# Simulation of Geometric Brownian Motion

I generate 10000 random binomial paths for a stock whose price is from S(0) = 10 out to S(t) where t = 1 year. Assume geometric Brownian motion for the stock price with a drift of 15% per year and a volatility of 20%. I use 10000 equally spaced time steps of length along each path and $$\Delta S = \mu S \Delta t \pm \sigma S \sqrt{\Delta t}$$, where the + or – moves at each time step are generated at random with equal probability.

S = 10
S1 = []
mean = 0.15
sd = 0.2

for i in range(10000):
S = 10

for j in range(10000):
roll = np.random.rand()

if roll < 0.5:
S = S + mean*S/10000 + sd*S/100
else:
S = S + mean*S/10000 - sd*S/100

S1.append(S)


I use the above codes to simulate the GBM and my result is the same as the theoretical values. Mean should be $$S(0)e^{\mu t}$$ and the variance should be $$S^2(0)e^{2\mu t}(e^{\sigma^2 t} - 1)$$. But there is one question want me to calculate the inverse, which is $$G(t) = 1/S(t)$$. I consider that mean should be $$\frac{1}{S(0)}e^{\mu t}$$ and variance should be $$\frac{1}{S^2(0)}e^{2\mu t}(e^{\sigma^2 t} - 1)$$. Am I correct? But I'm not sure how to change my codes. I have tried to write down the following codes. But the result is far from the theoretical values mean = 0.116 and variance = 0.00055. Could someone explain where I'm doing wrong? Thank you very much.

S = 10
G1 = [1/10]
mean = 0.15
sd = 0.2

for i in range(10000):
S = 10

for j in range(10000):
roll = np.random.rand()

if roll < 0.5:
G = 1/S
S = S + mean*S/10000 + sd*S/100
G = G + mean*(1/S)/10000 + sd*(1/S)/100
else:
G = 1/S
S = S + mean*S/10000 - sd*S/100
G = G + mean*(1/S)/10000 - sd*(1/S)/100

G1.append(G)

• Hi: Vague outline for how to do this: $\frac{ds}{S}$ is continuous BW ( you discretized ) so I would do the same kind of discretization to figure out what the diffusion is for $\triangle(\frac{1}{S})$.. Then, you can run a similar kind of simulation but that will be for $\traingle(\frac{1}{S})$.. Couldn't figure out how to make a smaller triangle. I think that you can use Ito's lemma to figure out diffusion for $\frac{1}{S}$. Mar 6, 2021 at 20:59
• Hello, could you explain more about $\Delta \frac{1}{S}$? Thank you very much. I have tried several methods. But the results are all far away from the theoretical values mean = 0.116 and variance = 0.00055. Mar 6, 2021 at 22:50
• For the mean of 1/S, does your code produce: $E\left[\frac{1}{S_t}\right]=\frac{1}{S_0}\,e^{-\left(\mu-\frac{1}{2}\sigma^2\right)t+\frac{1}{2}\sigma^2 t}=\frac{1}{S_0}\,e^{-\mu t+\sigma^2t}$? Mar 6, 2021 at 23:00
• I'm trying to produce $\frac{1}{S(0)}e^{\mu t}$. My textbook shows that $E(S_t) = S(0)e^{\mu t}$. Mar 6, 2021 at 23:05
• That’s the mean of S right? The mean of 1/S will take a different form Mar 6, 2021 at 23:06

Just to explain the formulae for the mean and variance! We can start with the solution of the GBM SDE:

$$S_t=S_0\,e^{\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t}$$

then,

$$\frac{1}{S_t}=\frac{1}{S_0}\,e^{-\left(\mu-\frac{1}{2}\sigma^2\right)t-\sigma W_t}$$

The mean (and variance) are easily obtained via this identity for a Gaussian random variable Y:

$$E\left[e^Y\right]=e^{E\left[Y\right]+\frac{1}{2}V\left[Y\right]}$$

Applying this to S, and noting that:

$$E\left[\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right]=\left(\mu-\frac{1}{2}\sigma^2\right)t$$

$$V\left[\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma W_t\right]=\sigma^2t$$

we get the familiar expression for the mean of S:

$$E\left[S_t\right]=S_0\,e^{\left(\mu-\frac{1}{2}\sigma^2\right)t+\frac{1}{2}\sigma^2 t}=S_0\,e^{\mu t}$$

And applying the identity to the expression for 1/S, we get its mean as follows:

$$E\left[\frac{1}{S_t}\right]=\frac{1}{S_0}\,e^{-\left(\mu-\frac{1}{2}\sigma^2\right)t+\frac{1}{2}\sigma^2 t}=\frac{1}{S_0}\,e^{-\mu t+\sigma^2t}$$

Variance can be determined using the same identity.

• So, no discretization- simulation necessary. Just plug the terms in and obtain the expectations. Just making sure that that's correct ? Thanks. Mar 7, 2021 at 22:07
• @Cindy: I'm deleting amy answer since it's el-wrongo or atleast not necessary. Mar 7, 2021 at 22:08