Closed-Form Solutions for European and Digital Calls in the Hull and White Stochastic Volatility Model and Their Relation to Locally R-Minimizing and Delta Hedges

This paper derives an analytic expression for the distribution of the average volatility $\frac{1}{T-t} \int_t^T \sigma_s^2 ds$ in the stochastic volatility model of Hull and White. This result answers a longstanding question, posed by Hull and White (Journal of Finance 42, 1987), whether such an analytic form exists. Our findings are applied to obtain closed-form solutions for European and Digital call option prices. The paper also provides an explicit solution for the Delta hedge of a European call. Moreover, it is proved that the Delta hedge under the minimal martingale measure coincides with the locally $R$-minimizing hedge in the model considered here