( s T )] where E RN [ f ( s T )] is the expected final payoff, computed with respect to the risk-neutral probability: E RN [ f ( s T )] = N X j =0 N j ! q j (1- q ) N- j f ( s u j d N- j ) , with q = ( e rδt- d ) / ( u- d ). Let’s check this assertion for consistency and gain some intuition by making a few observations: What if the contingent claim pays the stock price itself? This is the case f ( s T ) = s T . It is replicated by the portfolio consisting of one unit of stock (no bond, no trading). So the present value should be s , the price of the stock now. Let’s verify that this is the same result we get by “working backward through the tree.” It’s enough to show that if f ( s ) = s for every possible price s at a given time then the same relation holds at the time just before. To see this, let “now” refer to any possible stock price at the time just before. We are assuming f ( s up ) = s up and f ( s down ) = s down and we want to show f ( s now ) = s now . By definition, f ( s now ) = e- rδt [ qs up + (1- q ) s down ] with q = e rδt s now- s down s up- s down . Simple algebra confirms the expected result f ( s now ) = s now . (As we noted in Section 2, this is no accident; it can be viewed as the defining property of q .) There is of course an equivalent calculation involving risk-neutral expectation. The formula for q in a multiplicative tree gives qu + (1- q ) d = e rδt 1 and taking the N th power gives N X j =0 N j !...
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