# Convert option inputs to standard Brownian motion

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? $$x = \frac{S-P}{vol \sqrt{t}}$$

• agreed, but is x the same as d1 for normal BS formula? – HJA24 May 26 at 19:02
• @HJA24 The cumulative normal distribution evaluated at that point is the probability that the price will end up at or above the strike. – stackoverblown May 26 at 20:11
• 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(). – noob2 May 26 at 20:20
• @stackoverblown okay, but I want the probability that it touches the strike at some point in time, not solely on the expiration date – HJA24 May 27 at 9:06
• @HJA24 Which will be the one minus the probability that the stock price never reaches the strike price at any point up to expiry. – stackoverblown May 27 at 17:30