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I want to simulate a geometric Brownian motion and we assume that the volatility of the stock can take just two values $\sigma_1=0.2$ and $\sigma_2=0.8$. We also assume that the jumps up from lower volatility $\sigma_1$ to higher volatility $\sigma_2$ occur as a exponential process with rate $\lambda_1=2$. Likewise, the jumps down from volatility $\sigma_2$ to lower volatility $\sigma_1$ occur as an exponential process with rate $\lambda_2=4$. I know how to simulate a geometric Brownian motion but i can't understand how I simulate the volatility. I must compare a number from the exponential distribution with what on every step to decide if I will make a jump or not?

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  • $\begingroup$ I think you need to compare a uniformly distributed random variate $u\sim U(0,1)$ with $z_i \equiv e^{-\lambda_i \Delta t}$, where $\Delta t$ is the time step in your simulation. If $u < z_i$, stay, else go to the other state. $\endgroup$ – Kermittfrog Nov 25 '20 at 8:34
  • $\begingroup$ I tried but doesn't work. $\endgroup$ – Hans Larsen Nov 25 '20 at 10:03
  • $\begingroup$ Well then, why don't you post your code and what you've tried so far in this forum? $\endgroup$ – Kermittfrog Nov 25 '20 at 12:26
  • $\begingroup$ So this is the code. I dont think that the condition in the second if statement is correct. $\endgroup$ – Hans Larsen Nov 25 '20 at 13:39
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Besides a couple of ways you might try to improve your code (which I will not do here); your jump check is not working correctly:

In a time step $\Delta t$, the process will jump with probability $\approx exp(-\lambda \Delta t) $. Hence, you need to compare

if (unifrnd(0,1) > exp(-lambda * dt))
  % jump occured
  % flip state
else
  % no jump occured
  % do not flip state
endif

HTH?

NB: You might want to simulate a vector of uniforms, and then iterate over elements to get the state $1$ or $2$. From this, you may compose a vector of volatiltities per time step.

I assume you are using Matlab or Octave? If that's the case, vectorisation is king!

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Just use the definition of conditional probability. With $\gamma_t \in \{-1,1\}$ an indicator returning 1 if $\sigma = 0.8$ and -1 otherwise, and $x_{1:T}$ the path of the Brownian motion over the time-period $[1,T]$ you have

$$p(x_{1:T}, \gamma_{1:T}) = p(x_{1:T} \vert \gamma_{1:T})p(\gamma_{1:T}).$$

In practice you just simulate, until time $T$, alternating event-times $\tau_i$ from $\text{Exp}(\lambda_1)$ and $\text{Exp}(\lambda_2)$ until $\tau_i > T$. Conditional on these times, you then just simulate your GBM conditional on $\gamma_t$. To make your life easier, choose your step-size such that you get very close to all the event-times $\tau_i$ when simulating the GBM.

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  • $\begingroup$ Can you explain it a bit more please? $\endgroup$ – Hans Larsen Nov 26 '20 at 10:32

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