Question

Please show work. The following are three-month call option prices: the call at strike 100 is trading at $ 5 and the call at strike 102 is trading at $ 2.5. The rate of interest (continuously compounded) is 3%.

1) Is there an arbitrage strategy is this market and how would you implement it?

2) Draw a cash flow table showing the outcome of your strategy at maturity for every possible stock price level.

Answer 1

1)

As a general rule, whenever the difference in strike prices of two similar options is less than the difference in their extrinsic values, an arbitrage opportunity exists. Here, the difference in strike prices is $2 (102 - 100), whereas the difference in their extrinsic value is $2.5 (5 - 2.5) and therefore, risk less profit can be made.

The way to implement arbitrage is to sell the lower strike ($100) call option for $5 and at the same time buy the higher strike ($102) option for $2.5. This nets an initial profit of $2.5 (5 - 2.5). It can be shown that for any stock price level at maturity, a finite amount of profit will be made.

2)

The cash flow table is as shown below:

The first column denotes the price of underlying stock at maturity for which arbitrary values have been chosen. There are two parts to cash flow from each of the option. One is the premium paid or received and the other is because of the difference in strike price and stock price at maturity. For the $100 call option, we act as the seller and therefore premium is positive and denotes cash inflow. At the same time when stock price moves above $100 strike, it denotes a loss for us hence, negative cash flows. It is exactly opposite in the case of $102 option, where we act as the buyer.

Net cash flow for an option is the sum of these two cash flows. The final column, net cash flow, is the sum of net cash flows from both the options combined. As can be seen, the profit that can be made from this arrangement ranges from $2.5 to $0.5 per one pair of options.