Where does using delta as a proxy for %ITM come from? I've heard and read this numerous times and looking at quotes it's definitely there, the Prob ITM % and delta are usually very close. Can anybody shed some light on this (apparent?) relationship?
I have benefited so much, from videos from (from East-to-West) Harvard, MIT, Yale, Michigan, Stanford, Berkeley, San Diego, .... One of my absolute favs is this one, from MIT: ENJOY!
Here's an attempt to explain it in a simplified way (without watching the option pricing video, which is better): In the BSM there are two major numbers d1 and d2. These are fed into chunks of the equation (and therefore the sensitivities) by passing them to the Cumulative Normal Distribution Function (N(dx) in the equation). This lends itself to a straight forward interpretation of the probability ITM. N(d2) in the equation represents the Probability of expiring ITM under the assumption asset drift is the risk-free rate. This will lead you to natural ask what N(d1) represents. Well, it's really not clear. But you'll say N(d1) is literally delta - you didn't help me! Here's my take on why delta became the representation despite N(d2) being more mathematically sound. A napkin calculation of ITM probability can be derived from delta with some handwaving: 1. A delta of .5 (basically ATM) has around a 50% chance of expiring ITM. You could think of it as "it could go either way before expiration really". 2. A delta of 1 has a very high probability of expiring ITM. This is reserved for deep ITM options. 3. A delta of 0 has a very low probability of expiring ITM. This is reserved for deep OTM options that require a miracle to turn a profit. So naturally you could extend this to any arbitrary delta given you've satisfied the endpoints and midpoint (OTM, ATM, ITM). If you're willing to accept some degree of mathematical uncertainty (and of course understand that combining deltas isn't exactly the same as combining probabilities), delta can be used in this way to approximate probability ITM. Hopefully this helps.
This is exactly how Steven Shreve derives the BS option pricing logic in his stochastic calculus book. I like this approach the best. But the lecturer is horrible. Glad he is not from MIT.
I only took one semester of calculus so I'm mostly just getting the gist of the MIT lecture but it is extremely interesting none the less. Thank you! Tell me if I'm in left field with this observation though: The idea that stock price is a random variable though doesn't make sense to me intuitively. I get it for math purposes because they don't use a specific function or value for stock price but they have to use something, so they just kind of stick that in there as a place-holding variable, since there is no definitive function that produces what the stock price is and/or will be because it would be backward-looking and therefore inaccurate, so the best thing to use instead is a random variable. Does that make any sense?
Sure -- that "random" nature is what drives the Brownian/Weiner content in BSM. There are three things that consistently mess with general discussions, though (and maybe work into your own dissatisfactions...) 1) Markets trend and do so regularly (read: "predictively"). Calling it "drift" (as many do) changes nothing. 2) Much random mechanics assume a Gaussian/Normal {symmetric} distribution, but overall market price movement is logNormal {shifted left}, while markets at peak are Pareto {shifted right} distributed."Havoc!" 3) The use of ANY of Normal, logNormal, Pareto, or even Weibull ["Yay!"] distribution assumes that prices in time T are unrelated to prices from time T-1: which is patently ridiculous: the best estimator of price in time T remains the price in T-1. Always. Upshot? Keep your skepticism intact. "Simplifying Assumptions" are not meant to remove material parts of reality, but merely to mask them a bit to allow focus on some other parts. Sometimes, it's really important to peel back the mask and look deeper inside.
That helps clarify it for me! Thanks for the feedback. I know there are many studies and whatnot that show that simple trend following type stuff like "if price was up last month it is most likely to be up next month" have been proven (for stocks), so using randomness for the calculations would by definition be inaccurate. I have no doubt that these types of models are the most accurate relative to other mathematical models that have been used, but it isn't predictive by any stretch of the imagination. I know that most people have moved beyond Black-Scholes and I'm sure that people have proprietary models that are much better but what I have seen that's out there is mostly using random variables as far as I know. My non-Diff-EQ's-knowing brain would think that you could refine a formula like that by using some kind of coefficient or maybe a randomized integral interval value (as a coefficient. Is this a thing?) to try to factor in trends. I'm sure someone has done this already though, it's very obvious and for someone who is a math/physics PhD it's probably child's play.
Everyone uses BSM just because it became the de-facto "official standard" for option lingo and its strengths and especially its weaknesses are well known. There are many alternative models that try to plug the holes in the BSM model. For instance Corrado-su introduces skew and kurtosis parameters to account for the non-normal distribution.