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corp_Ch22_09 - CLASS NOTES WEEK IX BM Ch.22 Valuing options...

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IRPGEN424 Corporate Finance Alex Kane 1 CLASS NOTES WEEK IX BM Ch.22 Valuing options
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IRPGEN424 Corporate Finance Alex Kane 2 Financial engineering The technique of creating and valuing derivative assets, including exotic derivatives (e.g., Asian, barrier, lookback, caput, quantos, digital). Financial engineering specialists must be highly skilled in math and statistics There is a number of graduate programs in financial engineering (e.g. MFE at Berkeley)
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IRPGEN424 Corporate Finance Alex Kane 3 Risk free portfolios and arbitrage A standard method in financial engineering is to identify a package of securities that make up a risk-free Pf A risk-free Pf is easy to value because the discount rate is observable. Risk-free portfolios most assuredly will be arbitrage-priced First concept: Option Delta Stock (S) and Call (C): If CF to long C across states of the world changes in fixed proportions to CF to long S, say 2 to 3, we can construct a risk-free Pf Write (sell) 3 calls and buy 2 shares This ratio (2/3) is called the hedge ratio or option delta
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IRPGEN424 Corporate Finance Alex Kane 4 CF from call and stock, and implied delta initial stock price S=50 initial call price = C stock price at the “up” stage u*S = 100 stock price at the “down” stage d*S = 25 CF from option C u = 50 CF from option C d = 0 notation: u =1+r u , d= 1+r d , f = 1+r f In this example, u=2, d=0.5 ( stock can double or halve) call delta=hedge ratio= C/ S ( C u –C d )/ (u*S–d*S) = 50/75 = 2/3 need 2 shares for 3 calls to set up arbitrage portfolio strike price X = 50
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IRPGEN424 Corporate Finance Alex Kane 5 Using call delta to Price by arbitrage The objective is to find C Investment strategy: write 1 call, buy shares (scale up if needed to round units) Payoff on maturity Portfolio Cost d*S = 25 u*S = 100 write 3 calls –3C –3Cd = 0 –3Cu = –150 buy 2 shares 2S=100 2u*S = 50 2u*S = 200 100–3C 50 50 We have engineered a risk-free pf, can discount by f=1+r f = 1.1 100–3C = 50/1.1 => C = (100 – 50/1.1)/3 = 18.18 This arbitrage-based solution does not depend on risk aversion or any equilibrium risk-return relationship
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IRPGEN424 Corporate Finance Alex Kane 6 Variables we didn’t use to value the call Note that the probabilities of up/down states are not necessary for the solution. Neither is the required return on the stock The probabilities and required rate of return are implied by the price of the stock, given the two possible outcomes. Put differently, probabilities and the required return are implicit in the stock price We can value the call from a no-arbitrage requirement (similar story: TV game)
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IRPGEN424 Corporate Finance Alex Kane 7 Variables in valuation of derivatives False intuition : a critical variable in the value of a derivative is the expected return on the underlying However: the expected return on the underlying is NOT on the list of relevant variables
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