stochastic processes - Getting the next price of a GBM (Geometric Brownian Motion)

I am writing a program that creates realizations of a GBM.

Starting from an initial price, I get the following price with this formula:

NewPrice = PreviousPrice * Exp(Volatility * N10 * Sqrt(DaysElapsed) + Drift * DaysElapsed)

Where:

• Volatility is the annual percentage volatility / 100 / sqrt(250)


• Drift the annual percentage Drift / 100 / 250


• N01 is a standard normal realization


• DaysElapsed are the days elapsed from previous price (this is a small fraction in my case)

I am not sure that I am doing this right. Is the above line correct ? Please, suggest the right code expression or other possible corrections. Thank you!

Answer

GBM is defined as

$$S_t = S_{t-1}\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)dt + \sigma dW_t\right)$$

So, in your notation, assuming your daily parameters:

$$S_{new} = S_{previous}\cdot\exp\left( \left({drift} - \frac{{volatility}^2}{2} \right)days + volatility \,\sqrt{days}\,N(0,1)\right)$$

So your formula was incorrect. The youtube you quote is only true for 1-year timesteps (while you have $days$ steps).