# Marketside chats #3: More options for non-dummies

December 19, 2013
• show you how to convert stock volatilities to something intuitive
• explain in basic terms the fancy concept of convexity
Please read Marketside chats #2: Options for non-dummies first for some of the other terminology.

Implied volatility (IV, a.k.a. vol)
The implied volatility of a stock (or index) is the standard deviation of its expected future annualized return. For example, Google options expiring in about a year show an IV of 24% at the time of this writing. If I buy a Google option now (either call or put), and the stock becomes more volatile some time soon (e.g. not after the option’s expiration, when nothing matters!), I will make money on my option on average. Another way to say it is that the expected value of payout of the option I bought is higher than the price I paid for it.
Assuming that volatility is roughly constant throughout that year, there’s a handy trick to convert that to daily price moves, which are more intuitive. There are 5/7*365-10 ~= 250 trading days in a year (*5/7 to exclude weekends, and -10 to exclude holidays). The way the math works out, standard deviation (and therefore IV) is proportional to the square root of time to expiration. Because 250 ~= 256 = 16^2, if we divide annualized IV by 16, we get the daily one. In our example, the option market is pricing Google’s 1-day standard deviation is 24% / 16 = 1.5%. (footnote 1) Since this is IV, it takes into account expected future price moves, not previous/historical ones (although they are often similar).
From this point on, you know the rest from basic statistics: 68% of the time, a day’s move will be within 1.5% up or down, etc. (footnote 2)
Removing uncertainty reduces volatility
The IV of an option will drop after an event that resolves uncertainty. Of course, the magnitude of this effect isn’t the same for all stocks on average.
Say A is a large company with stable earnings, and B is a small company whose distribution of earnings expectations by the market is very wide. E.g.
• A=Microsoft
• B=some biotech company whose product is a single drug which was released recently.
After an announcement of B’s earnings, B’s stock price will become much less volatile. The investing public now knows how that drug’s sales are going, and that was the make-or-break piece of information one needs to know about B.
However, after an announcement of A’s earnings, A’s stock price will not really become more or less volatile. A is already making a bunch of money on existing products. Therefore, B’s stock price will move (as a %) by more than A, since A’s earnings are much more likely to be within a tighter range.
Convexity
If you own a stock, you make $1 if it goes up by$1, and lose the same amount if it goes down by $1. This is so blindingly obvious that you might be wondering why we’re even mentioning it. However, options are different. In some simple cases, options can be priced with a closed formula called Black-Scholes. If you don’t want to take our word for it, or are curious for more, use e.g. this options calculator with the following inputs: • Stock & exercise price of$10,000 (both). Few stocks trade at $10,000, of course, but the actual levels don’t matter.$10,000 will give us numbers with more precision digits.
• Time to maturity (a.k.a. time to expiration) = 0.083 (about a month, i.e. 1/12 of a year)
• Interest rate 0.0000001% (that calculator disallows 0) so we can ignore rates, without loss of generality.
• Annualized volatility of 16. You now know what that is now!
These are the results you’ll get for Black-Scholes call option prices if the stock moves in a short period of time (footnote 2):
1. If the stock goes to $9,900 (=$10,000 – $100), with time to expiration being the same, then the option price will become 137.289. 2. If the stock stays at$10,000 the call option is worth $183.879 3. If the stock goes to$10,100 ($10,000 + 100):$239.086
In short, if the $10,000 stock price were to move by the equally likely amounts of +/-$100 (cases 1 and 3), the average of the 2 option prices is $188.188 >$183.879, which is the price of the option if there is no price movement in the stock. (footnote 3)
This obviously doesn’t constitute a proof, but it demonstrates that actual stock movement is good for you, on average, if you bought an option (either call or put). Of course, a downward movement in the stock (all else being equal) is bad for you if you hold a call option, but it’s less bad than an equal upward stock movement is good.
In options, this asymmetry is called gamma or convexity. Convexity goes hand-in-hand with optionality. What you see here is a micro manifestation of the fact that if the stock goes up by a lot, you gain a lot, but if it goes down by a lot, your losses are capped.

FOOTNOTES

(1) To be precise, the market usually expects volatility to be different throughout different parts of the year. Dividing by 16 would give you the volatility of an “average day in the year”, roughly speaking.
(2) Options can be seen as insurance. Roughly speaking, an insurance company charges buyers based on a higher probability assumption than the actual probability of an “insured event” (accident, etc.) occurring. They would not make money in the absence of such a buffer. If market participants on average hate the idea of being exposed to stocks going down a lot, they will pay up for that insurance. Therefore, if the market expectation (statistically) is for 20% annualized volatility AND the market dislikes volatility by some “reasonable” amount, the IV for an option could be 24%.
If this sounds vague, consider car rental companies that charge you ~$15/day for accident insurance = ~$5,500 annually ~= 1/3 of the price of their average (depreciated) car annually. This is equivalent to saying that you have a 1/3 chance of totaling their car if you were to drive it for a year. Statistically, this is rarely true. Alternatively, it’s equivalent to saying that you have a 1/15 annualized probability of totaling their car, but you are willing to pay for a 5x probability, e.g. to avoid the risk of having to come up with the cost of a car in case of an accident, to avoid the hassle of dealing with legal issues, etc.
(3) In reality, if the stock were to drop by 1% within a very short period of time, the market might get spooked and IV would go up. We are using 1% because it makes the point clearer; e.g. 0.1% would also work.
(4) In reality, stocks are best approximated as “lognormal”, i.e. the logarithm of a stock price follows a normal distribution. For example, +$100 and -$100 aren’t equally likely; *1.01 and /1.01 are. This makes sense; for instance, it’s easier for the $10,000 stock to go to$18,000 (price almost doubled), but if it goes to $10,000-$8,000 = $2,000, it means it dropped to 1/5 its value. Another qualitative argument: a$100 stock can go +$115 to$225, but cannot drop $115 to a negative number -$15!

In that case, a better set of 2 likely outcomes is $10,100 =$10,000 * 1.01 (already shows in case 3) and $9,900.990099 (=$10,000 / 1.01).  At that stock price, the call option is worth $137.708. The average of the 2 equally likely scenarios is ($137.708 + $239.086)/2 =$188.397, which is still bigger than the option price. So the argument still holds.

DISCLOSURE

The information provided here is for educational purposes only. Nothing in this article should be construed as a solicitation or offer, or recommendation, to buy or sell any security. Financial advisory services are only provided to investors who become Wealthfront clients. Past performance is no guarantee of future results.