Hi,I'm doing a back-of-the-envelope calculation for an unhedged (and perhaps unhedgeable)trade, and wondered if there's a nicer formula that illustrates on aspectThis is going to be a ham-fisted explanation but...Suppose you wish to sell a naked call option on an underlying you can't hedge. For thesake of the argument, you know exactly the distribution (lognormal) of the underlying,including its effective "volatility" (that is only really a measure of the standard deviation).As a measure of the compensation to cover the risk, you wish the expected profits (under the above distribution) to be a multiple of the expected losses. What premium do you have to sell the call for to make this true (and what is its implied volatility)?Well, you can work it out numerically, by saying that the value of any expected profits are upside = \int_{-infty}^{x+v} p(f)*[max(f-x,0)-v] dfx is strike, f is forward price that the underlying will take at expiry, v is premium charged and p(f) is the (known lgonormal) distribution.andthe expected value of possible for the losses is:downside= \int_{x+v}^{infty} p(f)*(f-x-v) df then say upside/downside = k (k>1) and solve for v.The neat thing I would like to be able to do is to relate the implied volatiltiy correpsonding to vto the effective volatility in p(f) via (a) the moneyness of the option and (b) the multiple k I feel there has to be an elegant approximation to this relation.Does anyone have any smart ideas how to find it?Muchas gracias for any pointers..Tom

Tom,you say you can't hedge the underlying. I assume you mean you cannot trade it, so you will be short a call with no chance of delta hedging it.I also assume you cannot trade other derivatives on that underlying, nor can trade related assets as a proxy hedge.In this case you can throw away all the standard option pricing theory. forget volatility, implied volatility, etc, they are completely irrelevant.You are telling us you know the exact pdf of the underlying at time of expiration.Hence the value of the call today will be the PV of the expected value of the payoff (integrated the payoff with the pdf).you could then sell it at a higher price and expecte statistically to make money, as if you sold a toin coss at 0.51 and expected a 0.01 profit.The next step would be to use a utility function in you integral.It seems to me there is nothing else you can do. But maybe I misunderstood you

I'm not sure I understand you, but let me take a crack at it anyway. To keep things simple, subtract off the mean and divide by the standard deviation so the forward price has a lognormal distribution with underlying normal mean zero and standard deviation 1. Adjust the premium (v) and exercise price (x) by the same transformation. There are three possibilities about the future (not forward) price (f):f < x, you make vx < f < x + v, you make x + v - ff > x + v, you lose f - x - vThe probability of the first case is N[ln(x)] where N is the standard cumulative normal distribution.The expected value of ln(f) in the second case is where n is the standard normal probability density functionThe expected value of ln(f) in the third case is I think this is enough for you to compute your answer.

QuoteOriginally posted by: AaronI'm not sure I understand you, but let me take a crack at it anyway. To keep things simple, subtract off the mean and divide by the standard deviation so the forward price has a lognormal distribution with underlying normal mean zero and standard deviation 1. Adjust the premium (v) and exercise price (x) by the same transformation. There are three possibilities about the future (not forward) price (f):f < x, you make vx < f < x + v, you make x + v - ff > x + v, you lose f - x - vThe probability of the first case is N[ln(x)] where N is the standard cumulative normal distribution.The expected value of ln(f) in the second case is where n is the standard normal probability density functionThe expected value of ln(f) in the third case is I think this is enough for you to compute your answer.Thanks Aaron,I did something similar and numericallly searched for the v that gave the correct ratio.The rough answer seemed to be that you needed to sell the calls at roughy twice the "fair"value (meaning fair price = expected profit) to give expected profits at three times expectedlosses for an ATM options ("ATM" in this sense meaning the strike is at the futureexpected mean). Raising the strike increased this ratio, which was what I wanted to demonstrate inthis (illustratative) calculation.What I was hoping to was some to approximate the calculation to get a function g of the form, (price need to achive payoff ratio)/(fair price) ~ g(moneyness, payoff ratio)Tinkering around with approximation methods (e.g expansions of N etc)didn't seem to give anything cleanly, but I suspect there's some way to get at itout there...Thanks again.

I was being a dufus....This is simple. The qualitative behavior all comes out in a simple Taylor expansion.v is premium charged. U is expected upside, D expected downside.To make upside a multiple of downsideU(v(k))=k.D(v(k))if k=1, v=v_fairHence v(k)=v_fair+ (dv/dk)*(k-1)Diff wrt. kdv/dk = D/ (dU/dv - dD/dv)When working out dU/dk since the payoff disappears at the integration limits:dU/dv = prob( f < x+v)dD/dv = probl(f > x+v)Hence, v=v_fair + [D/(prob( f < x+v)- probl(f > x+v))]*(k-1)+...which has the qualitative form I was looking for. QuoteOriginally posted by: twQuoteOriginally posted by: AaronI'm not sure I understand you, but let me take a crack at it anyway. To keep things simple, subtract off the mean and divide by the standard deviation so the forward price has a lognormal distribution with underlying normal mean zero and standard deviation 1. Adjust the premium (v) and exercise price (x) by the same transformation. There are three possibilities about the future (not forward) price (f):f < x, you make vx < f < x + v, you make x + v - ff > x + v, you lose f - x - vThe probability of the first case is N[ln(x)] where N is the standard cumulative normal distribution.The expected value of ln(f) in the second case is where n is the standard normal probability density functionThe expected value of ln(f) in the third case is I think this is enough for you to compute your answer.Thanks Aaron,I did something similar and numericallly searched for the v that gave the correct ratio.The rough answer seemed to be that you needed to sell the calls at roughy twice the "fair"value (meaning fair price = expected profit) to give expected profits at three times expectedlosses for an ATM options ("ATM" in this sense meaning the strike is at the futureexpected mean). Raising the strike increased this ratio, which was what I wanted to demonstrate inthis (illustratative) calculation.What I was hoping to was some to approximate the calculation to get a function g of the form, (price need to achive payoff ratio)/(fair price) ~ g(moneyness, payoff ratio)Tinkering around with approximation methods (e.g expansions of N etc)didn't seem to give anything cleanly, but I suspect there's some way to get at itout there...Thanks again.

Can u plis elaborate the differentiation wrt k..not clear how u got the result dv/dk = D/ (dU/dv - dD/dv)Thanks,Wasp

QuoteOriginally posted by: waspCan u plis elaborate the differentiation wrt k..not clear how u got the result dv/dk = D/ (dU/dv - dD/dv)Thanks,WaspImplicit differentiation.U(v(k))=kD(v(k))diff wrt k(dU/dv).(dv/dk)= D + k.(dD/Dv)/(dv/dk)re-arrange and specialise to the k=1 as required by the Taylor expansion.

I have another question in this regard :basically u are calculating the fair value of option by using standard deviation of stock price and adding some extra premium depending upon k and moneyness.But what happens if the impiled volatility prevailing in the market is lower than the standard deviation of the stock price. In this case calculated fair price itself will be higher than the premium available in the market.Can we think in terms tinkering with varying the time period while calculating the standard deviation ??? Any suggestion !!!!cheers,wasp

Well, actually the basis for my calculation is the optionality embedded inone aspect of a commodities structure. The standard deviation in question comes from a fundamental model, hence the inability to hedge. If you are talking about stocks surely the hedgingcan be performed directly and that particular problem disappears?I don't work with equities much, but if there's a systematic difference betweenimplied and historical volatility I would look to other sources than my calculation to find the answer...QuoteOriginally posted by: waspI have another question in this regard :basically u are calculating the fair value of option by using standard deviation of stock price and adding some extra premium depending upon k and moneyness.But what happens if the impiled volatility prevailing in the market is lower than the standard deviation of the stock price. In this case calculated fair price itself will be higher than the premium available in the market.Can we think in terms tinkering with varying the time period while calculating the standard deviation ??? Any suggestion !!!!cheers,wasp

Can the problem be addressed in this way :1 . If implied volatility is higher than the historical volatility teh there is no problem.2. If implied is less than the historical then can we use shorter time volatility rather than annualized one ? if my trades continue till say max 20 days then why should I use annualized volatilty ? instead i can use 20 days volatility whose absolute value will be definitely less than the annualized figure of implied volatility.Even i have a feeling that the above logic is not very convincing but then somebody plis point out what I am missing by taking different time period for two volatilities.cheers,wasp

I'm not sure I understand you. "Annualized" volatility refers to the units in which volatility is stated, it makes no difference to the calculation. I think you are concerned with the period over which historical standard deviation is measured.It's always true, of couse, that in pricing options we use a volatility scaled to the time until expiry. For a three month option, for example, we could use an annualized volatility divided by two, or we could just measure total volatility over the prior three months and not adjust it at all. In the first case, we could annualize volatility from the prior 30 years or 30 seconds.In general, you prefer to estimate historical volatility over the same period as the time to expiry of the option. That usually gives a reasonable balance between up-to-date data and enough data data to estimate reliably.However, historical volatility is not a good predictor of implied volatility in most markets. For most applications you need to go to some sort of stochastic volatility estimate, or some other more complicated model.