# The Sorcerer's Apprentice and Options

Goethe, the great German poet who lived between the 18th and 19th century, wrote a poem called "The Sorcerer's Apprentice". A century later, the French musician Paul Dukas, inspired by this work, wrote a symphony by the same name. In the 1940 animated movie "Fantasia", Walt Disney presents, among others, Goethe’s story with music by Dukas. In this part of the film, Mickey Mouse, the Sorcerer's Apprentice is tired of constantly carrying buckets of water to pour into a basin. Taking advantage of the temporary absence of his master sorcerer, he puts on his wizard hat and directs a broom carry the buckets for him. Everything goes well until the basin is full and the apprentice is unable to stop the broom. In a panic he chops the broom into pieces with an axe. Alas, each broken section of broom becomes a new broom, and now there is an army of brooms carrying water. Fortunately, the master returns with his quick use of magic, he returns things to the way they were.

With options, we must not act like the sorcerer's apprentice: we must master the magic wand.

Many investors would like to use options on stocks or ETFs to take advantage of the flexibility, diversification and profits offered by these amazing tools. There are dozens of options strategies just as there are many formulas in the magician’s book. Usually, investors use strategies like buying call options or selling covered calls, all allowed in RRSPs. We define these strategies as limited in risk but they are not always to your advantage. As in the magician’s book, sometimes it takes more than the simple application of a strategy to make money. It requires, for example, knowing the probability of success for this strategy, as the lifespan of these instruments is limited. Here are some examples:

## Example 1

On September 19, 2014, the XIU (S&P/TSX 60 Index ETF) closed at $22.17. The historical 30-day volatility is 7.75% (source: Montreal Exchange). An investor is convinced that the index is on the rise and that he would benefit with an investment equal to one thousand shares of XIU; albeit at a much lower cost. A thousand shares would cost $22,170 while the 10 December/22.50 call options have a premium of $0.39 per share. This translates to an outlay of $390 for 10 options (each option represents 100 shares).

The breakeven price of the XIU in this strategy is $22.89 (22.50 + 0.39). Given the fact that the XIU seems to be on the rise, can it generate a profit at the option’s maturity of December 19, 2014? The fundamental question is: what is the probability that the XIU will exceed the breakeven price of $22.89 and generate a profit?

The term probability invites the use of a probability calculator. There are a few available for free on the Internet. For example, the Option Strategist website offers a "Free Probability Calculator". This calculator is very easy to use. Simply enter the current stock price ($22.17), the breakeven price ($22.89), the number of calendar days until December 19th (the third Friday in December and last trading days of the options). This number of days is 89. Enter the volatility (7.75%) and finally, press the calculate button to get these results:

- Probability that XIU will be higher than $22.89 on December 19, 2014: 16.2%
- Probability that XIU will be lower than $22.89 on December 19, 2014: 83.8%.

According to these results, there is only one chance in five (or 20%) than the price will reach a level which allows us to make a profit. This call option strategy with a horizon of three months seems rather risky because there are four chances in five (80%) of losing the premium if the position is held until expiry.

## Example 2

Since it does not seem like an attractive strategy in the short term, let’s try with a much longer expiry: for example, March 2017 with an exercise price of $23.00. The premium is high at $2.00 per share or $2,000 for 10 call options but it's still less than $22,170 for the purchase of a thousand XIU.

With this option the number of days until expiry (March 17, 2017) is 908. The new data is entered into the probability calculator: current price $22.17, target price $25.00 (23.00 + 2.00), number of days 908, volatility 7.75%. The result is:

- Probability that XIU will be higher than $25.00 on March 17, 2017: 16.2%
- Probability that XIU will be lower than $25.00 on March 17, 2017: 83.8%.

This result is more disappointing than the previous one: despite a much later expiry, the target price is so high that the probability of achieving it is one in six (16.2%). One reason is that the volatility of XIU is low. An investor could choose a more volatile stock (volatility is published on the Montreal Exchange website) and they would see the resulting difference in probability.

## Example 3

The impression that arises from these two strategies is that it seems better to do the opposite: instead of buying, one should sell call options with an opening sale (short) covered by the stock. In this case, the investor must have one thousand shares of XIU (now at $ 22.17) in order to sell 10 call options. This strategy is generally more advantageous if it is used in the short term and repeated often.

The investor sells 10 October/22.25 calls at $0.18 per contract and receives $180. Using the probability calculator with the current price of $22.17, breakeven of $22.43 (22.25 + 0.18), 26 days between now and the expiry of October 17 and volatility of 7.75%, we get the following results:

- Probability that XIU will be higher than $22.43 on October 17, 2014: 28.6%
- Probability that XIU will be lower than $22.43 on October 17, 2014: 71.4%.

The investor has a one-in-three chance that the price will be higher than the breakeven point and two out of three chance that it will be lower.

But no matter where the share price will be when the options expire, the investor has collected a premium of $0.18 per share, or 8/10 of 1% in 26 calendar days (0.18/22.17) or 11.40% on an annual basis. If, in addition, the shares close at 22.25, there will be an additional profit of $0.08 (22.25 – 0.17) on the shares.

The probability calculator is a tool that the investor has every interest in using to streamline their approach and better control the power of options. One must not forget that options have a limited lifespan and investors cannot reduce their risk over time, as can be done with stocks which are perpetual by definition.