0
$\begingroup$

I'm dealing with option pricing models and there is a statement that says the variance of underlying asset price is propotional with time $π‘‰π‘Žπ‘Ÿ(𝑆_{π‘š+1})=𝑆_π‘š^2𝜎^2Δ𝑑$ where $\Delta t = \frac{T}{N}$ and $\sigma$ is volatility. How this equality can be explained/proved?

$\endgroup$
1
  • 1
    $\begingroup$ This holds true for stochastic processes having time homogeneous Independent Increments. Are you familiar with this property? For such a process the variance over two days is the sum of the variance of day 1 and the variance of day 2 (by independence). These two variances are the same by the assumption that the process is time homogeneous (does not change in time). So the two day variance is twice the 1 day variance. $\endgroup$
    – Alex C
    Commented Jun 10, 2019 at 13:07

1 Answer 1

1
$\begingroup$

Exactly what @Alex C said. It's the time homogeneous diffusion proprety. You can't state such an argument in models where volatility is no longer time homogeneous ( that's being time independant and depending only on the underlyings).

$\endgroup$

Your Answer

By clicking β€œPost Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.