Hi @Jonathon! Why we use square root of time when we switch to the scaled random walk? From what I see increments are additive, so to get to the same position at time 1 we need to do NxSize steps and not sqrt(N)xSize steps. I gues this is to keep properties of the process the same, but this is not too intuitive. If the answer is long - please steer me into the right direction. Thanks!
Hi Sergey, the reason we take the square root of n, is because Brownian Motion accumulates variation at rate one per unit time. Hence the process scales with the square root of time/(size steps). The proof is long, my favourite source is Steven Shreve, Stochastic Calculus for Finance II, page 101-107 (2008 edition).
Wanted to see the reason behind 1/sqrt(n), too, my thoughts were: If the [variance Var(x)] increases at one per time, the [average observed squared outcome E(x^2)] does as well, since Var=E((x-µ)^2) and µ=0. So for one step per t, the unscaled RW_0 has E_0(x^2)=t , then we want to speed up and do n steps per t. Sped up Var_1(x) and E_1(x^2) will be n-fold the original then. To bring it back to the original(0) distribution, divide by n and rearrange so that we know how we must scale x: E_1(x^2)/n = E_0(x^2) Then we can get the factor in the expected value expression and thus scale the x^2 by 1/n E_1(1/n * x^2) = E_0(x^2), or, equivavently, x by sqrt(1/n): E_1((x/sqrt(n))^2) = E_0(x^2) We can see that replacing x by x/sqrt(n) yields the original property. But might be invalid for some reason.
I am currently pursuing the FRM (cleared FRM Level 1) and also learning Python! Which books would you suggest me to refer for building stronger basics??
How can you use the normal distribution in your example for Brownian Motion with n = 100 steps and time t = 10 when "n" is not at all high enough to assume a normal distribution? It should have been a Binomial distribution instead. No? The binomial won't converge to normal at just small n. Law of Large Numbers.
It is a foundation used in many models, such as, iirc, the Black-Scholes model . Double checked, and yes, Black-Scholes assumed that stock prices follow geometric Brownian motion. What is shown here, Brownian motion, is not geometric Brownian motion, but you should understand Brownian motion before understanding geometric Brownian motion (and going from Brownian to Geometric Brownian is a very small step.) Of course, I imagine quants and such use proprietary more complicated things on top, but they are presumably still using something closely related to geometric Brownian motion, and so knowing Brownian motion is required. Of course, Brownian motion shows up in non-financial contexts as well. For example, the motion of small particles sitting in some water, is the origin of the idea. Brownian motion / the Wiener process (nearly synonymous) is used in electrical engineering when modeling (an integral of) white noise, and in control theory . It is also the basis of a formulation of the of the path integral formulation of quantum mechanics? Though that last example isn’t so much “in industry” I guess.
