Replacing this in Eq. (13.71) gives

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

Simplifying and regrouping, we obtain

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

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

${\sigma}_{i}^{u}=\frac{1}{2}\mathrm{ln}\left[\frac{{L}_{i}^{uu}}{{L}_{i}^{ud}}\right]$ (13.79)

The result for the down state will ...

Get *Principles of Financial Engineering, 3rd Edition* now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.