Principles of Financial Engineering, 3rd Edition

(13.76)

Replacing this in Eq. (13.71) gives

$Var(L_i|j=u)=E^{\tilde{P}}\left[{\left(\ln (L_i)-\frac{1}{2}[\ln (L_i^{uu})+\ln (L_i^{ud})]\right)}^2|j=u\right]$ (13.77)

(13.77)

Simplifying and regrouping, we obtain

$Var(L_i|j=u)={\left(\frac{1}{2}[-\ln (L_i^{uu})+\ln (L_i^{ud})]\right)}^2$ (13.78)

(13.78)

This means that the volatility at time i, in state u, is given by

${\sigma }_i^u=\frac{1}{2}\ln \left[\frac{L_i^{uu}}{L_i^{ud}}\right]$ (13.79)

(13.79)

The result for the down state will ...