Intution behind buying the option when implied vol seems low

Hi All: I've started reading "volatility trading" by Euan Sinclair and it's a very nice book. It's not so theoretical but instead focuses on the practicalities when dealing with trading options. I've never worked in options ( so just understand the basics from textbooks such as cox and rubinstein, baxter rennie etc ) and I've often had the following question and I figured ( since reading this book, reminded me of my question ) I would ask here. The book touches on the topic but, so far, doesn't give the intuition I'm looking for. So, here goes.

You have say a stock XXX. You calculate the implied vol of one of one of its calls. It doesn't matter which one.

Next you find that the volatility that you estimate over the life of the call is MUCH, MUCH, MUCH greater than the implied vol of the call. So, you buy the call and hedge your position by selling the correct number of shares of the stock. You modify your hedge as needed and do this until the option expires. Now, according to Sinclair, ( and of course this is true ), the end result should be that you generate some profit if your volatility estimate was a decent estimate in hindsight.

The part I don't understand regarding the profit is the following. In a world where the implied vol was equal to the true volatility that occurred, the hedging cost should be equal to the value of the option. So, what goes on in the case where your forecast is greater than the implied vol and, in hindsight was pretty close to correct ? Does it mean that the hedging cost is less than the value of the option so hedging doesn't cost as much so you end up profiting ?

Conversely, if your forecast is MUCH, MUCH less than the implied vol, then the standard profiting attempt is to sell the call and buy the stock in the appropriate amount. But, hopefully if I can understand the first case described above, then I will understand the case where one sells the call.

Also, if this question is covered in cox and rubinstein or baxter and rennie (it's been so long since I looked at them that I could have forgotten), I can check those out. Thanks for any insights-wisdom.

• At which chapter/page is this discussed ? – Gabe Jul 10 at 21:40
• @Gabe: It's probably discussed in a few places ( since I haven't finished the book ) but the place where I read it was in the middle of page 6 in chapter 1. It's a pretty well known way to make money from forecasting volatility. I just always wondered why it worked and Brian explained it quite nicely. C and R may also explain it so when I have time I'll go back and look there also. – mark leeds Jul 11 at 2:05
• So the claim is that if you are long a vanilla European option and you bought it at a given implied volatility $\sigma_1$, and then you delta-hedge the long position (but only delta hedge), then if the future realized volatility turns out to be higher, say $\sigma_2>\sigma_1$, you actually make a profit? – Gabe Jul 11 at 9:59
• @Gabe: On page 6, Sinclair doesn't go into the details that you did but that's correct. He just assumes that your details are known by the reader. – mark leeds Jul 11 at 12:47
• I haven't looked into options in awhile, but another thing to keep in mind is, if the IV of a call is less than future realized vol, there also is the question of whether or not to hedge with the delta of the low IV call or delta of a theoretical call priced at your forecasted volatility. I forgot what book or resource showed that there is slightly different payoff profile between the two methods, even though both lead to positive expected profit. – confused Jul 29 at 7:42

Note: what you call true volatility is often termed realized volatility.

When you purchase a call or a put, each time the underlying increases in value, your hedge modification consists of selling a little bit. When it drops in value, you buy a little bit. Those buy-low/sell-high elements are a replication strategy that, as you note, would be expected to match the cost of your option purchase, all else being equal.

When realized volatility is higher than the implied volatility that drove your purchase price, you get "extra" hedge trading opportunities because the stock is moving up and down more than expected. Those trades pay off more than what you paid for the option, and are the source of profit.

(Great choice of book, by the way)

• Brian B: I think that I somewhat get what you're saying in that the stock is moving up and down more than expected. But I'm not sure that see how it moving up and down more makes the replication strategy more profitable. Is that because, since you sell when it increases in value, you'll be profiting more on that side and, since you buy when it decreases in value, you'll be paying less for it on that side ? If so, I get it. Thanks. – mark leeds Jul 10 at 19:22
• Oh, as far as the book goes, it's a unique book. I think it gives you some insight into how a quantitative but not totally systematic trader thinks about things. I haven't finished it ( not even half way through ) but I definitely would give it a thumbs up. – mark leeds Jul 10 at 19:27

The implied volatility is not G_d given, but comes from the collective judgement of market participants. If you are right and the market is wrong (unlikely in my personal experience but perhaps true for you) then you can make money when you are proven right by future developments (i.e. by future realized volatility being close to your prediction and higher than the market expected).

In other words the creation of options has opened up a new field for human beings to compete in making predictions (just as they have always tried to predict future events).

• Thanks noob2. I see what you mean by not G_d given but I'm still not clear on where the profit comes from. – mark leeds Jul 10 at 19:17
• Based on Brian's answer, it seems to come from the hedges generating more profit which I think is equivalent to the hedge costing less ? – mark leeds Jul 10 at 19:33
• Yes, the hedge becomes profitable (instead of break even) if the day to day price changes are systematically bigger than $\sigma$ used in the delta hedging calculation. – noob2 Jul 10 at 19:58