Gated Neural Networks for Option Pricing: Rationality by Design

We propose a neural network approach to price EU call options that significantly outperforms some existing pricing models and comes with guarantees that its predictions are economically reasonable. To achieve this, we introduce a class of gated neural networks that automatically learn to divide-and-conquer the problem space for robust and accurate pricing. We then derive instantiations of these networks that are ‘rational by design’ in terms of naturally encoding a valid call option surface that enforces no arbitrage principles. This integration of human insight within data-driven learning provides significantly better generalisation in pricing performance due to the encoded inductive bias in the learning, guarantees sanity in the model’s predictions, and provides econometrically useful byproduct such as risk neutral density. Introduction Option pricing models have long been a popular research area. From a theoretical perspective, new option pricing models provide an opportunity for academics to examine financial markets’ mechanics. From a practical viewpoint, market makers desire efficient pricing models to set bid and ask prices in derivative markets. The earliest and simplest pricing model, Black–Scholes (Black and Scholes 1973) gives a rough theoretical estimate of European option price. Since then many studies attempted to find better option pricing models by relaxing the strict assumptions in Black– Scholes. The models proposed by economists usually start from a set of economic assumptions and end up with a deterministic formula that takes as input some market signals (e.g., moneyness, time to maturity, and risk-free rate). In contrast, machine learning studies solve option pricing in a data-driven way: as a regression problem, with similar inputs to econometric models, and real market option prices as outputs. The complicated relationship between input and output (e.g., a Black–Scholes like formula) is learned from a large amount of data rather than derived from econometric axioms. Progress in data-driven option pricing can be driven by improvements in model expressivity, as well as integrating selected econometric axioms into a data-driven model as inductive bias. In this paper we achieve excellent option pricing results by contributing on both of these lines. Copyright c © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Regression models trained by machine learning techniques, such as kernel machines and neural networks, generalise well to out-of-sample cases as long as the training data is sufficient. Such data-driven methods give good option price estimates (Malliaris and Salchenberger 1993), and can even surpass formula derived from economic principles. One drawback of existing data-driven approaches is that they seek a unique solution for all options. However, learned pricing models fail on certain options, for example, some overestimate deep out-of-the money options (Bennell and Sutcliffe 2004), or underestimate options very close to maturity (Dugas et al. 2000). To alleviate these issues, (Gradojevic, Gencay, and Kukolj 2009) proposed a ‘divide-and-conquer’ strategy, by first grouping options into sub-categories, and building distinct pricing models for each sub-category. However, this categorisation is done by manually defined heuristics, and may not be consistent with market conditions, and their changes in time. In this paper, we propose a novel class of neural networks for option pricing. These implement a divide-and-conquer method where option grouping is automatic and learned from data rather than manual heuristics. Therefore, it can dynamically adjust both option classification and refine the per-class pricing model as the market changes with time. Experiments on S&P 500 index options show that our approach is significantly better than others. A limitation of all the above machine learning-based methods is that while they may fit the data well (e.g., mean square error), they do not enforce some economic principles, thus ruling out their suitability for pricing in practice. E.g., option prices have theoretical bounds, the violation of which makes investors gain risk-free profit (so-called arbitrage). This motivates another less-studied approach to improving data-driven option pricing: opening up black box models to integrate economic axioms as constraints into learning algorithms (Dugas et al. 2000). From an economic perspective, this is designing a NN to make economically meaningful predictions, and from a learning perspective it is providing domain-specific inductive bias to improve generalisation and avoid overfitting. In this paper, we derive a class of gated neural networks with stronger economic rationality guarantees than existing work. In particular our neural price predictor is the first learning based approach to carry a valid risk neutral density function, i.e., a valid probability distribution over the future asset price in risk neutral probability space. The terminology risk neutral roughly implies no arbitrage but its rigorous definition is out of scope of this work, see (Jeanblanc, Yor, and Chesney 2009) for details. Our contribution is three-fold: (1) We propose a neural network with superior option pricing performance. (2) We evaluate our method against several baselines on a largescale dataset: it includes 5139 trading days and 3029327 option contracts – this is 70 times larger than previous studies (Dugas et al. 2000; Gradojevic, Gencay, and Kukolj 2009). (3) Our neural network model is meaningful in that it enforces all the necessary requirements for an economically valid (no arbitrage) call option pricing model. This results in a valid risk neutral density function, from which users can extract many metrics, e.g., variance, kurtosis and skewness, that are crucial for risk management purposes. Related Work Econometric Methods Asset pricing is a very active research area in finance and mathematical finance. The oldest and most famous model for option pricing is Black– Scholes (Black and Scholes 1973). The biggest criticism of this model is its incompatibility with the volatility smile behaviour in real markets due to its constant volatility assumption. The volatility smile exists due to the fact that real-world distributions are often fat-tailed and asymmetric. Stochastic volatility models, (e.g. (Heston 1993)), aim to model the above smile behaviour through allowing randomness of volatility, compensated for by introducing random volatility process (Heston 1993). Another stream of research suggests including jumps which represent rare events in the underlying process to alleviate the smile problem. These models are called Levy models (Merton 1976; Kou 2002; Madan, Carr, and Chang 1998; Barndorff-Nielsen 1997; Carr and Geman 2002) and are able to generate volatility skew or smile. A comprehensive theoretical explanation of asset pricing models can be found in (Jeanblanc, Yor, and Chesney 2009). This paper tackles the skew/smile problem in a more data-driven way: it learns from market prices so that a model that fits the market prices well is expected to carry the same smile structure. There are many methods for implementing option pricing models including: Fourier-based (Carr and Madan 1999), Tree-based (Cox, Ross, and Rubinstein 1979), Finite difference (Schwartz 1977) and Monte Carlo methods (Boyle 1977). In this paper, we employ the fractional FFT method (Cooley and Tukey 1965) for our benchmark option pricing models as their characteristic functions are known. Neural Network Methods There is a long history of computer scientists trying to solve option pricing using neural networks (Malliaris and Salchenberger 1993). Option pricing can be seen a standard regression task for which there are many established methods and neural networks (rebranded deep learning) are one of the most popular choices. Some researchers claim that it is an advantage of neural network (NN) methods that they do not make as many assumptions as the econometric methods. However, NNs are not orthogonal to econometric methods. In fact, some NN methods leverage classic econometric insights. For example, (Garcia and Gençay 2000) proposed a neural option pricing model with a Black–Scholes like formula. (Dugas et al. 2000) chose specific activation functions and positive weight parameter constraints such that their model has the second-order derivative properties required by economic axioms. These studies suggested that introducing econometric constraints produces better option pricing models compared to vanilla feed-forward NNs. A good survey of this line of work can be found in (Garcia, Ghysels, and Renault 2010). While some NN methods have benefited from econometric insights, these methods have always tried to find a universal pricing model for all options in the market. However, it has been shown that for deep out-of-money options or those with long maturity, NN methods perform very badly (Bennell and Sutcliffe 2004). This is unsurprising because NN methods usually produce a smooth pricing surface that fails to capture these awkward and low-volume parts of the market. (Gradojevic, Gencay, and Kukolj 2009) tried to address this issue by categorising options based on their moneyness and time to maturity, and training independent NNs for each class of options. Their grouping of options is based on fixed manual heuristic that is suboptimal, and does not adapt to the changing market data over time. Our method is a neural network that exploits a similar divide-and-conquer idea, however it jointly learns the inter-related problems of separating options into groups and pricing each group. Providing this increased model expressivity challenges our previous goal of building in econometric axioms to ensure meaningful predictions, because rationality constraints are harder to enforce in this more complex model. Thus we apply significant effort to contribute both a more expressive neural learner, and stronger rationality constraints guarantees to existing work.