Calibrating financial market models via optimal transport

任何金融市场模型在用于实践之前，都需要进行校准，以便复制并解释观察到的市场数据的结构。经典的Black-Scholes模型假设一只股票的波动率是恒定的，然而，通过反转看涨期权的分析公式并利用观察到的市场价格，我们可以计算出“隐含波动率”曲面。这一表面通常会出现扭曲和微笑，因此让恒定波动率假设显然是不现实的。杜皮尔提出了一个所谓的“局部波动率”公式，其中波动率函数取决于时间和股票价格。该公式将欧式看涨期权的价格与波动率函数联系起来。然而，它需要以罢工和到期的连续不断的方式获得市场价格。此外，在随机利率的背景下，它还要求利率的整个期限结构。由于在现实中，只有有限多的数据点可用，因此需要某种形式的内插。许多技术，无论是否参数，在过去都被用来帮助完成这一关键的插补任务。在这个项目中，我们建议使用最优的运输技术。我们使用最优运输的动态公式，即我们的模型概率度量被约束为半鞅度量，我们的目标是最小化给定的凸性成本函数，该凸性成本函数将惩罚我们选择使用的任何模型的结构偏差。此外，我们在市场数据中强制匹配条件，并使用已知的解析公式来改变模型参数以匹配市场数据。这自然导致了极小化问题的偏微分方程式，因此进行了对偶论证，并得到了对偶问题，需要对文献[1]中的对偶论证进行一些修改。我们的目标是将[1]和[2]中使用的技术应用于本地随机波动率校准和利率为零的SPX/VIX的联合校准，并将其推广到随机利率的设定。稍后，目标可能是在多曲线环境下与(LIBOR/EURIBOR)市场模型进行联合校准。这可以按顺序完成，即我们首先校准市场模型，然后使用该模型分别校准基础。或者，我们可以考虑联合校准市场模型和标的--这将需要让市场模型依赖于标的本身，或者修改成本函数，以具有迫使市场模型联合校准的惩罚。额外的复杂性将带来数字上的挑战，因为添加更多继承自利率市场模型的状态变量，将增加结果HJB方程的维度，从而使PDE无法通过通常在二维中使用的方法进行数值求解。因此，另一个目标是找出市场模型中是否有某种结构可以用来解决这样的偏微分方程，或者什么技术可以用来数值解决那里的问题--目前的研究途径包括神经网络近似。这个项目是跨学科的，因为它结合了来自最优运输、金融模型校准、非线性偏微分方程的数值计算和潜在的机器学习的技术。该项目属于EPSRC“运筹学”研究领域，并得到巴黎银行的参与。推荐人：Ivan Guo，Gregoire Loeper，&Shiyi Wang。“通过最优运输对局部随机波动模型进行校准”。Arxiv预印本arxiv：1906.06478(2019年)。[2]郭伊万等人。通过最优运输对SPX和VIX进行联合建模和校准。在SSRN 356899(2020年)上提供

Any financial market model, before it can used in practice, needs to calibrated so that it replicates and accounts for the structure of observed market data. The classical Black-Scholes model assumes that a stock evolves with a constant volatility function, however by inverting the analytic formula for a call option and using observed market prices, one can compute the "implied volatility" surface. This surface typically displays skews and smiles, so making a constant volatility assumption a clearly unrealistic one. Dupire introduced a formula for so-called "local volatility" where the volatility function depends on time and the stock price as well. The formula links the price of a European call option to the volatility function. However, it requires access to market prices at a continuum of strikes and maturities. Further, in a setting of stochastic interest rates, it also requires the whole term structure of the interest rates. Since, in reality, only finitely many data points are ever available, some form of interpolation is required. Many techniques, parametric or not, have been used in the past to help with this crucial interpolation task. In this project, we propose to employ optimal transport techniques. We use the dynamical formulation of optimal transport, that is where our model probability measures are constrained to be semimartingale measures, and we aim to minimise a given convex cost function that will penalise deviations from the structure of whichever model we choose to use. In addition, we enforce matching conditions in the market data and use known analytical formulae to change the model parameters to match the market data. This naturally leads to a PDE formulation of a minimisation problem, so a duality argument is carried out, and the dual problem is attained, with some adaptation to the duality argument in [1] required. The resulting problem will require the numerical solution of an HJB equation to compute the optimal parameters via a policy iteration method.We aim to apply the techniques used in [1] and [2] on Local-Stochastic Volatility calibration and the joint calibration of SPX/VIX where the interest rate was assumed to be zero, and extend them to the setting of stochastic interest rates. Later on, a joint calibration with the (LIBOR/EURIBOR)-market model in a multi-curve setting could be the objective. This could be done sequentially, that is we calibrate a market model first, then using that calibrate the underlying separately. Or we could consider jointly calibrating the market model and the underlying - which will either require making the market model depend on the underlying itself or by modifying the cost function to have a penalty that forces the market model to jointly calibrate. The extra layer of complexity will present numerical challenges as adding more state variables, inherited from interest rates market model, will add dimensions to the resulting HJB equation and therefore render the PDE numerically unsolvable through methods generally used in two dimensions. Thus, another objective is to figure out if there is some structure that can be used in the market model to solve such a PDE or what techniques can be applied to numerically solve the problem there - avenues of investigation currently include neural network approximations.This project is interdisciplinary in that it brings together techniques from optimal transport, financial model calibration, numerics for non-linear PDEs and potentially machine learning. It builds on previous works but extends them into novel directions.This project falls into the EPSRC "Operational Research" research area and has involvement from BNP Paribas.References:[1] Ivan Guo, Gregoire Loeper, & Shiyi Wang. "Calibration of local-stochastic volatility models by optimal transport". arXiv preprint arXiv:1906.06478 (2019).[2] Ivan Guo et al. "Joint modelling & calibration of SPX and VIX by optimal transport". Available at SSRN 356899 (2020)