I recommend that you use my solution approach, which is less prone to errors than using complex notations and formulas in the textbook.
Problem 9. The textbook Equations 9. This is how to arbitrage on the calls. To arbitrage, buy low and sell high. We use T to represent the common exercise date. This denition works whether the two options are American or European. If the two options are American, well nd arbitrage opportunities if two American options are exercised simultaneously.
If the two options are European, T is the common expiration date. At T , we get non negative cash ows so we may get some free money, but we certainly dont owe anybody anything at T. This is clearly an arbitrage. The textbook Equation 9. However, the strike call is currently selling for 16 in the market.
To arbitrage, buy low the strike call and sell high the strike call. This is how to arbitrage on the puts. In other www. However, the strike put is currently selling for 14 in the market.
To arbitrage, buy low the strike put and sell high the strike put. They are equivalent to the textbook Equation 9. If the above conditions are violated, arbitrage opportunities exist. To arbitrage, we buy low and sell high.
The cheap asset is the diversied portfolio consisting of units of K1 -strike option and 1 units of K3 -strike option. In this problem, the diversied portfolio consists of half a strike call and half a strike call.
The expensive asset is the strike call. Simultaneously,we sell two 55strike call options. This denition works whether the options are American or European. If the options are American, well nd arbitrage opportunities if the American options are exercised simultaneously.
If the options are European, T is the common expiration date. Transaction buy two portfolios buy a strike call buy a strike call Portfolio total Sell two strike calls Total. This is arbitrage.
The above strategy of buying units of K1 -strike call, buying 1 units of K3 -strike call, and selling one unit of K2 -strike call is called the buttery spread. The cheap asset is the diversied portfolio consisting of units of K1 -strike put and 1 units of K3 -strike put.
In this problem, the diversied portfolio consists of half a strike put and half a strike put. The expensive asset is the strike put. Simultaneously,we sell two strike put options. Sell two strike puts 2 The above strategy of buying units of K1 -strike put, buying 1 units of K3 -strike put, and selling one unit of K2 -strike put is also called the buttery spread. This is similar to Problem 9. In this problem, the diversied portfolio consists of 0.
Simultaneously,we sell ten strike call options. Transaction buy ten portfolios buy two strike calls buy eight strike calls Portfolio total Sell ten strike calls Total.
Transaction buy ten portfolios buy two strike calls buy eight strike calls Portfolio total. Since we cant buy half a fraction of an option, well buy 10 units of the portfolio i. Simultaneously,we sell ten strike put options. Transaction buy ten portfolios buy two strike puts buy eight strike puts Portfolio total Sell ten strike puts Total.
The cheap call is the strike call; the expensive call is the strike call. Once again, we buy low and sell high. The payo is: www.
However, our payo is always non-negative. So we never lose money. Its important that the two calls are European options. If they are American, they can be exercised at dierent dates. Since we cant buy a partial option, well buy 3 units of the portfolio i. Simultaneously,we sell three strike calls. The payo at expiration T : www. If the stock pays dividend, then early exercise of an American call option may be optimal. Suppose the stock pays dividend at tD. Time T Pro and con for exercising the call early at tD.
If you exercise the call immediately before tD , youll receive dividend and earn interest during [tD , T ]. Youll pay the strike price K at tD , losing interest you could have earned during [tD , T ].
If the interest rate, however, is zero, you wont lose any interest. You throw away the remaining call option during [tD , T ]. Had you waited, you would have the call option during [tD , T ] If the accumulated value of the dividend exceeds the value of the remaining call option, then its optimal to exercise the stock at tD. As explained in my study guide, its never optimal to exercise an American put early if the interest rate is zero.
The only reason that early exercise might be optimal is that the underlying asset pays a dividend. If the underlying asset doesnt pay dividend, then its never optimal to exercise an American call early. Since Apple doesnt pay dividend, its never optimal to exercise early.
The only reason to exercise an American put early is to earn interest on the strike price. The strike price in this example is one share of AOL stock. Thus its never optimal to exercise the put. If the Apple stock price goes to zero and will always stay zero, then theres no benet for delaying exercising the put; theres no benet for exercising the put early either since AOL stocks wont pay dividend.
Exercising the put early and exercising the put at maturity have the same value. If, however, the Apple stock price goes to zero now but may go up in the future, then its never optimal to exercise the put early. If you dont exercise early, you leave the door open that in the future the Apple stock price may exceed the AOL stock price, in which case you just let your put expire worthless. Theres no hurry to exercise the put early.
If Apple is expected to pay dividend, then it might be optimal to exercise the American call early and exchange one AOL stock for one Apple stock. However, as long as the AOL stock wont pay any dividend, its never optimal to exercise the American put early to exchange one Apple stock for one AOL stock. This is an example where the strike price grows over time.
If the strike price grows over time, the longer-lived valuable as the shorter lived option. The longer-lived call is cheaper than the shorter-lived call, leading to arbitrage opportunities. To arbitrage, we buy low Call 1 and sell high Call 2. If ST2 e0. From T2 to T1 , e0. This is not an arbitrage. Sell a put, receiving P b b 4. The payo at T is:.
The payo is always zero. Sell a call, receiving C b 2. Buy a put, paying P a a 4. Buy one stock, paying S0. The payo is zero. According to the put-call parity, the payo of the following position is always zero: 1. Buy the call 2. Sell the put 3.
Short the stock 4. Lend the present value of the strike price plus dividend The existence of the bid-ask spread and the borrowing-lending rate dierence doesnt change the zero payo of the above position. The above position always has a zero payo whether theres a bid-ask spread or a dierence between the borrowing rate and the lending rate. If there is no transaction cost such as a bid-ask spread, the initial gain of the above position is zero.
However, if there is a bid-ask spread, then to avoid arbitrage, the initial gain of the above position should be zero or negative. We are told to ignore the transaction cost. In addition, we b are given that the current stock price is The dividend is 0.
To nd the expiration date, you need to know this detail. In CBOE, the expiration date of an equity option is the Saturday immediately following the third Friday of the expiration month. To verify this, go to www. Click on "Products" and read "Production Specications. Then the expiration date is November Then the expiration date is January 22, We don't recognize your username or password.
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