DeepOption: A novel option pricing framework based on deep learning with fused distilled data from multiple parametric methods

Highlights

• A new option pricing and delta-hedging framework based on deep learning is proposed.

• The framework can process small and imbalanced datasets containing real option data.

• Fused distilled data from conventional parametric option pricing methods are used.

• The biased data problem is tackled with label augmentation and transfer learning.

• We propose a single model applicable to all situations, regardless of moneyness.

Abstract

The remarkable performance of deep learning is based on its ability to learn high-level features by processing large amounts of data. This exceptionally superior performance has attracted the attention of researchers studying option pricing. However, option data are more expensive and less accessible than other types of data and are imbalanced because of the liquidity of options. This motivated us to propose a new option pricing and delta-hedging framework called DeepOption. This framework, which is based on deep learning, can improve the performance even when applying imbalanced real option data. In particular, the framework fuses simulated big data, known as distilled data, obtained using various traditional parametric methods. The proposed model employs the following three-stage training approach: Our model is pre-trained using big distilled data after it is fine-tuned using real option data through transfer learning. Finally, a delta branch is added to the model and trained. We experimentally evaluated the proposed method using three sets of real option data, namely S&P 500 European call options, EuroStoxx50 call options, and Hang Seng Index put options. Our experimental results on option pricing demonstrate that our proposed model outperforms parametric methods and other machine learning methods. Specifically, our model, which uses pre-training with distilled data, reduces the overall mean absolute percentage error (MAPE) by more than 50%, compared with that of a deep learning model using only real option data without pre-training.

Introduction

Options are one of the most important derivatives applied to risk management. An option offers the buyer the right, as opposed to an obligation, to trade the underlying asset at a fixed price. These rights enable buyers to trade at a certain price; thus, accurate pricing is especially crucial for investors by helping them to make the right decisions in terms of their perceived risk. However, option pricing is highly complicated owing to numerous internal and external factors, including stock price, strike price, option type, time-to-maturity, and macroeconomic factors. Considering the significance of risk management and the difficulty of accurate pricing incorporating the many crucial factors, option pricing is one of the most challenging and critical issues in finance.

Option pricing is a methodical process that entails the calculation of theoretically fair prices under certain conditions [1]. The basic factors that affect the pricing are the strike price, spot price of the underlying asset, volatility, time-to-maturity, and risk-free rate. The option price can be calculated using both parametric and non-parametric methods [2]. Black and Scholes initiated the application of parametric methods to option pricing [1]. Although the Black–Scholes (BS) model assumes constant volatility over the maturity period, it is widely used in practice because it is easy to apply and only slightly different from other methods with respect to the pricing of at-the-money options and options with short maturity. However, the BS model has limitations in that the method is based on constant volatility. Thus, advanced option pricing methods using stochastic volatility have been studied to reflect more realistic market environments and overcome the limitations of the BS model. Typical advanced models include Heston's random diffusion process [3] and Merton's jump diffusion process [4]. In addition, more advanced models, such as the Lévy process [5] and variance gamma process [6], have been actively studied.

Parametric methods that apply numerical approaches to option pricing have also been developed. Numerical methods are used when it is difficult to solve a model in a regular form owing to a large number of variables or when the BS model is difficult to apply when a closed-form does not exist. Typical numerical option pricing methods include the Monte Carlo method (MC), finite differential method (FDM), and binomial tree method (BI) [7]. The BI method, developed by Cox, Ross, and Rubinstein [8], is straightforward, converges to the BS model within the limit of a small period, and offers greater flexibility for incorporating other factors such as dividends and the interest rate, which can change prior to maturity [9]. Boyle [10] used the MC method for option pricing. As the dimensionality of the problem increases, this approach becomes more attractive than others because the error convergence rate is independent of the dimensionality of the problem. In addition, the restriction of the MC method is lower than those of other numerical approaches. Subsequently, more efficient methods of generating random numbers, such as the quasi-Monte Carlo method [11], the variance reduction method (e.g., the antithetic variates [12] and control variates [13] methods), and the importance sampling method [14] for error reduction have been proposed. The FDM, which was first employed by Schwartz [15], determines the option price by approximating the differential equations to difference equations. The FDM obtains accurate numerical solutions by solving the partial differential equations, where it quickly converges to a stable solution compared to other methods. Since the ultimate goal of the FDM is to solve partial differential equations, it is studied with the goal of solving new models developed through the progress of the BS model.

In addition to these approaches, deep neural networks (DNNs) with multiple hidden layers, which learn highly complex nonlinear relationships among input data, have demonstrated outstanding performances. Therefore, DNNs have been applied in many areas, including speech recognition, visual object recognition, object detection, drug discovery, and genomics [16]. Advances in neural networks have led to the development of non-parametric DNN-based option pricing methods [7,17,18]. Unlike parametric methods, neural network-based option pricing methods do not require many assumptions and are easy to implement, as long as sufficient data are available.

Option pricing models using an artificial neural network were first introduced by Hutchinson et al. [19]. Unlike their predecessors, such models do not specify the way in which inputs affect the option prices; instead, they allow the network to explore the relationships using data. The results showed that the neural network model for option pricing and delta hedging achieved superior performance compared to that of BS, which employed historical volatility. Lajbcygier and Connor [20] proposed a hybrid neural network to estimate the difference between the price obtained using conventional parametric methods and the real market option price by applying a bootstrap method for trading. Gençay and Qi [21] overcame one of the shortcomings of a neural network (i.e., overfitting) using Bayesian regulation, early termination, and bagging. This approach yields better performance when the proposed methods are applied to out-of-sample pricing and delta-hedging. Amilon [22] proposed a neural network model for predicting the bid price and ask price using daily call option data. Gradojevic et al. [18] proposed a method using a modular neural network consisting of three to nine modules based on moneyness and time-to-maturity; that is, they used different modules depending on these two factors. Liang et al. [7] forecasted option prices using an improved version of a conventional option pricing method. This allowed them to forecast option prices with the aid of neural networks and a support vector machine. Das et al. [23] proposed a hybrid parametric and machine-learning model to price European index options with homogeneity, for example, categorizing option data based on moneyness and time-to-maturity.

Unlike traditional machine-learning algorithms, deep learning (DL) tends to become more accurate as data availability increases [24]. Given sufficient data with a distribution similar to that of out-of-sample data, a DL model can learn more features based on the training data. Furthermore, it is possible to use large and deep networks, which have a superior representation capacity, and thus learn high-level representations without being concerned about the overfitting problem. Neural network architectures and hyperparameters are critical because basic DL algorithms learn features by themselves from the training data. In addition, large and good-quality training data are essential when using DL. However, the availability of daily financial data is rather limited, and a relatively small amount of data is available compared with other fields.

Option data are more expensive and less accessible than other types of data, including other financial data [25]. Option data are mainly obtained using databases such as Bloomberg and Wind Database, but these databases are expensive for individuals to use. Even if individuals can buy option data separately without using a database, the longer the period to acquire more data, the higher the purchase cost. In addition, owing to the liquidity of options, option data may be imbalanced with respect to moneyness, even if they are obtained at a high cost. Moneyness (i.e., in-the-money (ITM), at-the-money (ATM), or out-of-the-money (OTM) option) is an important issue in option pricing because it is the main indicator that helps option traders choose correct options based on specific situations. Imbalanced data also present a challenge in fields other than option pricing. A model trained using imbalanced data predominantly learns the patterns of classes that constitute the majority, whereas the patterns in small classes are insufficiently learned. Several methods have been developed to solve the problem presented by imbalanced data, including resampling, changes in evaluation metrics, and synthetic data augmentation. Synthetic data augmentation is one of the most efficient methods to tackle the imbalance problem. However, it is difficult to apply data augmentation to pattern-sensitive financial data because using the method incorrectly could cause more serious problems. Therefore, in an attempt to solve the problem of financial data scarcity, Baek and Kim [26] proposed ModAugNet, which enables feature augmentation such that the representation capacity of the DNN is sufficiently large and overfitting is prevented. Jeong and Kim [27] used pre-training methods with index component stocks to develop a trading system with a deep Q-network.

In the field of option pricing, previous approaches have employed moneyness to divide the data into sections to which different models can then be applied [19] or allow different modules to be used depending on the moneyness and maturity [18]. Other possible approaches involve the use of only a fraction of the data, OTM [17], or the use of data simulated based on the BS model [28]. However, methods applicable to all situations are more useful when applied as a single model rather than as a model that can be used in specific situations. This is because training a single model is more cost-effective. In addition, the division of data or the use of only a portion of the data is inappropriate in DL, where the amount of data is important. Therefore, models that use only data simulated by the BS model cannot be realistically applied to the pricing of real market data because the pricing obtained with this model are not close to real market prices as a result of the numerous assumptions made.

The motivation for our research is as follows. First, to improve the pricing performance over that of parametric methods and other machine learning methods, we consider the application of DL in the area of option pricing. Second, in situations in which the real market data are either insufficient or imbalanced, the use of a single model rather than multiple models will be helpful in reducing the computational cost and amount of training required. Third, combining various existing option pricing methods in the proposed framework and fusing the data they generate will enable the knowledge acquired by each method to be mimicked; this will both increase the data and ensure a more balanced dataset. Specifically, the distilled data will allow the DNN model to mimic the knowledge acquired by various parametric option pricing methods such as BS, BI, MC, and FDM. In other words, using supervised learning, the proposed framework will be able to imitate a function with fused input and output information from various traditional option pricing models. Thus, transferring knowledge from these various option pricing models can improve real option forecasts.

In addition, to estimate the option prices for all types of moneyness using a single model, the generation of distilled data enables us to create balanced data with respect to moneyness. This solves the problem caused by the imbalanced nature of real data. Furthermore, learning the outputs of diverse models simultaneously has a regularization effect, which improves the prediction performance. Ultimately, we developed a combined delta-hedging module and an option pricing module based on DL because delta can be used to measure the major risks associated with the options. In particular, delta is the ratio comparing the change in price of the underlying asset with the corresponding change in the option price, which is the first derivative of the option price with respect to the price of the underlying asset. Therefore, our delta-hedging model estimates differentiation through the use of a DNN.

In this study, a novel option pricing and delta-hedging framework called DeepOption is proposed, which is applicable to all situations with respect to moneyness. This framework consists of an option pricing module and a delta module, and both modules are designed using a DNN. We use real market option data and simulated big data. The latter are generated using four parametric option valuation methods—BS, BI, MC, and FDM—and are subsequently fused. We refer to these simulated big data as (big) distilled data. DeepOption is a three-stage training model that not only improves real option price forecasts but also overcomes the problem of imbalanced data. In addition to estimating option prices, DeepOption estimates the value of delta for hedging purposes. Specifically, the first stage of the proposed framework entails pre-training the DNN model for option pricing using the distilled data. This pre-training stage with the distilled big data is necessary to help the DNN improve the representation capacity. The second stage of DeepOption is the main training stage of the option pricing module. We fine-tune the model using real market option data to learn the patterns of real option prices. Since the amount of distilled data is about tens to hundreds of times the amount of real market option data, the process of training DeepOption will be less successful if we were to use these two types of data together. The final training stage of DeepOption is to train the delta module. Specifically, the delta module is trained to estimate delta by using the estimated option price, which is the output of the option pricing module, and the stock price as inputs. In addition, this module can learn gamma for the delta-gamma hedging strategy, and can be applied to other Greek estimation modules by changing the input variables. This is explained in Section 3.2.2.

We evaluated our model by analyzing its performance with respect to pricing and hedging. Empirical tests using out-of-sample data were conducted on S&P 500 call options, EuroStoxx50 call options, and Hang Seng Index put options, which are globally important benchmark indicators. We calculated the mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) to measure the pricing accuracy of our proposed method and compared it with those of other approaches. We verified the effectiveness of our model in terms of pricing performance in three different ways. First, we compared the error in the option price estimated by the models using various combinations of four types of distilled data to determine the best combinations. Second, we compared the proposed method with a baseline DNN model that is not pre-trained with distilled data and has the same architecture to verify the effect of distilled data. Finally, we compared the performance of our proposed method with those of the parametric methods and other machine learning methods. We evaluated the hedging performance using delta hedging, which is one of the most common hedging methods. The hedging performance was evaluated by calculating the tracking error, which was then compared with those of other parametric methods. In addition, we conducted delta-gamma hedging to confirm that our proposed framework is applicable to other Greeks.

The remainder of this paper is organized as follows. Section 2 discusses the related methods used in this study. Section 3 describes the data and our proposed methodology. Section 4 compares the performance of the proposed method with those of other approaches. Finally, Section 5 provides some concluding remarks and directions for future research.