Calibrating financial market models via optimal transport

任何金融市场模型在实际应用之前，都需要进行校准，以便能够复制和解释观察到的市场数据的结构。经典的Black-Scholes模型假设股票以恒定的波动函数演变，然而，通过对看涨期权的解析公式进行反转并使用观察到的市场价格，可以计算出“隐含波动率”面。这个表面通常显示出倾斜和微笑，因此做出恒定波动的假设显然是不现实的。Dupire引入了一个所谓的“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，它需要获得连续的罢工和到期日的市场价格。此外，在随机利率设置下，还需要利率的整个期限结构。因为在现实中，只有有限的数据点是可用的，所以需要某种形式的插值。许多技术，参数或非，已经使用在过去，以帮助这个关键的插值任务。在这个项目中，我们建议采用最佳运输技术。我们使用最优运输的动态公式，这就是我们的模型概率度量被限制为半鞅度量的地方，我们的目标是最小化给定的凸成本函数，该函数将惩罚我们选择使用的任何模型结构的偏差。此外，我们在市场数据中强制匹配条件，并使用已知的分析公式来更改模型参数以匹配市场数据。这自然导致了最小化问题的PDE公式，因此进行了对偶论证，并获得了对偶问题，需要对[1]中的对偶论证进行一些调整。由此产生的问题将需要HJB方程的数值解，通过策略迭代方法计算最优参数。我们的目标是将[1]和[2]中使用的技术应用于局部随机波动率校准和假设利率为零的SPX/VIX的联合校准，并将其扩展到随机利率的设置。随后，在多曲线环境下，与LIBOR/EURIBOR市场模型的联合校准可能成为目标。这可以按顺序进行，即我们首先校准市场模型，然后分别使用该模型校准基础。或者我们可以考虑联合校准市场模型和基础——这要么需要使市场模型依赖于基础本身，要么通过修改成本函数来施加惩罚，迫使市场模型联合校准。由于从利率市场模型中继承了更多的状态变量，增加了HJB方程的维度，从而使得PDE无法通过通常在二维中使用的方法进行数值求解，因此额外的复杂性层将带来数值挑战。因此，另一个目标是弄清楚是否有某种结构可以在市场模型中用于解决这样的PDE，或者什么技术可以应用于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，因为它汇集了最优运输、金融模型校准、非线性偏微分方程的数值和潜在的机器学习等技术。它以以前的作品为基础，但将它们扩展到新的方向。该项目属于EPSRC“运筹学”研究领域，法国巴黎银行也参与其中。参考文献：b[1] Ivan Guo， Gregoire Loeper, and Shiyi Wang。“用最优输运校正局部随机波动模型”。[j] .中国农业大学学报（自然科学版）：1906.06478 （2019）Ivan Guo等。“SPX和VIX的最佳运输联合建模和校准”。SSRN 356899 （2020）

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)