Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 2 / d

Fix the horizon $T$ and define the half-volatility measure for a square-root stock claim byThe Novikov condition makes $Z$ a true martingale with $\mathbb{E}{Z}_T=1$, and $Z_T>0$ makes $Q$ equivalent to $P$. The Girsanov theorem identifies $W_s^Q=W_s-\frac{1}{2}{\int }_0^s{\sigma }_{u}du$ as a Brownian motion under $Q$.

The exact factorization from the previous part isUsing the Bayes formula for conditional expectation therefore givesThe random integrated variance need not have any independence property under $Q$. Its pathwise bounds still imply $e^{-b^2\tau /8}\le {\mathbb{E}}_Q[e^{-I_{t,T}/8}\mid {\mathcal{F}}_t]\le e^{-a^2\tau /8}$. Inverting the same price function proves $a\le \Sigma (t,T)\le b$ even for volatility adapted to the Brownian motion, on the positive-price event. The zero-price convention is as above. The auxiliary measure here is not asserted to be an equivalent martingale measure for the stock.