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Zhu and He [(2018). A new closed-form formula for pricing European options under a skew Brownian motion. The European Journal of Finance 24(12): 1063–1074] provided an innovative closed-form solution by replacing the standard Brownian motion in the Black–Scholes framework using a particular skew Brownian motion. Their formula involves numerically integrating the product of the Guassian density and corresponding distribution function. Being different from their pricing formula, we derive a much simpler formula that only involves the Gaussian distribution function and Owen's $T$ function. [ABSTRACT FROM AUTHOR]

In this paper, an additional factor is introduced into the Heston–Hull–White (HHW) hybrid model, which originally combines the Heston stochastic volatility model and the Hull–White stochastic interest rate model, to capture the correlation between the underlying price and the interest rate, while at the same time preserve the analytical tractability. With the analytical solution to the characteristic function of the underlying price being successfully derived, a closed‐form pricing formula for European options under the two‐factor HHW hybrid model is presented. Numerical experiments are also carried out to show the effect of the newly introduced factor. To further demonstrate the importance of introducing the correlation, we have also conducted an empirical study, comparing the performance of our model and that of the HHW model, based on European options written on S&P 500 index. [ABSTRACT FROM AUTHOR]

In this paper, we consider the pricing of European options under a regime‐switching Heston–Cox–Ingersoll–Ross (CIR) hybrid model, where the mean‐reversion levels of both the stochastic volatility and interest rate are assumed to change among different states. Albeit difficult, we have still managed to derive an semianalytical pricing formula for European options after the generalized moment generating function of this particular model is worked out. Numerical experiments are also carried out to demonstrate the accuracy of the newly derived formula as well as the influence of the introduction of the regime‐switching mechanics on option prices. Finally, through a preliminary empirical study, our model is shown to be superior to the Heston‐CIR model, which demonstrates the importance of introducing the regime‐switching mechanics. [ABSTRACT FROM AUTHOR]

We combine the rough Heston model and the CIR (Cox–Ingersoll–Ross) interest rate together to form a rough Heston-CIR model, so that both the rough behaviour of the volatility and the stochastic nature of the interest rate can be captured. Despite the convoluted structure and non-Markovian property of this model, it still admits a semi-analytical pricing formula for European options, the implementation of which involves solving a fractional Riccati equation. The rough Heston-CIR model is more general, taking both the rough Heston model and the Heston-CIR model as special cases. The influence of rough volatility and stochastic interest rate is shown to be significant through numerical experiments. [ABSTRACT FROM AUTHOR]

Stochastic volatility models are widely used in option pricing; however, most of these models assume a constant long-term mean of volatility that fails to account for the implied volatility term structure. In response, this article proposes an improved Stein–Stein model that enables the random variation of the long-term mean under the Stein–Stein model while also relaxing the zero correlation assumption between asset and volatility. One of the new model's chief advantages is that it offers an analytical solution to European option prices, thus speeding up the calibration process and allowing for easy implementation in practical risk management. An empirical analysis based on 50ETF options in the Chinese market demonstrates that the newly improved Stein–Stein model predicts market outcomes more accurately compared to the original Stein–Stein model. [ABSTRACT FROM AUTHOR]

In this paper, the impact of liquidity on the underlying asset is taken into account when pricing European options through a discounting factor which depends on two factors, i.e., market liquidity risk modeled as a mean-reverting stochastic process and the sensitivity of the underlying to market liquidity. A closed-form pricing formula for liquidity-adjusted European options is derived in the form of an infinite series using Karhunen–Loève expansion for the Ornstein–Uhlenbeck process. The convergence of the series solution is theoretically proved to guarantee closedness so that market practitioners can adopt the new formula when they need to account in market liquidity risk. The speed of convergence is demonstrated through numerical experiments. Finally, the accuracy of the newly derived formula is shown by comparing option prices calculated with our formula and those obtained from Monte-Carlo simulation, and various properties of our formula are also investigated. • A closed-form pricing formula for liquidity-adjusted European options is derived. • The series solution is obtained with Karhunen–Loève expansion for the OU process. • The convergence of the series solution is theoretically proved. [ABSTRACT FROM AUTHOR]