On the Calibration of the 3/2 Model

Abstract We consider the problem of calibrating the 3/2 stochastic volatility model to option data. In comparison to the characteristic function of the Heston model, the characteristic function of the 3/2 model can be up to 50 times slower to evaluate. This makes the standard least squares calibration with finite-difference gradients unreasonably slow. To address this problem we derive the analytic gradient of the characteristic function in compact form. We then propose a computational method for the analytic gradient formula which caches intermediate results across the partial derivatives, in addition to the strike dimension and the maturity dimension. Compared to the fastest method of calibrating the 3/2 model which we could find in the literature, the method proposed in this paper is more than 10 times faster. We also discuss the issue of apparent non-convexity in the least squares calibration of the 3/2 model for market data. To tackle it, we propose a regularized calibration where the regularization point is obtained using “risk neutral” MCMC estimation of the model. We find that this approach is particularly well suited for the calibration problem as it generates naturally a consistent damping matrix for the parameter estimates, in addition to being very fast.

**摘要** 我们考虑将3/2随机波动率模型校准到期权数据的问题。与赫斯顿模型的特征函数相比，3/2模型的特征函数求值速度可能慢达50倍。这使得使用有限差分梯度的标准最小二乘校准慢得不合理。为解决此问题，我们以紧凑形式推导了特征函数的解析梯度。然后，我们针对解析梯度公式提出一种计算方法，该方法除了在执行价格维度和到期期限维度外，还在偏导数之间缓存中间结果。与我们在文献中能找到的校准3/2模型的最快方法相比，本文提出的方法快10倍以上。我们还讨论了3/2模型针对市场数据进行最小二乘校准时明显的非凸性问题。为解决该问题，我们提出一种正则化校准方法，其中正则化点是通过对模型进行“风险中性”马尔可夫链蒙特卡罗（MCMC）估计得到的。我们发现这种方法特别适合校准问题，因为它除了速度非常快之外，还自然地为参数估计生成了一个一致的阻尼矩阵。