Probability of the High of Geometric Brownian Motion

by M.A. (Thijs) van den Berg


Contents

[edit] Description

This article describes the probability distribution of the high of a geometric Brownian motion. An equation is given that calculates the probability that the high is above a certain level H within a given timeframe 0<t<T.

Three scenarios of a stock price, of which one has a high above 180.
Three scenarios of a stock price, of which one has a high above 180.


[edit] Equation

\begin{align} P \left(\max_{0<t<T} S_t \ge H\right)&=\textstyle\frac{1}{2} \text{Erfc}(d_1)+\textstyle\frac{1}{2}\left(\frac{H}{S_0}\right)^{\frac{2\mu}{\sigma^2}-1}\text{Erfc}(d_2)\\ d_1 &= \frac{\ln \frac{H}{S_0} - (\mu-\textstyle\frac{1}{2}\sigma^2)T}{\sigma\sqrt{2T} }\\ d_2 &= \frac{\ln \frac{H}{S_0} + (\mu-\textstyle\frac{1}{2}\sigma^2)T}{\sigma\sqrt{2T} } \\ \end{align}
\max_{0<t<T} S_t\, The highest value of St in the time period 0<t<T
H The level of the high for which the probability is calculated
S_0\, The current value of the geometric Brownian motion
S_t\, The value of the geometric Brownian motion at time t
\mu\, The drift of the geometric Brownian motion
\sigma\, The volatility of the geometric Brownian motion
Erfc The complementary error function
ln Natural logarithm

[edit] Example equation usage

We want to know the probability that the high of a stock in the upcoming three years is above 180. The current stock price is 100, and its volatility 30%. Current interest rates are 5%, and we assume that the stock has a yield equal to the interest rate.

[edit] Parameter values

T = 3
H = 180
S_0\, = 100
\mu\, = 0.05
\sigma\, = 0.30

[edit] Substitution of values ...

\begin{align} d_1 &= \frac{\ln \frac{180}{100} - (0.05 - \textstyle\frac{1}{2}0.30^2)3}{0.30\sqrt{2*3} }\\ &= \frac{0.57279 - (0.05 - 0.045)3}{0.73485}\\ &= 0.77946 \end{align}

and

\begin{align} d_2 &= \frac{\ln \frac{180}{100} + (0.05 - \textstyle\frac{1}{2}0.30^2)3}{0.30\sqrt{2*3} }\\ &= \frac{0.57279 + (0.05 - 0.045)3}{0.73485}\\ &= 0.82029 \end{align}

thus

\begin{align} P &= \textstyle\frac{1}{2} \text{Erfc}(d_1)+\textstyle\frac{1}{2}\left(\frac{180}{100}\right)^{\frac{2 *0.05}{0.30^2}-1}\text{Erfc}(d_2) \\ &= \textstyle\frac{1}{2} 0.27032 + \textstyle\frac{1}{2} \left(1.8\right)^{0.1111} 0.24602 \\ &= 0.2665 \end{align}


The probability that the high of stock gets above 180 is thus 26.7%, and that is stays below 73.3%

[edit] Comments

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[edit] Igor Zomb said ...

Isn't this the same as probability of hitting the Up-and-Out barrier option. Joshi's book has a nice derivation of this formula using the reflected Brownian motion

--Igor Zomb 22:55, 19 July 2007 (CEST)

[edit] sitmo said ...

This is the probability that the highest value of the stock during a period will be above a given level. This is indeed related to barrier options (and lookback options).

--sitmo 13:23, 26 July 2007 (CEST)

[edit] Gerard D. Grey said ...

excellent article. as are the others I've read from this site. your numerical examples really help to illustrate the formulae.

--Gerard D. Grey 02:17, 3 November 2007 (CET)

[edit] Peter said ...

Hi, this a very interesting contribution indeed. I think there is a typing error in the example: 0.57279 should read 0.587786. How could the equitation be modified for touching a lower boundary price? Kind Regards Peter

--Peter 14:15, 11 November 2007 (CET)

[edit] Sitmo said ...

Peter, your right about the typo. A strange one, it doesn't look to resemble anything close. I'll fixed it soon & update the numerical. Thanks for finding this.

--Admin 23:29, 26 November 2007 (CET)

[edit] Yann said ...

I'm really wondering how Peter spotted this ! Do you really know ln(1.8) by heart ?

I would have liked to see the distribution of these probabilities for any h !

--Yann 18:40, 2 January 2008 (CET)

[edit] adschai said ...

Another variant of this is described in this equation http://www.sitmo.com/eq/245. I.e. the joint of high and low. However, what's the x in A and B term? Any help would be appreciated.

--adschai 17:08, 29 February 2008 (CET)

[edit] Admin said ...

adschai,

The A and the B term defined on the same page.

They are intermediate results that are taken out of the main equation to make it more readable.

--Admin 15:47, 12 March 2008 (CET)

[edit] Ozzie said ...

Err. Is Doob's (martingale) inequality another approach to arrive at the same result?

--Ozzie 15:11, 20 March 2008 (CET)

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