In this paper, we derive new closed-form valuations to European options under three-factor hybrid models that include stochastic interest rates and stochastic volatility and incorporate a nonzero covariance structure between factors. We make novel use of the empirically proven 3/2 stochastic volatility model with a time-dependent drift in which we are free to choose the moving reversion target. This model has been shown by many authors to empirically outperform other volatility models in maximising model fit. We also improve the valuation of European options under the Heston volatility and Cox, Ingersoll, Ross interest rate model, recently published in the literature, by replacing open-form infinite series with closed-form analytic expressions. For completeness, we also add a fuller covariance structure in this setting and detail closed-form valuations for options. The inclusion of nonzero covariances amongst the factors can significantly improve option pricing by allowing for a wider variety of market behaviour. The solutions are derived by firstly formulating the price of a European call option in terms of the corresponding characteristic function of the underlying price and then determining a partial differential equation for the characteristic function. By including empirically proven models into our analysis, the options formulae could provide more realistic prices for investors and practitioners.

在本文中，我们在三因素混合模型（包括随机利率和随机波动率，并在因素之间采用非零协方差结构）下得出了欧式期权的新封闭式估值。我们新颖地使用了经过时间证明的3/2随机波动率模型，该模型具有随时间变化的漂移，在其中我们可以自由选择移动的回归目标。许多作者已经证明，在最大化模型拟合度方面，该模型在经验上优于其他波动率模型。我们还通过用封闭式分析表达式代替开放式无穷级数，在最近发表于文献中的Heston波动率和Cox，Ingersoll，Ross利率模型下提高了欧洲期权的估值。为了完整性，我们还会在此设置中添加更完整的协方差结构，并详细说明期权的封闭式估值。在这些因素中包括非零协方差可以通过允许更多种类的市场行为来显着改善期权定价。通过首先根据标的价格的相应特征函数来公式化欧洲看涨期权的价格，然后确定该特征函数的偏微分方程，可以得出解决方案。通过将经过经验证明的模型纳入我们的分析，期权公式可以为投资者和从业人员提供更现实的价格。通过首先根据标的价格的相应特征函数来公式化欧式看涨期权的价格，然后确定该特征函数的偏微分方程，可以得出解决方案。通过将经过经验证明的模型纳入我们的分析，期权公式可以为投资者和从业人员提供更现实的价格。通过首先根据标的价格的相应特征函数来公式化欧式看涨期权的价格，然后确定该特征函数的偏微分方程，可以得出解决方案。通过将经过经验证明的模型纳入我们的分析，期权公式可以为投资者和从业人员提供更现实的价格。