Noobie question about probability of ITM (math, probability)

tommcginnis said:
WOW! It looks like a whole raft of new BSM/delta/probability videos have been posted to YouTube in the last couple of years! Could be great stuff; could be stinkers. (There are a stunning amount of stinkers -- warning.) Bionic Turtle is a great source for all sorts of math --
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Bionic Turtle is a beast man... he has so much content. A lot of it goes over my head but I"ll watch sometimes for fun
 
jamesbp said:
It is the probability of the strike being touched ( regardless of option type )
... use the OTM delta ... which should always be less than 0.50
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So, P(Touch) for a 0.75 delta is 25%, or (2*25%) 50%?

This is very interesting. I haven't seen the actual P(Touch) in a while, but would love to map this "twice(OTMδ)" rule-of-thumb and see how they track. Any sources handy?
 
Wait..Whats happening here???

I must be missing something...

This nifty little formula just shows the corresponding put probability of being touched..
 
ironchef said:
Please explain: So, if delta is = 0.51, the probability of touch is 1.02?
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For delta near 50, if the current price is exactly the strike, then the probability of touch is 100%. Otherwise, the delta approximation is:

2 * min(delta,1 - delta)

This is a pretty crude approximation. A better approximation for probability of touch is:

2 * min(dual-delta,1 - dual-delta)

Delta is N(d1) or the 1st derivative of option price with respect to S (underlying stock price). Dual-Delta is N(d2) or the first derivative of option price with respect to K (strike). Dual-Delta is the probability, under the risk-neutral measure, of the option expiring ITM. You can see that in the second term of the BSM call formula, the term involving K. That term is the discounted K (strike, or the amount the call buyer has to pay if he exercises) times the probability (N[d2] or dualdelta) that he will actually exercise.

R code to compute exact probability of touch under risk-neutral measure:

Code: 
bsProbOfTouch <- function(S,K,T=1,r=0,q=0,sigma=0.1)
{ if(K == S) {
  return(1)
  } else
  { mult <- sign(S - K)
  rr <- (r - q) - sigma^2 / 2
  pt1 <- pnorm((mult * log(K / S) + rr * T) / (sigma * sqrt(T)))
  pt2 <-((K / S)^(2 * rr / sigma^2))
  pt2 <- pt2 * pnorm((mult * log(K / S) - rr * T) / (sigma * sqrt(T)))
  return(pt1 + pt2)
  }
}

Of course, risk-neutral implied probabilities may have little relationship with real-world probabilities. So caveat emptor.
 
Last edited:
Kevin Schmit said:
For delta near 50, if the current price is exactly the strike, then the probability of touch is 100%. Otherwise, the delta approximation is:

2 * min(delta,1 - delta)

This is a pretty crude approximation. A better approximation for probability of touch is:

2 * min(dual-delta,1 - dual-delta)

Delta is N(d1) or the 1st derivative of option price with respect to S (underlying stock price). Dual-Delta is N(d2) or the first derivative of option price with respect to K (strike). Dual-Delta is the probability, under the risk-neutral measure, of the option expiring ITM. You can see that in the second term of the BSM call formula, the term involving K. That term is the discounted K (strike, or the amount the call buyer has to pay if he exercises) times the probability (N[d2] or dualdelta) that he will actually exercise.

R code to compute exact probability of touch under risk-neutral measure:

Code: 
bsProbOfTouch <- function(S,K,T=1,r=0,q=0,sigma=0.1)
{ if(K == S) {
  return(1)
  } else
  { mult <- sign(S - K)
  rr <- (r - q) - sigma^2 / 2
  pt1 <- pnorm((mult * log(K / S) + rr * T) / (sigma * sqrt(T)))
  pt2 <-((K / S)^(2 * rr / sigma^2))
  pt2 <- pt2 * pnorm((mult * log(K / S) - rr * T) / (sigma * sqrt(T)))
  return(pt1 + pt2)
  }
}

Of course, risk-neutral implied probabilities may have little relationship with real-world probabilities. So caveat emptor.
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Thank you. This is very useful because probability of touch is an important factor for me to consider.

I have downloaded historical data and would like to compare probability of touch with actual outcome. I can code in VBA excel but not R.

I think I can do the formula in excel.

Regards,
 
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