Quiz: Probability and ITM?

MrAgi1 · Apr 2, 2021

A quick question guys,

We know approximately;
probability of touch=2*probability of expiring in the money
and
probability of expiring in the money=delta

Now as an example, A stock ABC is currently trading at 100$. A 30-days call on ABC with strike 110 has a delta of 20 and A 30-days put on ABC with strike 90 has a delta of 20.

What is the probability that the stock price at least would touch:
a) Both 90 and 110 before expiration(order is irrelevant)?
b)Both 90 and 110 before expiration(but must first touch the price 110 before touching price 90)?

I actually made this question up and I am not sure I know the solution. Just curious about how to solve such a problem.

Thanks in advance for your contribution.

Overnight · Apr 2, 2021

MrAgi1 said:
A quick question guys,
...

I'll take a stab at it...

a) Both 90 and 110 before expiration(order is irrelevant)?
b)Both 90 and 110 before expiration(but must first touch the price 110 before touching price 90)?..

The answer is 50/50 on all fronts.

Options do not drive stock prices. Rather, stock prices drive options.

*shrugs*

GotherL · Apr 2, 2021

Overnight said:
Options do not drive stock prices. Rather, stock prices drive options.

*shrugs*

Max pain theory , gamma squeeze?

ffs1001 · Apr 2, 2021

Its nearly midnight my local time, but I'll post my calculation and thinking tomorrow.
Great question, BTW

panzerman · Apr 2, 2021

Just a quick go at the math:

1 - 0.4 - 0.4 = 0.2 or 20% for questions A or B

Same Lazy Element · Apr 2, 2021

panzerman said:
Just a quick go at the math:

1 - 0.4 - 0.4 = 0.2 or 20% for questions A or B

Isn't it A simply P(touch_low | touch_high)? P(touch_low | touch_high) = P(touch_low) * P(touch_high) = (0.2 * 2) * (0.2 * 2) = 0.16, no? This, of course assumes no skew or term structure and assumes delta ~ risk neutral probability.

B is much harder, since it's conditional on one hitting before the other. I can't think of a back of the envelope way to estimate it.

PS. NYS just legalized Mary Jane so I might not be fully coherent

guru · Apr 3, 2021

I’d think that (b) would have half the chance of (a), since either the first strike will be touched and then the second strike, or vice versa. Though this may also depend on whether you’d discard 110 being touched multiple times before 90 is touched.

Kevin Schmit · Apr 3, 2021

A is the price of a binary that pays $1 if the price touches both barriers (U and L), and zero otherwise. You can price this as two DTKO's, the call struck at L and the put struck at U, normalized by U minus L. An iterative formula for DTKO's is in Haug. Serveral closed-form solutions have been published but I don't have them handy. Haug code is based on Ikeda, IIRC.

As Guru pointed out, B is approximately half of A.... Or would be, if the question weren't so ill posed. DualDelta (N(d2), not Delta (N(d1), is what you want. Since the call and put vols are different (.3715 and .475 under ZIRP), the change from Delta to DualDelta is not parallel (.22 amd .24 for DD). so it matters which one is hit first (the 1d GBM first passage time is not symmetric).

ffs1001 · Apr 3, 2021

MrAgi1 said:
b)Both 90 and 110 before expiration(but must first touch the price 110 before touching price 90)?

@MrAgi1 , the scenario (b) is easier to work out first. In simple terms, here goes.

MrAgi1 said:
probability of touch=2*probability of expiring in the money
and
probability of expiring in the money=delta

Based on the above, probability of touching 110 = 2* 0.2 (delta of 110) = 0.4.
Once the stock is at 110, now we need to calculate the probability of touching 90. However, the delta of the 90 strike is no longer 20, as the stock has moved up to 110 from the starting position of 100. It will be lower. We have to make an assumption on it's value. Lets assume it is now 7.

So, now, the probability of touching 90 when the stock is at 110 = 2 * 0.07 = 0.14.

Probability of stock touching 110 first and then 90 = 0.4 * 0.14 = 0.056 (or 5.6% chance)
This is the answer to (b).

The answer to (a) is simply double this value, so 0.112 (11.2%).

MrAgi1 · Apr 3, 2021

Same Lazy Element said:
Isn't it A simply P(touch_low | touch_high)? P(touch_low | touch_high) = P(touch_low) * P(touch_high) = (0.2 * 2) * (0.2 * 2) = 0.16, no? This, of course assumes no skew or term structure and assumes delta ~ risk neutral probability.

B is much harder, since it's conditional on one hitting before the other. I can't think of a back of the envelope way to estimate it.

PS. NYS just legalized Mary Jane so I might not be fully coherent

Your explanation makes sense to me. However, there are different answers on this thread.