Delta and probability question
- Thread starter nravo
- Start date
You left out a left parens:
d1 = (ln(price/strike) + (interestRate + (volatility^2)/2) * timeLeft) / (volatility * sqrt(timeLeft))
Not true. Examine the second term in the numerator in
your formula above, it doesn't go to zero even if the interest
rate is zero (in BS models zero interest rate corresponds
to the currency interest rates being equal in G-K models).
Vol squared further divided by two and further multiplied
by a time-to-expiry much less than one is a small number
but it does not go to zero.
Even under assumptions of zero drift, Delta is only a decent
approximation of probability very close to expiry in low vol
and low interest rate environments. It is never exactly
equal.
Neither the delta nor the "probabilty," each calculated by
plugging in BS implied vols, is a good estimator of true
probability due to the fact that drift has magically dropped
out of the BS equation. A better method is to fit an implied
probability distribution over actual bid ask prices, as is now
done for the calculation of the VIX. For long expiries, such
as LEAPS on US equity indices, you will find that the peak
of the derived PDE is considerably to the right of the current
index price -- a manifestation of expected positive drift.
.
If you have questions about delta, then it sounds like it's time for you to review the basics. Pick up a Natenberg book or check out some of these sites. They will steer you on the right path:
www.888options.com
www.theoptionsinsider.com
www.oicoptions.com
www.cboe.com
www.888options.com
www.theoptionsinsider.com
www.oicoptions.com
www.cboe.com
That's what i tried to expain yesterday. But nravo didn't get the message.
Delta shows the relation of how much you cover an exposure compared to the spot market. If the delta is 50 it means your options premium will move only 50% of the move it makes in the spotmarket. So to completely cover yours options by a spot hedge you only need to take a 50% position in the spotmarket to neutralize the loss (or profit) of your option position.
I did get the message, but it is assumed we all know on this board what delta is, per se. I was inquiring about using delta informally, so to speak, for a second purpose, making a BOE compuation about probability, and how imprecise it may be. A couple people got my drift on here, and I thank them for the answers.
Sbelmont, this was the 10th post you tout these sites that answer only the most basic Q's. Not sure what your agenda is but we know'm by now. Thanks.
Almost, but not quite.Quote from FullyArticulate:
Delta is close, within 5% generally. The formulas, however, are *different*.
d1 = ln(price/strike) + (interestRate + (volatility^2)/2) * timeLeft) / (volatility * sqrt(timeLeft))
Delta = Normalize(d1)
p1 = ln(targetPrice / price) / (volatility * sqrt(timeLeft))
Prob = Normalize(p1)
In other words, delta includes the interest rate:
(interestRate + (volatility^2)/2) * timeLeft)
but probability does not.
In a period with extremely high interest rates (or high volatility), the difference will become much more noticeable.
One other thing to keep in mind--if there is a dividend or some other corporate action coming, delta and probability will have absolutely nothing to do with each other.
Whereas to calculate d1 interest rates are added so as to include the value of acquiring the underlying now, to calculate d2 (or Prob) interest rates are discounted so as to give the present value of paying the exercise price at option expiry.
So the formula is actually written;
p1 = ln(price/strike) + (interestRate - (volatility^2)/2) * timeLeft) / (volatility * sqrt(timeLeft))
Or it could equally be written;
p1 = d1 - volatility * sqrt(timeleft)
And then as you say;
Prob = Normalize(p1)
For near time expiries in a low interest rate environment the Delta is good enough. As you go out in time and / or interest rates rise then the difference between the two values becomes more significant.
nravo
valueline.com/edu_options/rep11.html
Consider that at any point in time you are looking at a snapshot.
Then consider the delta gamma relationship.
At the snapshot in time (with a PERFECT option pricing model)
over time the sum of ALL option probabilities would equal the actual outcomes. It is as if in each specific option pricing instance we are looking at an average probability of outcome. The end result of total expectations will be accurate over time but the individual out comes which make up this total will be quite different than the original expectations or probabilities.
Definitely look more closely at the delta gamma relationship.
Also, you might want to take a look at this book.
www.amazon.com/gp/search/ref=sr_adv...srank&mysubmitbutton1.x=0&mysubmitbutton1.y=0
There is also a interesting chapter in one of the market wizard books I think you would find very interesting. I will look through my notes and see if I can find it for you.
Good luck with what you are working on.
Nutsneal
valueline.com/edu_options/rep11.html
Consider that at any point in time you are looking at a snapshot.
Then consider the delta gamma relationship.
At the snapshot in time (with a PERFECT option pricing model)
over time the sum of ALL option probabilities would equal the actual outcomes. It is as if in each specific option pricing instance we are looking at an average probability of outcome. The end result of total expectations will be accurate over time but the individual out comes which make up this total will be quite different than the original expectations or probabilities.
Definitely look more closely at the delta gamma relationship.
Also, you might want to take a look at this book.
www.amazon.com/gp/search/ref=sr_adv...srank&mysubmitbutton1.x=0&mysubmitbutton1.y=0
There is also a interesting chapter in one of the market wizard books I think you would find very interesting. I will look through my notes and see if I can find it for you.
Good luck with what you are working on.
Nutsneal
You're absolutely right. My typo.
I regretted that error as soon as I went to bed last night, but figured I'd have time to fix it this morning. I'm too slow.
All of this assumes risk neutral probabilities, which aren't "real world". The question was solely if a quick "back of the envelope" calculation made delta roughly equal to the risk-neutral probability. Since this is a commonly tauted "rule of thumb", the exercise of finding out how close RNP is to delta would seem to be useful to everyone.
So, the answer is that RNP is pretty close to delta, except in very high volatilities, or very high interest rates.
you know...I've been using TOS's probability formula (1sigma=stock price*volatility*(SQRT(Days to expiration/365))) and it is different than if one used the delta as a measure of probability. Where something that is like 1.3 sigmas out using TOS's formula is sometimes shown to have a delta of like 11....that being said, neither way is accurate and is just a general guide.