1. The stock price of CH4 trading is $100 and in each 3 month period will...

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1. The stock price of CH4 trading is $100 and in each 3 month period will either increase by 25 percent or fall by 20 percent. A 6-month call on CH4 stock has an exercise price of $90. The risk- free 3-month interest rate is 1 percent. [NOTE: this interest rate is not annualised so that if you invest $1 in a risk-free bank account, after 3 months the bank account contains $1.01.] a. What is the value of the CH4 call? b. Now calculate the option deltas for the second 3-month period if the stock price rises to $125 or falls to $80. C. Does the call option delta vary with the level of the stock price? Explain intuitively why. d. Suppose that in month 3 the CH4 stock price is $80. How at that point could you replicate an investment in the stock by a combination of call options and risk-free lending? Show that this strategy does indeed produce the same returns as from an investment in the stock. 31. a. Note: at this point we do not need to specify whether this is an American or European option as they both have the same value (because an American call is never exercised early when the underlying does not pay dividends). The possible prices of CH4 Trading stock and the associated payoffs for a call option with a strike of $90 (shown in parentheses) are: 100 80 (?) 38. (?) 225 125 (?) 64 (0) 100 (10) 156.25 (66.25) Let p equal the artificial risk-neutral probability of a rise in the stock price: p (0.25) + (1 - p)(-0.20) = 0.01 p = 0.47 If the stock price in month 3 is $80, then (given that it will not be exercised as its strike is $90) it will be worth: [(0.47 x $10)+(0.53 x $0)]/1.01 = $4.7 Similarly, if the stock price is $125 in month 3, then, if it is exercised, it will be worth ($125 - $90) = $35. If the option is not exercised, it will be worth: [(0.47 x $66.25) + (0.53 x $10)]/1.01 = $36.1 Therefore, the call option will not be exercised. Finally, the value of the option today is: [(0.47 x $36.1)+(0.53 x $4.7)]/1.01 = $19.3 b. (i) If the price rises to $125: Delta 66.25-10 156.25-100 = 1.0 (ii) If the price falls to $80: Delta = 10-0 100-64 0.28 c. The option delta is 1.0 when the call is certain to be exercised and is zero when it is certain not to be exercised. If the call is certain to be exercised, it is equivalent to buying the stock with a partly deferred payment. So a one-dollar change in the stock price must be matched by a one-dollar change in the option price. At the other extreme, when the call is certain not to be exercised, it is valueless, regardless of the change in the stock price. d. If the stock price is $80 at 3 months, the option delta is 0.28. Therefore, in order to replicate the stock, we buy 1/delta=3.6 calls and lend, as follows: Initial Outlay Stock Price = 64 Stock Price 100 Buy 3.6 calls -16.63 +0.00 +36.00 Lend PV(64.00) -63.37 +64.00 +64.00 -80.00 +64.00 +100.00 This is equivalent to: Buy stock -80.00 +64.00 +100.00 1. The stock price of CH4 trading is $100 and in each 3 month period will either increase by 25 percent or fall by 20 percent. A 6-month call on CH4 stock has an exercise price of $90. The risk- free 3-month interest rate is 1 percent. [NOTE: this interest rate is not annualised so that if you invest $1 in a risk-free bank account, after 3 months the bank account contains $1.01.] a. What is the value of the CH4 call? b. Now calculate the option deltas for the second 3-month period if the stock price rises to $125 or falls to $80. C. Does the call option delta vary with the level of the stock price? Explain intuitively why. d. Suppose that in month 3 the CH4 stock price is $80. How at that point could you replicate an investment in the stock by a combination of call options and risk-free lending? Show that this strategy does indeed produce the same returns as from an investment in the stock. 31. a. Note: at this point we do not need to specify whether this is an American or European option as they both have the same value (because an American call is never exercised early when the underlying does not pay dividends). The possible prices of CH4 Trading stock and the associated payoffs for a call option with a strike of $90 (shown in parentheses) are: 100 80 (?) 38. (?) 225 125 (?) 64 (0) 100 (10) 156.25 (66.25) Let p equal the artificial risk-neutral probability of a rise in the stock price: p (0.25) + (1 - p)(-0.20) = 0.01 p = 0.47 If the stock price in month 3 is $80, then (given that it will not be exercised as its strike is $90) it will be worth: [(0.47 x $10)+(0.53 x $0)]/1.01 = $4.7 Similarly, if the stock price is $125 in month 3, then, if it is exercised, it will be worth ($125 - $90) = $35. If the option is not exercised, it will be worth: [(0.47 x $66.25) + (0.53 x $10)]/1.01 = $36.1 Therefore, the call option will not be exercised. Finally, the value of the option today is: [(0.47 x $36.1)+(0.53 x $4.7)]/1.01 = $19.3 b. (i) If the price rises to $125: Delta 66.25-10 156.25-100 = 1.0 (ii) If the price falls to $80: Delta = 10-0 100-64 0.28 c. The option delta is 1.0 when the call is certain to be exercised and is zero when it is certain not to be exercised. If the call is certain to be exercised, it is equivalent to buying the stock with a partly deferred payment. So a one-dollar change in the stock price must be matched by a one-dollar change in the option price. At the other extreme, when the call is certain not to be exercised, it is valueless, regardless of the change in the stock price. d. If the stock price is $80 at 3 months, the option delta is 0.28. Therefore, in order to replicate the stock, we buy 1/delta=3.6 calls and lend, as follows: Initial Outlay Stock Price = 64 Stock Price 100 Buy 3.6 calls -16.63 +0.00 +36.00 Lend PV(64.00) -63.37 +64.00 +64.00 -80.00 +64.00 +100.00 This is equivalent to: Buy stock -80.00 +64.00 +100.00