I need help with questions about the current value of American call options. I have attached an example problem. Excel is ok.

B1 65 B2 7 B3 60 B4 35.5 Two-Step Option Trading Problem Problem: A stock price is currently $50. Over each of the next three-month periods it is expected to go up by 6 percent or down by 5 percent. The riskfree interest rate is 5 percent per annum with continuous compounding. What are the current values of a six-month European call option and a sixmonth European put option with strike prices of $51, respectively, using a two-step binomial tree? How will you trade to make profits if the call option's current market price is $3 and the put option's current market price is $0.50. Solution (European Call): Stock European Call 56.18 5.18 53 50 C11 50.35 C00 47.50 0 C10 45.13 p= 0 e r t d e.05.25 0.95 = = .5689 ud 1.06 .95 C00 = e .05.25 .5689 e .05.25 (5.18 .5689) = 1.6351 Solution (European Put): Stock European Put 56.18 0 53 50 P11 50.35 47.50 P00 0.65 P10 45.13 5.875 1 p= e r t d e.05.25 0.95 = = .5689 ud 1.06 .95 P = e .05.25 [.5689 0 + 0.65 .4311] = 0.2767. 11 P = e .05.25 [.5689 0.65 + 5.875 .4311] = 2.866. 11 P00 = e .05.25 [.5689 .2767 + .4311 2.866] = 1.3756. Trading strategy with call trading for $3: Short one call and buy = [2.91 0] /[53 47.5] = .5291 shares of stock for net current cash flow of 3-50(.5291)=-23.45, funded by borrowing $23.45 from bank at 5% per year, continuously compounded, for 6 months. If the stock price rises to 53 at 3 months, then readjust the shareholding to = [5.18 0] /[56.18 50.35] = 0.8885 shares by buying 0.8885-.5291 = .3594 number of extra shares at 53 per share by borrowing an additional sum of 53(.3594)=19.05 at 5% per year for three months. If the stock price falls to 47.5 at 3 months, then readjust the shareholding to = [0 0] / [50.35 47.5] = 0 shares by selling off .5291 shares at 47.5 per share and deposit in bank 47.5(.5291) = 25.13 at 5% per year for three months. At maturity buy the call to cover the short position on it and sell the shares of stock if held. Covering the call is necessary only if the stock price is 56.18, when 5.18 is paid to the call holder who will exercise the call. In this case, selling 0.8885 shares will generate 0.8885(56.18) = 49.92 out of which 5.18 is paid to call holder and 23.45 e.05.5 + 19.05 e.05.25 = 43.33 is repaid to bank to close the borrowed position, leaving 49.92-5.18-43.33=1.41. If the stock price is 50.35 at maturity and 0.8885 shares are in the account, then proceeds will be 50.35(0.8885) - 43.33 = 1.41. If the share price is 45.125 at maturity, no shares are in the account. The bank balance is -23.45 e.05.5 .+ 25.13 e.05.25 = 1.41. 2 Thus, there is no cost of the trading strategy at time zero and a sure receipt of 1.41 at maturity irrespective of stock price. Trading strategy with put trading for $0.5: Buy one put and buy -(.2767-2.866)/(53-47.5) =.47078 (delta is -.47078) shares of stock for net current cash flow of -.5 - .4708*50 = =-24.04, funded by borrowing $24.04 from bank at 5% per year, continuously compounded, for 6 months. If the stock price rises to 53 at 3 months, then readjust the shareholding to - (0-.65)/(56.18-50.35) = .1115 shares [Delta-up = -.1115] by selling off 0.4708-.1115 = .3594 number of shares at 53 per share to generate a sum of 53(.3594)=19.05 which can be deposited at 5% per year for three months. The net cash position here is 19.05-24.04*exp(.05*.25) = -5.292. If the stock price falls to 47.5 at 3 months, then readjust the shareholding to (.65-5.875)/(50.35-45.13) =1.000 shares [Delta-down = -1.000] by buying (1.0000-.4708) = .5292 shares at 47.5 per share by borrowing 47.5(.5292) = 25.14 at 5% per year for three months. The net cash flow here is -25.1424.04* exp(.05*.25) = -49.482. Close all positions at maturity, i.e., sell the put and sell the shares of stock held. When stock price is 56.18 at maturity, closing positions will fetch 0+.1115*56.18 = 6.26 minus (24.04 e.05.5 .- 19.05 e.05.25 ) or 5.36 4.92 to be paid to the bank to close the borrowed position, leaving 6.26-5.36=0.90. If the stock price is 50.35 at maturity, then there is a long put option worth 0.65 and either (i) .1115 shares worth .1115*50.35 = 5.61 minus a net borrowed amount of (24.04 e.05.5 .- 19.05 e.05.25 ) = 5.36, i.e., a net cash flow of 5.61-5.36+.65 = .90; or (ii) .8954 shares worth 1.0000*50.35 = 50.35 minus a net borrowed sum of (24.04 e.05.5 .+ 25.14 e.05.25 ) = 50.10, i.e., 50.3550.10=.25. This gives a net cash flow of 0.25+0.65=0.90 when the stock price is 50.35. If the share price is 45.13 at maturity, the put is worth 5.875, and 1.000 share is worth 45.13 and net borrowed sum due to the bank is (24.04 e.05.5 + 3 25.14 e.05.25 ) = 50.10, which gives a net cash flow of 5.875+45.13-50.10 = 0.90. That the cash flow is the same 0.90 for each different stock price at maturity shows that we have a fully hedged position. 4