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Understanding Stock Options: Examples and Exercises

## Example 1

Example 1. Now, let us consider the risk taken by a seller, or writer, of the call, who simply writes the call and waits to see what happens. The call writer takes in the initial option “premium” of\$0.40 and then waits to see what happens to the stock price. If the stock price rises to \$10.80, the person to whom the option was sold will exercise the option, and the call writer will have to paythem \$0.80, for a total profit of 0.40 0 0.80 = =\$0.40 or, said another way, a loss of \$0.40. If, onthe other hand, the stock price falls (to \$9.20), the option expires valueless and the call writer needpay nothing, retaining the initial premium of \$0.40.So, this call writer makes \$0.40 if the stock price falls and loses \$0.40 if the stock price rises.Now, let us consider another call writer who decides to buy the stock at the same time aswriting the call on it. Her cash flow is more complicated. At the beginning of the trade, she takesin \$0.40 from the option premium but pays out \$10 to buy the stock, for a total expense of \$9.60.Now, if the stock price rises to \$10.80, the call she has written is exercised. However, she already owns the underlying stock and so is “covered”. She must sell her stock for \$10 to the option holder but still makes a profit of \$0.40 on the whole trade: \$10 0 \$9.60 = \$0.40.If the stock price falls to \$9.20, the call she wrote expires valueless but, were she to sell the stock here, she would end up losing \$0.40: :\$9.60 + \$9.20 = =\$0.40.
In contrast to her “naked” colleague, the covered call seller makes \$0.40 when the stock price rises and loses \$0.40 when the stock price falls. This is very interesting! Two investors selling exactly the same product are exposed to exactly opposite risks, depending on what else they happen to have in their portfolio.To underline this point, what if an investor entered into both trades simultaneously? In other words, they sold two calls and bought one share. That investor would pay \$10 0 \$0.80 = \$9.201to initiate the trade. If the stock fell to \$9.20, they would get to keep the \$0.80 option premium but would lose \$0.80 on the stock trade, leaving them with no profit or loss (“flat”) on their entire position.
If, on the other hand, the stock rose to \$10.80, they would have to pay out \$0.80 on each of the two options they wrote, but would have a stock worth \$10.80 to sell, for a total final portfolio value of \$10.80 0 \$1.60 = \$9.20. Thus, this investor is completely indifferent to what the stock market does, as they make or lose no money, no matter whether the market rises or falls.

## Example 2

Example 2. Now, consider almost exactly the same example as we have done above, but now with the call selling at a different value. Suppose that the call struck at \$10 and expiring at T written on a stock currently trading at \$10, which, at time T, will trade either at \$10.80 or \$9.20. However, the general investing public believes that a rise is 60% likely to happen but a fall just 40% likely, so the option is thought to be worth 0.6 × (10.80 0 10) + 0.4 × 0 = \$0.48. An investor can make certain profit by purchasing the stock and writing two options, for a total cash flow of ?10 + 2 × 0.48 = =\$9.04.
If the stock price rises to \$10.80, they must pay the option holders 2 × 0.80 = \$1.60, but, after selling their stock, they have total cash of 10.80 0 1.60 = \$9.20, a \$0.16 profit. If, on the other hand, the stock price falls to \$9.20, the two options expire valueless and the stock can be sold for \$9.20, again for a \$0.16 profit. So, in this case, the investor makes \$0.16, no matter what happens!

Exercise: Show that an investor can make money if options can be sold for anything greater than \$0.40.

Example 3. On the other hand, suppose that, because market sentiment is “bearish,” call options struck at \$10 are available for \$0.30. An astute investor can lock in certain profits by purchasing two options for \$0.60 and short sells the share at \$10 (short selling a share is like having}1 shares).
He therefore obtains 10 0 2 × 0.30 = \$9.40 in cash. If the stock price rises to \$10.80, he exercises his two options for cash proceeds of \$1.60 and “covers his short” for \$10.80. This uses 10.80 0 1.60 = \$9.20 of the \$9.40 cash he obtained at the beginning of the trade, leaving him with \$0.20, which he can keep, as he has no outstanding positions in the stock or the option.
If the stock price falls to \$9.20, then his two options expire valueless, but he can cover his short for \$9.20 of the \$9.40 he took in, again leaving him with a profit of \$0.20.

Exercise: Show that the investor can make money if options can be sold for anything less than \$0.40.
So, money for nothing is available to astute traders if the K = \$10 option is sold at any price other than \$0.40. This suggests that, within the confines of this very simple stock price model, the correct price for the option is \$0.40, no matter the investor’s risk-return trade-off and no matterwhat they think the stock price is going to do.