I know this is a very elementary question but... when modeling asset prices through a stochastic process as in

$$dS_t=S_t μ dt+S_t σdW_t,$$

where the following is a wiener process $$dW_t=σN(0,1)dt^{1/2}$$

how does the mean $μ$ and volatility $σ$ scale with the time interval $dt$?

If I am forecasting for 1 year in the future, in 1 day steps, $dt$=1/250=.004

But what if the mean parameter I have calculated for $μ$ and $σ$ is for daily returns, would the equation still hold? ie, I take a 20SMA of returns for the last 20 days, so my return is already daily. In all the literature I have read $μ$ and $σ$ are already annual which in my case is not helpful because I look at daily returns for assets not annual returns that have been scaled down to daily.


First, your statement that $dW_t=\sigma\,dt^{1/2}$ is incorrect. In fact, it's not even meaningful (you can see this by noticing that the expression on the left-hand-side is an "increment" of Brownian motion, and hence random, while the expression on the right-hand side is deterministic). What you mean to say is that $W$ is a Brownian motion, and hence $\text{E}(W_t)=0$ and $\text{Var}(W_t)=t$. Or better, the quadratic variation of $W$ is $<W>_t=t$. It's important to remember that the "increment" $dW_t$ is simply a notational convenience - it's not really a well-defined concept, since the paths of Brownian motion have infinite first variation.

Now, coming to your question, the in asset pricing models the parameters $\mu$ and $\sigma$ are usually specified as the annual drift rate and the annual volatility. If instead, you have estimated a daily drift rate $\mu_d$ and a daily volatility $\sigma_d$, then you can scale them as follows $\mu=252\mu_d$ and $\sigma=\sqrt{252}\sigma_d$, where we're using the convention that a year contains 252 trading days.

Regards Hardy

  • $\begingroup$ Hi Hardy, I appreciate your patience and time in helping answer my question. I edited my post. I total forgot about the random component in $$dW_t=σN(0,1)dt^{1/2}$$ drawn from a normal distribution. In regards to the second part of your answer, so essentially what your doing is scaling the daily mean/vol to annual so they can be inputted into the model. Am I correct? If so, what is the relationship between $μ$,$σ$, and $dt$.. $\endgroup$ – jessica Dec 13 '13 at 19:22
  • $\begingroup$ If I have an annual mean of $μ=10%$ and vol of $σ$=20% and I am trying to simulate potential asset paths over a 6 future month period in 2 months steps, so 3 steps total, $dt$ would be 1/3 but that would erroneously scaling an annual figure to quarterly returns and vol. $\endgroup$ – jessica Dec 13 '13 at 19:22
  • $\begingroup$ $\mu$ scales with dt, and $\sigma$ scales with $\sqrt(dt)$, so for your example above, $\mu = 10*\frac{2}{12}$ and $\sigma = 0.2*\sqrt(2/12)$, and $dt = \frac{1}{6}$, not 1/3 $\endgroup$ – experquisite Jan 12 '14 at 18:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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