How to find the probability of touching between two dates

[Socratic]

I'm trying to figure out how to calculate the probability of a stock touching a certain price target specifically between two dates I choose, without it being touched before.

For example, suppose I want to find the probability of a stock price touching some out the money target next week between Wednesday to Friday, without ever touching that target anywhere between now to Wednesday. How do I do it?

Anyone has an idea?

 

[tomorton]

You might be able to guess this but I don't think I'd go so far as to say 'calculate'. Maybe there is another way to where you want to be - why do you want to be able to do this?

 

[Socratic]

I think there should be something more than just guessing about it (given we assume certain value for future volatility) just as we can calculate the probability of touching somewhere between the current date and a future date as here:

Monte Carlo Option Probability Calculator | Option Trading Probability Calculator

The reason for getting the right idea about probability is because the time when a price reaches a target may make a big difference as to whether your options position is profitable or not.

 

[Martinghoul]

If you have a few option prices for the two maturities, you may be able to calculate the *risk-neutral* probability of touching (under a variety of assumptions).

 

[Cadavre]

I'm trying to figure out how to calculate the probability of a stock touching a certain price target specifically between two dates I choose, without it being touched before.

Last decade or so there always seems to be a short squeeze put in play by the invisible hand. Typical gambit is currency manipulation and sovereign devaluation right around option expiration time by Central Banks and so called Economic Advisory Boards known affectionately as the PPT.

You'd have to monitor the tick and / or review intraday histories to avoid the trade after the designated high or low water benchmark was "touched".

Sounds similar to so called stop loss floor strategies that trading systems based on optimized "back test" models use.

 

[tomorton]

Well, yes, that's just what I was about to say.

 

[Socratic]

Martinghoul: right, I can calculate the probability of touching at some point in between now and the the nearer date and I can calculate the probability of touching at some point in between now and the farther date. Yet what I want to calculate is the probability that the touching does not take place before the nearer date yet does take place sometime in between the two dates.

Cadavre: It seems very complicated to do back-testing on complicated strategies, all the more so when the historical options prices aren't all available. What I need I guess is some mathematical formula or computer modeling designed to give probabilistic answer to such a question, much like the link referred to above purports to do.

 

[Shakone]

I've probably misunderstood what you're trying to do, and the assumptions. If you know what the probability of it hitting before time T, then you also know the probability of it not hitting by that time. So if you know the prob of it hitting before T+whatever, and it not hitting before T, don't you already have the info you need?

 

[Martinghoul]

Martinghoul: right, I can calculate the probability of touching at some point in between now and the the nearer date and I can calculate the probability of touching at some point in between now and the farther date. Yet what I want to calculate is the probability that the touching does not take place before the nearer date yet does take place sometime in between the two dates.

I think Shakone has beaten me to the punch here.

 

[Socratic]

I've probably misunderstood what you're trying to do, and the assumptions. If you know what the probability of it hitting before time T, then you also know the probability of it not hitting by that time. So if you know the prob of it hitting before T+whatever, and it not hitting before T, don't you already have the info you need?

Yes, both the probabilities "not hitting before T" and "hitting before T+whatever" are known. Yet I can't just multiply the two, because the events are not independent. Whether or not it hits by T has a probabilistic effect on whether or not it hits by T+whatever. If it hits by T the probability of it hitting by T+whatever is 1, and if it doesn't hit by T then the probability is unknown to me: sometime not hitting means it went far away from the target and sometimes it means it used the time to approach the target.

 

[Cadavre]

An accurate forward priced Call / Put Probability surface?

 

[Shakone]

Yes, both the probabilities "not hitting before T" and "hitting before T+whatever" are known. Yet I can't just multiply the two, because the events are not independent. Whether or not it hits by T has a probabilistic effect on whether or not it hits by T+whatever. If it hits by T the probability of it hitting by T+whatever is 1, and if it doesn't hit by T then the probability is unknown to me: sometime not hitting means it went far away from the target and sometimes it means it used the time to approach the target.

You don't multiply them, you subtract one from the other. Because one event is a subset of the other.

Think of it another way. Suppose you simulate 100 paths. The paths you're interested in are those that do hit the level before T+whatever, but not the ones that have hit it before T. So you count the number of ones that hit by T+whatever and subtract all those that have hit by T. This leaves you with those paths that hit between T and T+. Doesn't it?

 

[Socratic]

You don't multiply them, you subtract one from the other. Because one event is a subset of the other.

Think of it another way. Suppose you simulate 100 paths. The paths you're interested in are those that do hit the level before T+whatever, but not the ones that have hit it before T. So you count the number of ones that hit by T+whatever and subtract all those that have hit by T. This leaves you with those paths that hit between T and T+. Doesn't it?

Considering you example, it seems intuitively right. I'm not yet sure though, because according to probability theory, non-mutually exclusive events satisfy the following formula

p(A or B) = p(A) + p(B) - p(A and B)

If the events are mutually exclusive then the term p(A and B) becomes 0 and the equation remains

p(A or B) = p(A) + p(B)

This is equivalent to

p(A) = p(A or B) - p(B)

Now your suggestion seems to correspond to this as follows

p(hitting only between T and T+whatever) = p(hitting before T+whatever) - p(hitting before T)

where

A=hitting only between T and T+whatever.
B=hitting before T.

This is because p(A or B) simply covers all possible events in which the target is reached before T+whatever.

The problem seems to me to be that the probability number given for p(hitting before T) doesn't subtract from it the cases where the same sample that hits before T also hits in between T and T+whatever and so do not represent a mutually exclusive event. So as long as p(hitting before T) isn't in itself restricted to merely the cases where it hit the level before T and NOT between T+whatever, we seem to be missing a tern in the formula. Obviously, the probability calculators I use do not look forward to subtract the cases it hits the same target in the future.

 

[Shakone]

If A is a subset of B, then P("B but not A") written as P(B\A) is equal to P(B)-P(A)

let B be the event that we hit before T+whatever. Let A be the event that we hit before T.

Then every case where we hit before T, we have also hit by T+whatever. Obvious right, because we hit it early. This means that A is a subset of B. Any outcome that is in A, is also in B.

So the probability that we hit between T and T+whatever, is the probability that we hit before T+whatever, but not before T, i.e. P(B\A)=P(B)-P(A).

It's late for me, I may be losing my mind...I don't understand your last paragraph. It's irrelevant whether it hits before T and also hits between T and T+whatever. It hit. It's an outcome that hit before T, and an outcome that hit before T+. It's been accounted for.

 

[Shakone]

You know what. The best way to deal with these thigns (in my opinion) is that you simulate the sample paths. Give some dynamics, Monte Carlo the thing, and you can get the probabilities straight out. Hard to argue with those 😉

 

[Socratic]

Problem is I never tried to simulate anything like that so I'll need some source to learn that form. Is there any website you know that teaches it? It might a good idea to know just for the fun of checking some crazy occurrences

Anyway now that I gave it another thought I'm inclined to be more convinced you're right. Maybe it's just a mistake on my part because by my own assumption the expression "p(A and B)" turns zero anyway because one of the terms in it sufficiently prevents it from being exemplified together with the other (even if the other doesn't exclude it alone). I mean, the expression "p(A and B)" in

p(A or B) = p(A) + p(B) - p(A and B)

turns zero since A was "hitting only between T and T+whatever." and there is no such single event combined with both it and B (=hitting before T)

 

[Cadavre]

Consider this proposal: US Equities trade as a derivative of the USD FX. (an inverted dollar carry).

Ideas jobs production assets revenues sales (etc) are down across all sectors. Yet, as if by magic, US indexes bubble up to magnificence of past glory like a giant underlying Elliot event is just over the horizon and heading this way on a fast horse.

Bolted together a forward [currency] rate swap engine back the the day. The specs came from a well credentialed geek. Recollection is that machine could reliably peg an FX, as well as LIBOR, to the basis point, six months out.

The markets back then still had some basis in reality. Assets priced to business product events. The inverse sovereign carry trade was in the closet - so this approach may not be as affable now as it was then.

Reality check: FX Spikes are being telegraphed ..
Ann Rand suggested "We can ignore reality but we cannot ignore it's consequences".

1) The Euro Zone will change
2) That change will cause demand for USDs
3) The strong USD will deflate US equity value and collapse US equity and commodity indexes.
4) The very famous so called super committee in the US will fail to deliver a workable austerity plan.
5) Without a US austerity plan S&P will downgrade, as promised, US credit rating.
6) The rating downgrade will weaken the USD
7) US indexes, thanks to weak USD, again climb to unbelievable glory.
8) Followed by the final and largest wave of the Elliot Wave set resulting in a total collapse of the global fiat currency apparatus.

Within items 6, 7, 8 there will be opportunities for profit (and we know the date the "Super Committe" will anounce it has failed) , albeit fiat profits, but profits, nonetheless for those with nose to wheel and ear to ground. After item 8, fiat exchange tokens worthless.😢

Reality check fact: China just a few months back said it's pencil pushers had determined the true US GDP was more like 5 Trillion than 14 Trillion.

There is no business like fiat printing business 😈 !

Accepting equities trade more as derivatives of the FX Swap Market then the solution may be as simple as forward (let's say) 3M LIBOR curve. Once the forward LIBOR is in hand it would be easy to hash a set of forward currency spreads.

With forward curve and spreads in hand, some work would need to be done with a critter called "Beta Weighted Delta". BWDs, done right, are said to prescribe the trading behaviors of equity / index pairs. IOW: When "A" expresses this vector "B" expresses that vector.

Next determine the BWD for the equities or indexes being considered paired with their sovereign FX index.

Use the forward FX curve, apply the BWD and, viola, you can forward price the market for the equities being considered.

Well, maybe, in theory, anyway! 😴

 

[scose-no-doubt]

[1- P(T)] x [P(T1)] ? 😕

Would that even work? Does that only work between two given strikes?

 

[Hotch]

Buy Hull, bit of c++, bit of stoch calc, bit of math finance, take you a month max.

 

[Cadavre]

[1- P(T)] x [P(T1)] ? 😕

Would that even work? Does that only work between two given strikes?

Option probabilities rate the likely hood, based on tick at moment probability is calibrated, that the market will price the option in / out money before expiration.

Next tick could change everything.

A less than precise probability surface: 18 day 105 SPY OTM PUT (S=109.43):

Pegging the probability of the above PUT's moneyness on a given date at a given spot is not the problem.

To payoff the market has to deliver spot - or so we been told!