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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.

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1 Answer 1

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

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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 $dt$.. –  jessica Dec 13 '13 at 19:22
    
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. –  jessica Dec 13 '13 at 19:22
    
$\mu$ 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 –  experquisite Jan 12 at 18:45

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