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校准，以便复制和解释观察到的市场数据的结构。经典的Black-Scholes模型假设股票的波动率函数为常数，但通过对看涨期权的解析公式进行反演，并利用市场价格，可以计算出隐含波动率曲面。这个表面通常显示倾斜和微笑，因此使恒定波动率的假设显然是不切实际的。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，它要求在连续的罢工和到期日获得市场价格。此外，在随机利率的设置，它还需要整个利率期限结构。由于实际上只有1000多个数据点可用，因此需要某种形式的插值。许多技术，参数化或非参数化，在过去已经被用来帮助这个关键的插值任务。在这个项目中，我们建议采用最佳的运输技术。我们使用最优运输的动力学公式，即我们的模型概率测度被约束为半鞅测度，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，并使用已知的分析公式来改变模型参数以匹配市场数据。这自然会导致最小化问题的PDE公式化，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。本文将文献[1]和[2]中的方法应用于局部随机波动率校准和SPX/VIX联合校准，并将其推广到随机利率的设定。稍后，在多曲线环境中与（LIBOR/EURIBOR）市场模型进行联合校准可能是目标。这可以按顺序进行，也就是说，我们首先校准市场模型，然后使用该模型单独校准基础。或者，我们可以考虑联合校准市场模型和标的资产--这需要使市场模型依赖于标的资产本身，或者通过修改成本函数来强制市场模型联合校准。额外的复杂性将带来数值挑战，因为从利率市场模型继承的添加更多的状态变量将为所得的HJB方程增加维度，因此通过通常用于二维的方法使PDE在数值上不可解。因此，另一个目标是弄清楚是否有一些结构可以用于市场模型来解决这样的偏微分方程，或者可以应用什么技术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，因为它汇集了最佳运输，金融模型校准，用于非线性偏微分方程和潜在的机器学习的数值。该项目是在以往研究的基础上进行的，但又将其拓展到了新的方向。该项目福尔斯属于EPSRC的“运筹学”研究领域，并得到了法国巴黎银行的参与。参考文献：[1] Ivan Guo，Gregoire Loeper，& Shiyi Wang。“Calibration of local-stochastic volatility models by optimal transport”. arXiv预印本arXiv：1906.06478（2019）。[2]Ivan Guo等人，“通过最佳运输对SPX和VIX进行联合建模和校准”。可在SSRN 356899（2020）上获得

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)