Sebastien Mollaret
Artificial intelligence in finance can offer a solution for old, unresolved issues. Although this may have been a very generic tagline a few years ago, today, this promise is now fulfilled. Sebastien Mollaret, Quantitative Researcher at JP Morgan, explores two major developments in quantitative finance aided by deep learning.
Machine learning has recently been a growing area in quantitative finance and especially the field of deep learning. It now allows solving high-dimensional problems, which were too complex to solve using traditional methods due to the so-called curse of dimensionality (Bellman 1954) – namely, as the dimensionality grows, the complexity of the algorithm grows exponentially. Let's focus on two major and recent developments in the literature, which will without any doubt lead to future applications for practitioners to price financial derivatives:
In quantitative finance, a lot of non-linear option pricing problems can be expressed as non-linear parabolic PDEs or as BSDEs, which are equivalent thanks to the non-linear Feynman-Kac formula. To cite only a few examples, we have American options, the uncertain volatility model, transactions costs, different rates for borrowing and lending and credit valuation adjustment, which are well described in Guyon and Henry-Labordere’s work (2013).
1/ Express the PDE as its equivalent BSDE using the non-linear Feynman-Kac formula:
Let (Ω,F,P) be a probability space, W a standard d-dimensional Brownian motion, Y and Z stochastic processes defined for all t ∈ [0,T] by:
2/ Transform the BSDE into the corresponding stochastic control problem:
Examples of PDEs of dimension 100 are studied and compared to either closed-form solutions or approximated ones using other numerical methods (E, Han, and Jentzen). Applications to credit value adjustment (CVA) and initial margin (IM) are presented in Henry-Labordere (2017) and to optimal posting of collateral in Henry-Labordere (2018).
Let's consider the problem of calibrating a stochastic volatility model (classic examples are Heston and Bergomi models), which means finding the set of parameters θ to match market option prices.
where one could also use some other metric and add weights to force better calibration for most liquid options.
The idea presented in Horvath, Muguruza, and Tomas (2019) is the following:
Horvath, Muguruza, and Tomas (2019) presents the application to the calibration of rough stochastic volatility models, which are known to be computationally expensive due to fractional Brownian motions not being Markov processes. Given the above framework, one can thus compute as many prices as needed to train and perform well on the test set since this step is performed offline. We then end up with a standard optimisation problem, which can be quickly solved using standard optimisers like Levenberg-Marquardt.
Bellman, R. The theory of dynamic programming. Bulletin of the American Mathematical Society, Volume 60, Number 6 (1954), 503-515.
E, W., Han, J., Jentzen, A. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Available at arXiv: https://arxiv.org/abs/1706.04702 (2017).
Guyon, J., Henry-Labordere, P. Nonlinear Option Pricing. Financial Mathematics Series CRC, Chapman Hall (2013).
Henry-Labordere, P. Deep primal-dual algorithm for BSDEs: Applications of machine learning to CVA and IM. Available at SSRN: https://ssrn.com/abstract=3071506 (2017).
Henry-Labordere, P. Optimal Posting of Collateral with Recurrent Neural Networks. Available at SSRN: https://ssrn.com/abstract=3140327 (2018).
Horvath, B., Muguruza, A., Tomas, M. Deep Learning Volatility. Available at SSRN: https://ssrn.com/abstract=3322085 (2019).
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