Chapter 2 ASM Arbitrage

[Elk]

Here is an example (2C) from the ASM Manual, 4th Edition, pg. 33

Two 1 year European call options on the same stock are priced as follows:
Strike price 40, Premium 10 and Strike Price 45, Premium 4. The risk free rate is 8%. You take advantage of arbitrage by buying one 45 strike call and selling c 40 strike calls, where c is the lowest possible value that results in no possibility of loss when interest is ignored. After a year, the stock price is 46. Determine your profit including interest.

Answer: Your initial gain is 10c-4. We already saw above how to perform an arbitrage for c=1. Therefore, c<=1. Note that S=45 is the worst case. For each 1 the stock price is above 45, you gain 1 for the option you bought and lose c<=1 for the options you sold, which increases your profit. Whereas for each 1 the stock price is below 45, you gain c on the option you sold, until the stock price is 40 and all options are worthless. So, in the worst case, your net cost at expiry is c(45-40)=5c. We want no possibility of loss, so lets set the total of initial gain and final gain equal to 0.

(10c-4)-5c=0, c=4/5. You sell 4/5 of a 40 strike call. Your initial gain is 10 (4/5)-4=4. At expiry, your gain is (46-45)-(4/5)(46-40)=-3/8. With interest, your net gain is 4e^.08-3.8=.5331.

Here are my questions:
1) How do we know that C<=1?
2) If the stock price is below 45 (and more than 40), why do you gain c on the options you sold, since the person you sold the call to at strike 40 would exercise the call, preferring to buy it at the 40 strike price rather than at spot price greater than 40?

[yorklee44]

For #1, if c = 1, arbitrage exists since you'd buy one 45-strike call at $4 and sell one 40-strike call at $10, getting an immediate profit of $6.
If S <= $40, both options are worthless and your profit is $6.
If $40 < S < $45, the 45-strike option is worthless and you lose between $0 and $5 for the 40-strike option, so your profit is between $1 and $6.
If S > $45, you lose $5 more on the 40-strike option than you earn on the 45-strike option, so your profit is $1.

If c > 1, then you lose more on the 40-strike option as the price of the stock increases, and you might end up with a loss. Try c = 2 and see what happens to your profit for S > $45.

So c <= 1.


For #2, they're starting at the point where S = 45. That's the worst case scenario, where you lose $5c. They're saying that for each $1 that the stock price drops below $45, you gain $c back, so that when S reaches $40, you would incur no loss on the option you sold.

Hope that helps.