Standard deviation is the square root of the variance. Variance of fBM=> sigma^2* t^(2*H) w/ H= Hurst parameter and H is in (0,1). Volatility = sigma*t^H. If one wants an uncorrelated process/ Markovian process, choose H=1/2. Thus, sigma*sqrt(T)==> volatility scales with the square root of time.
You would, *if* returns were perfectly correlated from period to period: var(2t) = 2var(t)+2var(t) = 4var(t); i..e., doubling the period would quadruple the variance and double the std dev. But this makes the "typical" assumption that returns are independent (i.i.d.). And, under this onerous (unrealistic) assumption, variance scales with time; therefore volatility scales with SQRT[time]. If we thought in terms of variance, it might be more natural: the variance does multiply by time. good q.
Assume the multi-fractional stochastic process, f(v(t)), is a measurable function of a nonlinear multifractional stochastic process=> dv(t)=n*M(v(t))*(w-v(t))dt+p*N(v(t))*dBh(t). Under the assumption of constant sigma, volatility scales with the f(v(t))th power of time. IID is the strongest possible assumption. IID implies strict stationarity, ergodicity,strong time reversibility etc. Systems that exhibit IID dynamics are completely trivial.
Old but gold. Very helpful. Also the comments.
You all brought me a big step closer on my research‘s.
Standard deviation is the square root of the variance. Variance of fBM=> sigma^2* t^(2*H) w/ H= Hurst parameter and H is in (0,1). Volatility = sigma*t^H. If one wants an uncorrelated process/ Markovian process, choose H=1/2. Thus, sigma*sqrt(T)==> volatility scales with the square root of time.
You would, *if* returns were perfectly correlated from period to period: var(2t) = 2var(t)+2var(t) = 4var(t); i..e., doubling the period would quadruple the variance and double the std dev.
But this makes the "typical" assumption that returns are independent (i.i.d.). And, under this onerous (unrealistic) assumption, variance scales with time; therefore volatility scales with SQRT[time]. If we thought in terms of variance, it might be more natural: the variance does multiply by time. good q.
Assume the multi-fractional stochastic process, f(v(t)), is a measurable function of a nonlinear multifractional stochastic process=> dv(t)=n*M(v(t))*(w-v(t))dt+p*N(v(t))*dBh(t). Under the assumption of constant sigma, volatility scales with the f(v(t))th power of time. IID is the strongest possible assumption. IID implies strict stationarity, ergodicity,strong time reversibility etc. Systems that exhibit IID dynamics are completely trivial.
But shouldn't ∆T be 5 - 2 instead of 5/2? Since it is delta and that's what delta normally do?
while calculating VaR, shouldn't it be added by mean value as well.
Great tutorial!