European call option issued on a bond governed by a geometric or a fractional geometric Ornstein-Uhlenbeck process
Volume 1, Issue 1 (2014), pp. 95–108
Pub. online: 27 June 2014
Type: Research Article
Open Access
Open Access
Received
5 March 2014
5 March 2014
Revised
25 April 2014
25 April 2014
Accepted
5 June 2014
5 June 2014
Published
27 June 2014
27 June 2014
Abstract
European call option issued on a bond governed by a modified geometric Ornstein-Uhlenbeck process, is investigated. Objective price of such option as a function of the mean and the variance of a geometric Ornstein-Uhlenbeck process is studied. It is proved that the “Ornstein-Uhlenbeck” market is arbitrage-free and complete. We obtain risk-neutral measure and calculate the fair price of a call option. We consider also the bond price, governed by a modified fractional geometric Ornstein-Uhlenbeck process with Hurst index $H\in (1/2,1)$. Limit behaviour of the variance of the process as $H\to 1/2$ and $H\to 1$ is studied, the monotonicity of the variance and the objective price of the option as a function of Hurst index is established.
References
Beleza Sousa, J., Esquivel, M.L., Gaspar, R.M.: Machine learning Vasicek model calibration with Gaussian processes. Commun. Stat., Simul. Comput. 41(6), 776–786 (2012). MR2877834
Deakin, A.S., Davison, M.: An analytic solution for a Vasicek interest rate convertible bond model. J. Appl. Math. 2010 (2010). MR2588207
Huang, G., Deng, G., Huang, L.: Valuation for an American continuous-instalment put option on bond under Vasicek interest rate model. J. Appl. Math. Decis. Sci. 2009 (2009). MR2530015
Liptser, R.S., Shiryayev, A.N.: Statistics of Random Processes I, General Theory. Springer (1977). MR0474486
Lo, C.F.: Lie-algebraic approach for pricing zero-coupon bonds in single-factor interest rate models. J. Appl. Math. 2013 (2013). MR3056212
Mamon, R.: Three ways to solve for bond prices in the Vasicek model. J. Appl. Math. Decis. Sci. 8(1), 1–14 (2004). MR2042166
Norros, I., Valkeila, A., Irtamo, J.: An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4), 571–587 (1999). MR1704556
