# On the valuation of variance swaps with stochastic volatility

Journal Article

### Abstract

• This paper is an extension to a recent paper by Zhu and Lian (2011) , in which a closed-form exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston (1993)'s  two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved. © 2012 Elsevier Inc. All rights reserved.

• 2012

### Citation

• Zhu, S. & Lian, G. (2012). On the valuation of variance swaps with stochastic volatility. Applied Mathematics and Computation, 219 (4), 1654-1669.

### Scopus Eid

• 2-s2.0-84867581199

### Web Of Science Accession Number

• http://ro.uow.edu.au/infopapers/2304

• 15

• 1654

• 1669

• 219

• 4

• Germany

### Abstract

• This paper is an extension to a recent paper by Zhu and Lian (2011) , in which a closed-form exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston (1993)'s  two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved. © 2012 Elsevier Inc. All rights reserved.

• 2012

### Citation

• Zhu, S. & Lian, G. (2012). On the valuation of variance swaps with stochastic volatility. Applied Mathematics and Computation, 219 (4), 1654-1669.

### Scopus Eid

• 2-s2.0-84867581199

### Web Of Science Accession Number

• http://ro.uow.edu.au/infopapers/2304

• 15

• 1654

• 1669

• 219

• 4

• Germany