Pricing Vulnerable Options with Stochastic Volatility and Stochastic Interest Rate

  • Chaoqun Ma
  • Shengjie YueEmail author
  • Hui Wu
  • Yong Ma


This paper considers the pricing issue of vulnerable European options when the price process of the underlying asset follows the GARCH diffusion model with stochastic interest rate. Based on the proposed model, we obtain an approximate solution for the vulnerable European option price via means of Fourier transform. In addition, the Greeks of vulnerable option price are derived explicitly. Besides, the approximate solution of vulnerable option price can be quickly computed by using the fast Fourier transform (FFT) algorithm. The results of Monte Carlo simulations indicate that FFT is accurate, fast and easy to implement. More important, the pricing model also reveals that: (i) a negative correlation of volatility with the spot return creates a fat left tail and thin right tail in the distribution of continuously compounded spot returns. Thus, for in-the-money options, the vulnerable option prices of the proposed model are higher than those of Klein (J Bank Finance 20(7):1211–1229, 1996). While for deep-out-of-the-money options, the vulnerable option prices of the proposed model are smaller; (ii) the higher long-run mean of the underlying asset price’s instantaneous variance, the higher vulnerable option price; (iii) the long-run mean of the stochastic interest rate exerts a positive effect on the value of vulnerable European option.


Vulnerable European options Characteristic function GARCH diffusion model Stochastic interest rate Fast Fourier transform 



This research is partially supported by the National Natural Science Foundation of China (Grant Nos. 71431008, 71521061, 71601075, 71850012, and 71790593), Major special Projects of the Department of Science and Technology of Hunan province (Grant No. 2018GK1020) and the China Postdoctoral Science Foundation (Grant No. 2017M612768).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Business School of Hunan UniversityChangshaChina
  2. 2.China Merchants BankShenzhenChina
  3. 3.College of Finance and StatisticsHunan UniversityChangshaChina

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