I thought Nitro sounded right but thought it was something I should know for sure so I tried a quick example. Stock = 100 Strike = 100 vol = 20 T= 3 mos rate = 0% ATM straddle price from BS = 7.99 Implied vol should be 1 std of continous return. From the normal distribution there is a 50% chance the return will be +/- 0.7 standard deviations from the mean. An annual vol of 20% gives a 3 month vol of 10%, so this gives a 50% chance of a return between 7% and -7% or a stock price between 107.25 and 93.23. The range is 14.01, much bigger than the straddle price. Maybe the straddle price bakes in the expected gamma scalping profits like Nitro said. Or maybe this is not a valid analysis because of non-normality issues. But with fat tails the the range containing 50% of the price distribution would be even bigger. Anybody have an alternate explanation? Here's the disclaimers about stuff I am blowing off - if anybody thinks it changes the answer please chime in. I am assuming a mean return of 0 even though I think there is supposed to be some drift that I don't feel like figuring out - I made the interest rate 0 to make this less of an issue. I also think I remember seeing some term in BS reducing the expected return, maybe related to Jensen's inequality, which I am ignoring.
I also remembered I said I was going to post the study that shows put-call parity violations reflect informed trading and predict stock price movements, here it is: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=968237 I also think somebody was asking about how trading pressure in individual options affects their implied vols (and skew). It looks like this paper says something about it, but I haven't read it. http://pages.stern.nyu.edu/~lpederse/papers/DBOP.pdf
Correct for the most part, but the straddle break-even at expiration exceeds the range implied by your 20% annual vola figure: 92.01/107.99.
That is because BlackD used the median absolute deviation (MAD at about 0.7 sigma) when he should have been using the average absolute deviaton (AAD) of sqrt(2/pi) or about 0.8 sigma. That would have given him a range of 92.00/108.00/ He should also run it around the forward price, but since the risk free rate is so low, there is not much difference. Edit: changed 2*pi to 2/pi
Yessir on both counts as he's light on sigma. I was referring to his error of halving the atm straddle range. The atm straddle has always been used to mark the vola to a premium reference.
The expected range (range is Max-Min) is equal to roughly three times the price of the straddle. In your case, the range for three months should be ~ $21.
You've made a mistake. This rule about AAD that is about 0,8 sigma will hold just for a normal distribution. If stock prices or returns are not, one can't use this rule.
Thanks shortie. I appreciate it. I hope the market is treating you well. You deserve it if she is doing it. Also do not be surprised if you seem me less around here from time to time. This forum can be an addiction, so I have to watch out for the time spent here. Also, I have many other things that would make more difficult to show up more often. Cheers! PS: What happened to your shorting of Ford? I recalled your 100K share short position of sometime back. Did you cash or did they cash?
Thanks for the replies. But I still think this is a bit puzzling. Atticus: I didn't halve the ATM straddle. The BS prices for the put and call were both around 4 and added up to 8. And to ignore the forward price issue I set the example interest rate to 0. But if you meant I should have doubled it when comparing it to the expected expiration range calculated directly from the implied vol that makes sense, I see that now (brain fart!). Emilio: I think this is what I was referring to when I said I was blowing off the Jensen's inequality part, I am just trying to get close to check the intuition. And the range does not come out even b/c I am using continous compounding. MasterAtWork: The way I think about this is that the market knows returns are not really normal, but when using the BS model you have to squeeze the "real" distribution into the closest fit normal distribution. But when you are using this implied vol outside the BS model, is there a particular adjustment you would make to it or do you just need to understand it is only an approximation? So that leaves me straddle breakeven prices of 92.01/107.99 (thanks Atticus) and the vol taken literally says 50% of the time the price at expiration will be in a range of 93.23/107.25 ( ignoring Jensen's inequality which I think would actually make it even tighter than the straddle breakevens). This looks sort of in the ballpark of the straddle showing the expected move at expiration, but a little more. If that is right then this brings up 2 questions: 1) Is the difference from to the expected gamma scalping profit Nitro mentioned, which should make the straddle cost more than half the expected range at expiration? Or approximation error or something else? I am having a hard time with the idea that the component parts are priced according to a no-arb model but the price of the combined position could allow expected gamma scalping profits, I must be missing something. 2) RiskFreeTrading- where does the range you give come from and how does it square with this calc? Are you just using a rule of thumb that adjusts for fat tails? I would think option prices/IVs would go up to account for this too. Sorry if I am being anal about this, it is not as straightforward as I thought and seems like something pretty fundamental.