I want to know the probability that the strike price of an option is touched. My input values are:

P = price
S = strike
v = vol
t = time to expiration

According to the answer of Hans in the following topic, the probability of touching a barrier when the underlying has zero drift is equal to:

$0.5 * (1-\text{erf}\Big(\frac{x}{\sqrt 2}\Big))$

How can I rewrite x in terms of my input values?


There should be a factor of a half in front. No? $$x = \frac{S-P}{vol \sqrt{t}}$$

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  • $\begingroup$ agreed, but is x the same as d1 for normal BS formula? $\endgroup$ – HJA24 May 26 at 19:02
  • $\begingroup$ @HJA24 The cumulative normal distribution evaluated at that point is the probability that the price will end up at or above the strike. $\endgroup$ – stackoverblown May 26 at 20:11
  • $\begingroup$ The function erf() in your formula can (if you want) be converted to the function N() used in Black Scholes using a well known identity. Most Finance people use N(), mathematicians like erf(). $\endgroup$ – noob2 May 26 at 20:20
  • $\begingroup$ @stackoverblown okay, but I want the probability that it touches the strike at some point in time, not solely on the expiration date $\endgroup$ – HJA24 May 27 at 9:06
  • $\begingroup$ @HJA24 Which will be the one minus the probability that the stock price never reaches the strike price at any point up to expiry. $\endgroup$ – stackoverblown May 27 at 17:30

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