I think there are three separate issues here.The first is the statistical issue of nuisance parameters. You want to estimate the volatility of the underlying. You need to know the expected return in order to calculate historical volatility, which you want to use as an estimate of underlying volatility; but you don't care about the expected return for its own sake. You observe a time series with an actual return that you know is much different from the underlying expected return. If you use the actual return as expected return, you get an unreasonable actual volatility.The standard answer to this problem is it doesn't happen very often. If you happen to get an unlikely outcome, you have trouble making predictions. Bayesians have no trouble with this, they can compute a reasonable volatility estimate anyway because they don't use the observed parameters as estimates of the underlying ones.The second issue is the nature of Brownian motion. In theory, volatility can be observed directly, it does not need to be estimated. It is impossible for a realized path of a Brownian motion to have a volatility different from the underlying volatility (the actual return can vary, but not the actual volatility). In the example, you are observing only discrete points from the path. Strictly speaking, there is nothing to estimate, the volatility is observable.It's like a coin that has already been flipped, but the result has been concealed from you. For some purposes, like a bet between two people who don't know the result, it can be treated the same way as a coin that has not yet been flipped. But for other purposes, this is will lead to error, for example if you are betting with someone who does know the outcome.Finally, there is the difference between risk-neutral and actual probability measures. This should only affect the drift rate, not the volatility. However, as you point out, any realized volatility will be different for the two measures unless the actual expected return is the risk-free rate. Therefore, the same data can give different estimates for the same volatility, under different measures. It's not a mathematical contradiction, but it is a practical one. I think it springs from our inadequate understanding, and will someday be resolved.

I will provide a different perspective, largely based on how to practicallly deal with limited information, say in the case of your IPO option:First of all, no matter what method you use, 10 days of data is not enough to get any meaningful estimate of volatility, except possibly for a contractual settlment of a 10-day variance swap you traded at the IPO. Philosophically, I have a problem with using a single stock's historic data for any kind of option pricing, no matter how long the stock has been trading, although I admit I sometimes use it as one of many inputs to pricing an option when there are no available implieds.Implied volatility is completely different from historic volatility, partly because they cover completely disjoint periods, but also because an option's price gives more weight to the volatility when the option has high gamma than when it has low gamma. So even stochastic vol models do not equate implied volatility of an option with expected realized volatility because of the need for gamma weighting.The problem of pricing an option on a stock with little or no market data is similar to the investment banker's problem of pricing the stock of the IPO before the market trades it. A better analogy would be to quoting an interest rate for a 100 year loan when you only have a yield curve going out to a few years, and historical data for the short rate going back well fewer than 100 years. Do you simply extrapolate the yield curve, calibrate a model to historic short rate data, or tack a swag risk premium on the longest rate you have? The answer of course is that you use whatever rate you expect will make you money on the trade.Similarly, you could use comparables when selling options on new stock, but that only gets you so far. The vol you use when selling an option is purely a function of how competitively you want to sell that vol given what other positions are on your book, and at what level you don't mind losing the trade. I once got a quote for vol at 35 bid - 50 offer (and not on some iffy IPO, but a major emerging market stock index), and understood that was as tight a market as other desks could make for me...

I'm probably a little late in the thread however isn't RMS the term used for the calc of volatility utilising quadratic variation? ie ([<In(S)>/t]^0.5)*No of trading days. I'd be suprised if this is a new concept to quants.

Esrtwhile, just found this old topic and have found it fascinating and informative. But for the risk of revisiting old ground, when you say:"If he now does nothing all day but watch the stock go up and down without delta hedging, he more or less has a coin flip for his P&L.If it finishes below 110.5 (his call strike) his P&L takes a hit. If it finishes above he gets a windfall gain."Are you sure about this? Let's suppose (for simplicity) that going into the final day the call is exactly ATM (so using the approximation for ATM calls and puts would have a value around 0.44 (=2/5*110.5*0.16*squareroot(1/256))and we have fully delta hedged upto this day, so flat P+L so far, and now (approx) short stock half the number of calls bought.Suppose that we don't delta hedge intraday on the final day. Wouldn't the final day payoff be a V with the bottom of the V being a loss of 0.442 (lose all the call's initial value and stock unch), and break even if the final day move is at least +/- 0.885 (ie. less than 109.6 or greater than 111.4) ?ie. Not necessarily a windfall gain if mkt up final day - only if it rises enough, and no P+L hit if mkt falls enough on final day. We only have a P+L hit if mkt closes near ATM. In other words, going into the final day with no further hedging we are taking a real big bet on the absolute size of the mkt swing (long straddle held to expiry). From a theoretical trading standpoint is it worth delta hedging until the call's value is all but negligible, then our P+L loss is virtually certain but negligible? In trying to balance up the earlier discussion that buying OTM calls is a losing strategy, though true most of the time, it seems it will occasionally reap an extremely large gain(either way). So presumably shorting OTM calls which move ITM, delta hedging up till the final day and NOT hedging intraday on the final day is a very risky strategy..Also from a trading perspective would it be an idea to compare the initial premium paid with the ATM call/put approximation with 1 day remaining (ie.0.442 above) when deciding at the outset if it is worth delta hedging a long OTM call. For example, if you paid 0.7 presumably delta hedging upto the final day would reduce your likely loss given the annoying convergence to ATM, whereas if you paid only 0.2 for the call delta hedging would probably increase your loss most of the time. In the example you've given the call will only rise until the final day so implicitly we must be starting below 0.442. ie. We know at the outset that it isn't worth delta hedging our long call.It's also making me think when we value an OTM call should we make an 'adjustment' using a % of the final day ATM value (eg. x% of 0.442 here) based on our 'view' of the final day P+L volatility..QuoteOriginally posted by: erstwhileYes - I was thinking it is time to end this before everyone gets bored! Hopefully not too late...OK - the first warning signal to me when I was talking to this trader, and I probably have an advantage here having managed traders in the past, is a trader who is effectively telling me that the basic european option replication method devised by Merton does not work in some cases. Having had large option books with strikes and maturities all over the place it would be pretty worrying if you could suddenly take losses due purely to the replication method failing!The trader is telling me that for deep OTM calls it fails, as you can get a 1% per day up move which is actually zero vol, not 16 vol. Thinking about it, if it fails in equity, it must also fail in currency options. But a USD call is a put on something else like JPY. Therefore by symmetry the method must also fail for deep OTM puts! So should it also fail for 1% a day down moves? How far OTM must these options be for the basic replication method to be useless?? This is not sounding right. We are not talking about real world effects causing the method to fail - we are judging the trader's claim: the replication method is wrong because even though you put in the right vol, and do the hedged the model tells you to do, you lose money.Clearly this trader has in the past bought OTM calls, mismanaged the hedge and lost money. His explanation to his boss must have been the explanation he gave to me. In other words his boss bought the explanation and the trader was not blamed. It was Merton's fault! The trader now believes this fiction, and happily repeats it to others.The second warning signal I got was his erroneous statement that 1% per day up movements will act like zero vol days to an option delta hedger. Let's first dispense with this rubbish.It is wrong to use the standard deviation to calculate the volatility appropriate to a delta hedged option book.What you want is what some people call "zero trend" vol, but what I call "root-mean-square" vol, as it helps me remember how to calculate it.To calculate RMS vol, do the following:(1) calculate log-ratios as usual(2) square the log-ratios(3) take the mean(4) take the square root(5) multiply by the square root of the number of trading days, i.e., sqrt(256)=16If you do a longer term graph of any normal stock index or other normal financial underlying, you will be almost unable to see any difference between the STDEV vol and the RMS vol.But the STDEV vol of a stock that goes up 1% per day is zero, and the RMS vol of a stock that goes up 1% per day is 16%!It is easy to see that the RMS vol is the correct one to use: the option trader is hedged at the end of the day and an up/down move of 1% leaves him P&L flat the next day in our case, regardless of whether it was an up or a down move yesterday.So the trader's "one percent up every day is actually zero vol" was rubbish. He was probably able to demonstrate it to his boss in a spreadsheet (or maybe even on Bloomberg?), and the boss simply agreed that it was bad luck that it was a zero vol period.So what about him losing money? Isn't it right that you would carry a short stock position, with the market going up 10% and then the option expires worthless?Yes, but the option has gained value every day to offset the short stock losses. The losses all occur on the last day, due to an unhedged option expiring out of the money.Let's look at the P&L account.If you calculate the daily P&L you will see that the P&L is flat every day up to the end of day 9 (the day before expiration). But on the 10th day, he comes in in the morning with the stock at a price of 100* (1.01)^9 = 109.4.If he now does nothing all day but watch the stock go up and down without delta hedging, he more or less has a coin flip for his P&L.If it finishes below 110.5 (his call strike) his P&L takes a hit. If it finishes above he gets a windfall gain.Let's say we now divide the last day into 10 segments, and have him delta hedge after each time period.He will again maintain his P&L and if he loses any money it will be in the last period when he stops hedging.Now we can divide the last period into 10 equal segments, etc.The loss would in this highly unlikely scenario be caused by being unhedged into a point-expiry, and being unlucky.It is doubtful that this is what actually happened to this guy.Keep in mind here that we are not looking at the real world. We are trying to judge whether his excuse for losing money was valid. His excuse is that mathematically, even in the Black-Scholes world, he would have lost money, as the replication method does not work.OK - part one of the problem is solved!We have determined that(1) As far as an option delta hedger is concerned, STDEV is not what you use to get historical vol. You use RMS vol. His claim that he experienced zero vol was nonsense.(2) It is wrong to say that even within the Black-Scholes world the standard Merton replication strategy fails. The P&L as a function of time is constant. If his option had been struck at 100*(1.01)^10 plus or minus a small amount, you would find that the P&L is continuous all the way into expiration.Part two was "how do you explain it to him"?You don't use words like "martingale" or "stochastic" or "moments".Here the easiest answer is to set up the P&L of the trade in a spreadsheet as a function of time. Set the thing up so that the option can have a slightly higher or lower strike and he would see that the P&L is absolutely constant through time.Then show him how the P&L would have decreased with time if the market really had been zero vol. It would decay away continuously, not be all lost on the last day or in the last hour.

Yes, this topic is interesting, and many people continue to not quite understand some of the concepts, given some of the PMs I have got.You are correct! The actual final day's PNL will be the same as if you had put on the entire position on the last day, paying 0.44 per option and then not delta hedging.You asked: "From a theoretical trading standpoint is it worth delta hedging until the call's value is all but negligible, then our P+L loss is virtually certain but negligible?" In order to answer this question you have to re-frame it very precisely. What does "is it worth delta hedging" mean? You need to think very carfeully about this point so that you can ask a more mathematical question. A famous finance quant once said that it didn't matter whether or not traders delta hedged, and that they could use any delta they liked. Traders were amazed at this comment, but here "theoretically" he probably meant that the overall strategy would on average have the same expected value. He didn't say that the PNL would have the same variance, which it obviously wouldn't. So if we say that we don't like a big swingy PNL, then maybe we should delta hedge. If the option is very small in size, you might want to take the hedge off completely and just gamble the premium. I don't really trust the standard Black-Scholes model to give a good hedge ratio once the (absolute value of the) delta is outside the range of 10-90%. I'm not saying i have a better model, but what I do is look at the exposures and ask what sorts of events would move the market enough to make these options become at the money.I used to have a trading book that consisted of long unhedged tiny premium options that had drifted out of the money, or which I had bought for risk management purposes, so as to gain some "wing exposure". But this is a matter of taste rather than a theoretical result. If your entire book consists of nothing but this, you may not have a very long trading career!In terms of pricing in the probability of getting one of these "point expiry" situations (option atm on the last day), you could easily do this if you were worried enough about the danger. You calculate the probability of this happening (usually pretty low) and multiply this by the value of a one day ATM option. You will probably often find that this number is less than the option bid-offer. I used to do things like this with barrier options that might have unmanageable gammas at the end of their life and near the barrier. The probability of actually ending up in the situation was initially nearly zero, but you could monitor this number and hedge as if you had a liability (of some reasonable amount to represent trading loss) if it actually happened. Then if the market almost gets to that point but misses, you get a windfall gain (as there never really was a liability). If there is a big swingy gamma mess at the barrier, you have all along been hedging as if you might have to pay extra cash out, so you are cushioned.Overall you can make adjustments like this on day one, but they will probably be tiny. Hedging as if you will take a loss if the market lands in the wrong region can be a clean way of cushioning (though not strictly hedging) against this or other risks.

Hmmm. Feel like a "thick" trader...I hedge the option. Every day the price go up 1%. Making my option more valuable as the price get closer to the strike. The loss on the hedge is corresponding. Falt P&L.On the final day the option did not go in the money, giving no payout. I buy back the final amount of stocks that I have shorted. Doesn't this all mean, that you started to sell stocks at the low price and in the end bought them back to a higher price. Assume r=0. Theta plays a trick here? The loss of option value due to time, you never get back in your hedge.

Just to clarify. I realize you don't buy all stocks back on the final day. But at the very end, your delta is zero. So you are back to neutral position in the stocks. Now the price was lowest at the first day. Meaning you can not have made any money on your hedge as you always bought back stocks at higher prices.

you might want to go check the numbers. your otm option should be itm by the time it expires if it moves up 1% every day.

I was not looking at the particular prices mentioned before.Simply look at a call option which is not itm at expiry.Summary simple example:day 1:buy call option, sell stockday 2:price increasere-balance hedge, buy back some stocksday3:price increasere-balance hedge, buy back some stocksday 4:final dayoption not exercisedbuy back the rest of the stocks (expensive now..)

i am telling you that if you find an otm option priced at 15.8% (1% per day) and the underlying moves 1% per day, you will break even. go through it. you will find it itm by expiration -- or worth 0 at purchase. if you find otherwise, please show me. the model accounts for this and doubting it myself, i went through it on excel. but if you find, differently, again, please show me.

Ok. I trust you!But if the price moves less up than 1% per day, it might not be itm at expiry.Have I then made the error by using the implied vola when pricing and the realized vola simply was lower in the end?

no need to trust me. always better to check for yourself. we are agreed, if realized<implied then p&l is bad. that is the error, implied>realized. direction is not important.consider the problem the other way -- you buy the call and the underlying goes down 1% every day. i think everyone is more comfortable that way (feels better even to me), but the p&l is no different. or if you bought the put and the underlying went down equivalent %. same thing.

On using the RMS vol:I figured a direct relation between RMS Vol and STDEV Vol:RMS^2=STDEV^2+ Annualized_Average_Return^2Why is this a better measure I don't fully understand apart from the fact that in this particular situation, the returns (1% every day) are not reprezentative for a normal distribution therefore BS won't do the job.Of course that if you assume another Vol, BS may do a better job OR MAY NOT. The same as, logically, TRUE implies only TRUE, while FALSE can imply either TRUE or FALSE.This reminds me of an old quiz: These three brothers inherit from their father 17 camels. The testament says that the oldest brother gets half, second gets 1/3, and the youngest gets 1/9. Of course they couldn't agree on the split so they brought this wise guy who came along with his own camel, and now they have 18, it splits to 9+6+2=17, and the wise goes along with his camel.Which is not necessarily the correct answer, but rather a funny number game.Correct me if I'm wrong. Thanks.David