Lecture%2023 - Lecture 23 Stock Options Options Pricing...

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Lecture 23 Stock Options Options Pricing continued A) Recall the binomial option model. Stock and Bond price t = 0 t =1 t = 2 B rB rrB uuS uS S udS dS ddS Call price t = 0 t =1 t = 2 B rB rrB Max(uuS – E , 0) C u C Max(udS – E, 0) C d Max(ddS – E, 0) Recall the hedge ratio is H = (C u – C d )/{(u-d)*S} Lets set S = 100, E = 90, r = 1.05, u = 1.1 and d = .9 and solve the two-period problem.
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Stock and Bond price t = 0 t =1 t = 2 B rB rrB 121 110 100 99 90 81 Call price t = 0 t =1 t = 2 Max(uuS – E , 0) = 31 C u C Max(udS – E, 0) = 9 C d Max(ddS – E, 0) = 0 What is C? To solve this first solve for C u and C d What is C u ? Given that t = 1, S = 110, H = (31 – 9)/{(1.1-.9)*110} = 1 Payoff at t = 2, given H = 1 is: (121 – 31) if stock rises and (99-9) if stock falls -C u + uS = payoff/(gross risk-free return) = 90/1.05 = 85.714 C u = uS – 85.714 = 110 – 85.714 = 24.29 What is C d ? Given that t = 1, S = 90, H = (9 – 0)/{(1.1-.9)*90} = .5 Payoff at t = 2, given H = .5 is: (99/2)-9 if stock rises and (81/2) - 0 if stock falls -C d + .5*dS = payoff/(gross risk-free return) = 40.5/1.05 = 38.57 C d = (.5*dS – 38.57) = 45 – 38.57 = 6.43 Note: When S rises to 110 the hedge ratio is 1 when S falls to 90 the hedge ratio is .5
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Now that we know C u and C d , what is the value of C ? Given that t = 0, S = 100, H = (24.29 – 6.43)/{(1.1-.9)*100} = .8929
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This note was uploaded on 04/12/2008 for the course ECON 435 taught by Professor Chabot during the Winter '08 term at University of Michigan.

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Lecture%2023 - Lecture 23 Stock Options Options Pricing...

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