6 Options II

00 7 option strike price k 2000 8 time to

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Unformatted text preview: 1) Then an investment of ½S0 – c = 50 – 30 = $20 would grow to $25 in one year, a rate of return of 22.3% (20e0.223(1) = 25). 2) This is riskless and better than the rate rf = 10% on T- bills, so arbs would rush to build portfolio H, which would mean selling lots of call options. The call premium would fall until c = $27.38 and the perfect hedge portfolio H earns the T- bill rate rf. c) Suppose that c = $20 < $27.38 (with constant rf and S0). 1) Short sell 1/2 share for $50 and buy the option for $20. Invest $30 cash. 2) If ST = $50, let option expire. Pay $25 to return ½ share and close short position. Ending wealth = 30e0.10(1) – 25 = 33.16 – 25 = $8.16. Ec 174 OPTIONS II p. 11 of 18 3) If ST = $200, exercise call option to buy one share for K = $125. Return ½ share to close short position. Ending wealth = ½(200) + 30e0.10(1) – 125 = $8.16. 4) You start with $0 and end with $8.16. The rush to buy calls will drive c up to $27.38. 3. Binomial Call Option Pricing (2- Period Example): {Hull §12.3} $110 $100 $121 (uu) $104.50 (ud,du) $95 Fig. 4 c(u) c(ud) = $0 c c(d) $90.25 (dd) c(uu) = $11 c(dd) = $0 a) Suppose that stock price changes twice a year (every 6 months). [Fig. 4] 1) Let S0 = $100, u = 1.10, d = 0.95, and rf = 10%/year. 2) Consider call option with K = $110 and T = 1 year from today (time t = 0) b) We can deduce c by backward induction. • Deduce c(u) and c(d) from end of year intrinsic values c(uu), c(ud), c(dd). • Then deduce c from c(u) and c(d). c) Illustration of backward induction for two price change a year. ! !! 1) We will need a “perfect hedge ratio” ℎ∗ = ! ! , where cu and cd are the call premiums !! !!! when stock goes up or down at the end of the 3- node tree branches. 2) Step 1 – deduce c(u). • h* = (11−0)/(121−104.5) = 2/3 • Form portfolio H: buy h* = 2/3 24 Today's date 1- Jan- 08 =now( ) or =date(yyyy,mm,dd) 25 Expiration date...
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