Stocks

Given the market data $S_{j,m}$ ( $j = 1, \ldots, d$; $m = -M,\ldots, 0$; time step = $\Delta$; $t=0$ is the present), discretization (50) is used to compute the estimators

$$\hat{\mu}_j = \frac{1}{M\Delta} \ln (S_{j,0}/S_{j,-M})$$ (52)

and

$$\hat{\sigma}_j = \sqrt{\frac{1}{M\Delta}\sum_{i=1-M}^{0} (\ln (S_{j,m}/S_{j,m-1}) - \Delta\hat{\mu}_j)^2}$$ (53)

for the parameters of the stock price dynamics.