This should be an easy one: how can I get the market-implied probabilities out of European option prices using the Black Scholes or Black 1976 framework? I looked up the question in two books ("Financial Claims and Derivatives", King 1999, and the Natenberg-Classic). The former says "N(d2) is actually the probability of the option ending up in the money." and the latter "the delta is approximately equal to the probability that the option will finish in-the-money".That N(d2) is the probability of the option ending up in the money is completely counter-intuitive. My at-the-money options always have a negative d2 value. Hence, the probability of breaking-even for the underlying until expiration is less than 50%?The delta version makes some sense intuitively, but I do not like "approximately" at all. Can anyone help?

that s correct N(d2) = prob({S>K})but d2 < 0 when the forward is lower than K or vol is very high - the first case is intuitive, the second is notyou might also want to look for dupire's formula which gives the implied distribution density at maturity :prob({S=K}) = d2C/dK2 / PV Factoressentially this formula says that the implied distribution is related to the price of butterfliese.

Before you get too worked up about these types of calculations, remember that the Black Scholes model will tell you the probability of exercise under the risk netural measure. This is probably not what you are really interested in.

I am not sure, David. What I am trying to do is construct an APT-type portfolio selection that uses, among other criteria, a "crashophobic" element. Say I am comparing asset classes, and I want to infer from an average volatility smile (using ask prices of each) the implied "crash" likelihood that traders are actually willing to sell (defining a crash, for example, as a decline by 50% or more over one year). I am sure that I will run into a lot of problems, but I do not want a flawed interpretation of BSM to be one of them. Are you with me?

Check out this Bank of England Page...http://213.225.136.206/statistics/impliedpdfs/main.htmI am no expert on options but have also heard delta is approximately the probability of in the money result.Now the 'approximately' might be because it is actually the probability of beating the risk free rate of return (as opposed to being in the money).Now how does that tie in with delta hedging?

Let’s use an example. Per 8/2, options with expiry on 16/6 for ES settled as following:S&P500 ES Options (Globex)1200 Call = 34.501080 Call = 132.25InputsRFR = 2.85%Underlying = 1206.50We get for the “ATM” Call:Delta 0.54N(d2) 0.52Implied Vol.: 11.1%We get for the 10% ITM Call:Delta 0.948N(d2) 0.861Implied Vol.: 16.5%N(d2) using ATM-Vol. = 0.951Is it save to say that the underlying has an implied risk-free probability of trading at or above 1200 of 52%, and above 1080 of 86.1%? What does “risk-free probability” actually mean? And where does that leave the Delta?It would be very useful to infer sth like “the implied probability of a 10% decline until June 16th is 2.8 times higher than under the assumption of log-normality”. (because of the volatility smile in the ES: (1-N(d2))/(1-N(d2*)) where d2* is d2 using ATM volatility) But is it proper?

I assume you mean implied probabilities of exercise (ie probability the option will end in the money). If that's the case you are indeed looking at N(d2). This is what the whole field of structural models of credit risk are founded on (KMV for example uses d2 but applies a proprietary distribution and not the normal distribution to it)N(d1) is a bit tricky to interpret as a probability, it is the probability scaled up to by the proportion the asset is likely to end up in the money....

Why then does the probability of exercise decrease with volatility?

I don't to confuse you even more but:Both N(d1) and Nd(2) are probabilities for the options to end up in money!They are probabilities calculated with two different probability measures.In fact there are as many probabilities for the option to end up in the money than there are probability measuresso I would say that implying probabilities from prices makes little sense unless you make up your mind on a particular numeraire as a reference.Take a simple example:Instead of talking about probabilities, let's talk about digital option prices (ie Arrow-Debreu prices).Consider these two types of digitals:- Digital 1 pays 100 in cash when spot ends up above strike at maturity- Digital 2 pays the stock value at maturity when spot ends up above strike at maturity. The stock today is worth 100The price of Digital 1 is 100*N(d2) whereas the price Digital 2 is S*N(d1) = 100*N(d1).So if cash is your reference asset then N(d2) is the probability for the stock be above the strike and the delta of Digital 1If the stock is your reference asset then N(d1) is the probability for the stock be above the strike and the delta of Digital 2Hope this helps