# 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

22 Pages Posted: 20 Jan 2007

See all articles by Christian-Oliver Ewald

## Christian-Oliver Ewald

University of Glasgow; Høgskole i Innlandet

## Klaus Reiner Schenk-Hoppé

University of Manchester - Department of Economics; Norwegian School of Economics (NHH) - Department of Finance

## Zhaojun Yang

Southern University of Science and Technology - Department of Finance

Date Written: 2007

### Abstract

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.

Keywords: Stochastic volatility models, incomplete markets, Delta hedging, locally R-minimizing hedging strategies, Malliavin calculus

JEL Classification: G13, C61, C63

Suggested Citation

Ewald, Christian-Oliver and Schenk-Hoppé, Klaus Reiner and Yang, Zhaojun, 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 (2007). Swiss Finance Institute Research Paper No. 07-11, Available at SSRN: https://ssrn.com/abstract=957807 or http://dx.doi.org/10.2139/ssrn.957807