L4v(1)-1 - ECON 3107 Lecture 4 Valuation Completing the...

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ECON 3107 Lecture 4: Valuation Completing the Markets Multiple Times Valentyn Panchenko
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Completing the Market Consider an example. Markets are incomplete: Q ( 3 × 2 ) = 20 43 20 28 20 28 Bond Stock Good Weather Fair Weather Bad Weather p S ( 1 × 2 ) = ( 19 35 ) Bond Stock Suppose that an investor asks an investment firm to create a product with the following payment: c ( 3 × 1 ) = 40 30 20 Good Weather Fair Weather Bad Weather The Questions: How to do it? What should the firm charge?
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Completing the Markets I The Problem: No matter how many bond and stock are chosen, the payments in the ”Fair Weather” state and the ”Bad Weather” state will be the same! I Suppose the firm will select 40 in ’GW’ and 30 in ’FW’ or ’BW’ to cover all outflows. Markets are sufficiently complete to do it: Q ( 2 × 2 ) = 20 43 20 28 Bond Stock Good W. Fair or Bad W. c ( 2 × 1 ) = 40 30 Good W. Fair or Bad W. n = Q - 1 · c = 20 43 20 28 - 1 40 30 = 0.566 67 0.666 67 Bond Stock p = p S · n = ( 19 35 ) 0.566 67 0.666 67 = 34. 10
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Completing the Markets I It costs 34.10 PA to purchase a portfolio, n , that would cover all future outflows: I The firm will receive 10 in ’BW’ state; I Investor will be assured to receive all the promised payments. I Since 10 BA equals to 3.9 PA, the firm will be happy to sell the product for 34.10PA. What if it offers the product for 32PA? Q ( 3 × 3 ) = 20 43 40 20 28 30 20 28 20 Bond Stock Product Good W. Fair W. Bad W. p S ( 1 × 3 ) = ( 19 35 32 ) Bond Stock Product p atom = p S · Q - 1 = ( 19 35 32 ) 20 43 40 20 28 30 20 28 20 - 1 = ( 0.56 0.18 0.21 ) Good W. Fair W . Bad W.
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Hedging at Minimum Cost Recall our original example. Market are incomplete: c ( 3 × 1 ) = 40 30 20 Good Weather Fair Weather Bad Weather Q ( 3 × 2 ) = 20 43 20 28 20 28 Bond Stock Good Weather
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