A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term m
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A closed-form pricing formula for European options under a new stochastic volatility model with a stochastic long-term mean Xin-Jiang He1 · Wenting Chen2 Received: 5 November 2019 / Accepted: 14 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Based upon the fact that a constant long-term mean could not provide a good description of the term structure of the implied volatility and variance swap curve, as suggested by Byelkina and Levin (in: Sixth world congress of the Bachelier Finance Society, Toronto, 2010) and Forde and Jacquier (Appl Math Finance 17(3):241–259, 2010), this paper presents a new stochastic volatility model, by assuming the long-term mean of the volatility in the Heston model be stochastic. An important feature of our model is that it still preserves the essential advantage of the Heston model, i.e., the analytic tractability, because a closed-form pricing formula for European options can be derived, which could not only facilitate the risk management process but also help save plenty of time in terms of model calibration. The effect of the newly introduced stochastic long-term mean is demonstrated through the numerical comparison with the Heston model. It is also shown that the current model can overall lead to more accurate option prices than the Heston model, through a carefully designed empirical study. Keywords Stochastic volatility · Stochastic long-term mean · Closed-form · European options · Risk management · Empirical studies JEL Classification G13
1 Introduction Nowadays, managing risk is becoming increasingly important for market practitioners. To cope with such an increasing demand, option derivatives are developed, which are very useful in measuring and managing financial risks, especially the volatility risk. With the sharp increase in the trading volume being observed, how to accurately and efficiently determine option prices is a widely pursued topic in quantitative finance and risk management.
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Wenting Chen [email protected]
1
School of Economics, Zhejiang University of Technology, Hangzhou, China
2
School of Business, Jiangnan University, Wuxi, Jiangsu, China
123
Mathematics and Financial Economics
A breakthrough was made in 1973 by Black and Scholes [7] and Merton [33], who proposed a simple and elegant model for the pricing of options. However, this model results in some biases in option prices, because some strong assumptions made to achieve analytical simplicity and tractability are not consistent with the real behavior of financial markets. In particular, the assumption of “constant volatility” is generally thought to be unsuitable because the implied volatility extracted from real market data usually exhibits a non-constant curve across different strike prices, i.e., the well-known “volatility smile” or “volatility smirk” [17]. In addition, the implied volatility also differs for similar options with different times to maturity, which is called the “term structure of volatility”. As a result, non-constant volatility processe
