#RuggeroRespigo on #Reddit #stockmarket How to compute the terminal price distribution of a stock at expiration based on options prices Im reposting this from where I wrote it in a comment in /r/options here: reddit/r/options/comments/2pzpjk/formula_for_iron_condors_based_on_gaussian/:TLDR: Option prices have a 1-to-1 relationship with implied vols, which have a direct mapping to the marketplace consensus of what the probability distribution of possible prices at expiration is for a given underlying asset. Theres a poor mans way to do this quickly by looking at the butterflies across the entire option chain, and you can see what the distribution looks like.The way to do this is to calculate the price of every butterfly on the curve. Heres a way to think about this.## SETUP ##Say you have a stock XYZ that is worth $10.00. It has options listed from the 0 strike all the way to the 25 strike. For argument now, assume that the probability that the stock goes above 25 before expiration is effectively 0.## THE LONG CALL CONDOR ##Now lets say I buy the 0 call, sell the 1 call, sell the 24 call, buy the 25 call. What does the option expiration graph look like? https://i.imgur/zhxvq14.pngYoull make $1 pretty much ANYWHERE on the curve.so how much should you pay for that condor? $1.00how much will you get on expiration? $1.00whats the probability youll get $1? 100%.So whats it worth? $1.00, dammit.This is the 0-1-24-25 call condor. (you could do it with puts too, doesnt matter.)## BREAKING THE CALL CONDOR INTO TINY BUTTERFLIES ##Now consider that another way to synthetically guarantee that position is to have a position that looks like:buy the 0-1-2 call fly buy the 1-2-3 call fly buy the 2-3-4 call fly buy the 3-4-5 call fly ... buy the 21-22-23 call fly buy the 22-23-24 call fly buy the 23-24-25 call fly If you do thatl youll own a bunch of little flies that looks like this: https://i.imgur/wkeCRvk.pngNote that each of these little flies peaks at a max value of 1. and as the terminal price increases, the peak fly decreases in payout and the next fly increases (with slope -1 and slope 1) so they continue to exactly offset each other across the entire range. so basically youll have 23 call butterflies (btw, i use fly and butterfly are interchangeably) that cover the entire range and GUARANTEE that youll have a payout of EXACTLY 1 at ALL POSSIBLE STRIKES.Likewise, owing each of these call flies is the SAME as ONLY owning the 0-1-24-25 call condor, because all the intermediate strikes cancel each other out. Demo for the lazy:long one 0-1-2 call fly : +0 call, -1 call -1 call +2 call long one 1-2-3 call fly: +1 call -2 call -2 call +3 call long one 2-3-4 call fly: +2 call -3 call -3 call +4 call as you see, the long and short positions offset each other except at the edges so adding up the position of being long every fly on the board is equivalent to owning the widest condor.## TURNING THIS INTO A TERMINAL PRICE DISTRIBUTION ##Why is this important? Since the fair prices for owning something that is worth exactly $1.00 with 100% certainty is $1.00, we expect the value of the wide call condor to be exactly 1.Further, weve now created a series of positions who are distinct, but can be COMBINED to create something that is worth a known, fixed amount at all possible outcomes.Now, if you look up the value of each of these call flies, itll be some number LESS than 1. The SUM of all these call flies will be 1. Now the individual PRICE of each individual call fly will be the probability that the stock will end at that price.So for example, with stock XYZ trading $10.00, the 0,1,2 call fly is way out of the money. you probably intuitively know that the odds of the stock ending there are pretty low (close to zero). so the call fly is probably going to be worth almost nothing. demo for the lazy:0 call value: 10.00 (basically ONLY intrinsic value for a deep ITM option)1 call value: 9.01 (almost entirely intrinsic value for a deep ITM call, which would be $9.00. however its a little closer to the ATM strike, so it has 1 cent of vol value)2 call value: 8.03 (hey this one is 8 bucks of parity value, but its closer so now it has 3 cents of vol value).So this fly is worth 10.00 - 9.01 -9.01 + 8.03 or 1 penny. (buy one 0-call, sell two 1-calls, buy one 2-call)if you buy this call fly, youre paying 1 cent. if the stock ends at the price of 1, youll make $1.00. So the market is implying a 1% chance of ending at that strike. You can repeat this exercise across the entire curve to get the terminal price distribution.In practice, a few things to note:you wont get to execute at midmarket. So if you use true bid/ask on a deep call fly, youll find that youll be paying a lot of money to own it because youre really just trading delta with the marketmaker. if you want to do this as an exercise in backing out the terminal price distirbution, pick a LIQUID stock (SPY), and use the midmarkets for every single one.if you DO execute, youll really come up with 2 distributions: the BUY distribution and the SELL distribution. (buy the 5 call against the 5 call offer, sell the 6 call twice by hitting the 6 call bid, buy the 7 call by lifting the 7 call offer AND the reverse)in real life the strikes wont be all neatly separate by 1. near the money they may be closer (1s), then as you get further away they may be $2.50 or $5.00 or $10 increments. if so, you need to adjust accordingly. So if the fly interval size around that strike is 5 times bigger, itll make that outcome 5 times more likely (5 times the price of a 1/5th sized fly). however, you just need to normalize it by dividing by 5. youll get a series of disparate points at different strikes, but then you can try to interpolate a smooth curve through them (its fair to assume the terminal price distribution is smooth, unless you have a specific reason to believe otherwise, like intended corporate action, some sort of outside influence, etc).
Posted on: Sun, 21 Dec 2014 23:59:04 +0000
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