Edited by: Jochen Papenbrock, Independent Researcher, Frankfurt, Germany
Reviewed by: Francesco Caravelli, Los Alamos National Laboratory (DOE), United States; Emanuela Raffinetti, University of Milan, Italy
This article was submitted to Artificial Intelligence in Finance, a section of the journal Frontiers in Artificial Intelligence
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Despite the current growing interest in Bitcoins—and cryptocurrencies in general—financial instruments, as well as studies related to them, are quite underdeveloped. Therefore, this article aims to provide a suitable pricing model for options written on this peculiar underlying. This is done through an artificial neural network approach, where classical pricing models—namely the trinomial tree, Monte Carlo simulation, and explicit finite difference method—are used as input layers. Results show that options written on Bitcoin turn out to be systematically overpriced when considering classical methods, whereas a noticeable improvement in price predictions is achieved by means of the proposed neural network model.
Stock options are a category of financial derivatives which became widely employed by investors and speculators during the last few decades. Nevertheless, investors may ineffectively manage their portfolios if they are not able to value options in a proper way. For this reason, a reliable methodology capable to yield an option's current price or forecast is fundamental for investors in order to produce a rigorous decision making. This is particularly true when considering non-mature and volatile markets like the cryptocurrency one.
The theory of option pricing is broad and involves various types of pricing techniques, largely parametric ones. The most widely known option pricing method is arguably the one defined by Black and Scholes (
Besides the Black-Scholes model and its modifications, other parametric models have been developed and became widely used, among which the (binomial and trinomial) tree models, the finite difference method and the Monte Carlo simulation. While tree models converge to the Black-Scholes one in case the time occurring between steps is small enough, other methodologies take into consideration pricing aspects that these two models do not. Indeed, the Monte Carlo simulation allows for random shocks other than those provided by the volatility and the movement probabilities of the tree models, whereas the finite difference method relies on a different simulation scheme. This is the reason why in this paper examines and includes tree models, the Monte Carlo simulation, and the finite difference method as pricing methodologies.
Alongside the category of classical derivative and option pricing models, non-parametric models, such as neural networks gradually emerged, mainly thanks to their improved predictive performance with respect to the former techniques. Yao et al. (
Research related to the cryptocurrency market, as the phenomenon itself, is relatively new. Despite that, there is a massive interest of the academic community in investigating this new market and its peculiar features from all points of view, with a particular focus on Bitcoin. Indeed, since Nakamoto (
Despite the quite wide set of studies in the cryptocurrency area, to the best of our knowledge there is not yet any research trying to address option pricing related to Bitcoin (or cryptocurrency) derivatives. The aim of this study is to propose a pricing methodology that is feasible to price cryptocurrency options. Without loss of generality, the paper focuses on european style Bitcoin put and call options which became recently available on the market. To this end, the study makes use of a two stage approach. The first stage consists of option pricing through parametric approaches, such as tree models, finite difference method, and Monte Carlo simulation. In the second stage, artificial neural networks are employed in order to combine the parametric option pricing approaches and capture the residual errors by learning schemes in the current status of the option market. Their performance is then compared to the conventional option pricing techniques obtained in the first stage. Results point to the predominance of the neural network models with respect to the conventional methods in pricing Bitcoin options and, therefore, in capturing their real price dynamics. As a robustness check, an out-of-sample analysis confirm the previous result, as well as a cross validation analysis through random sub-sampling reveals that—despite there is still some room for improvement—results are arguably stable and the neural network is a suitable model in order to price options written on Bitcoin.
The remainder of the paper proceeds as follows. Section 2 outlines the methodology employed. Section 3 describes and analyzes the data. Section 4 presents the results. Section 5 illustrates the robustness analysis conducted. Section 6 concludes.
This section briefly introduces the classical parametric option pricing techniques used in this paper: specifically, tree models, finite difference method, and Monte Carlo simulation. After that, I discuss the neural network model and the comprehensive approach for option pricing.
The following notation will be used.
Tree models are widely used not only to price European style options, but also closed-form American options, as they can account for the early exercise feature. Milestone references for binomial trees are the ones of Cox et al. (
In the binomial tree setup, the underlying asset price
where
A graphical representation of a
where
The trinomial tree (
In this case, the probabilities of up (
Among the advantages of using the trinomial trees, computational efficiency as well as precision are of our interest. Indeed, the trinomial tree should yield to more precise prices with less time steps if compared to the binomial counterpart.
Binomial tree.
Trinomial tree.
As extensively described in Brennan and Schwartz (
According to the finite difference method, the time to maturity
In the present case, the application uses the so called explicit finite difference method, which solves the differential equations in a forward way, as elucidated by Hull and White (
Where
The option price can then be derived as:
where the probabilities associated with an up, middle or down movement are respectively:
For a detailed explanation of the finite difference method, refer to Brennan and Schwartz (
The Monte Carlo simulation is used to obtain the underlying asset price at the option maturity by means of averaging a sufficiently high number of stochastic asset price paths, obtained by assuming that the underlying price follows a log-normal distribution, that is simulating
where
After that, option prices are found by discounting that average result backwards. In other words, given the payoffs at maturity
the resulting call and put prices are obtained as an average of the
where
Option prices dynamics depend on several variables as well as on an economic environment and rules that continuously change. Despite parametric methods mimic the behavior of real option prices, it may be argued that they do not fully reflect the actual market evolution of option prices.
To cope with that, similarly to Liang et al. (
The multilayer perceptron neural network model. The following notation is used: NN stands for the neural network model, TT corresponds to the trinomial tree, FDM represents the finite difference method, and MC for the Monte Carlo simulation.
It is well-known that the option market is a complex system with non-linear characteristics. This further motivates our approach, since the use of a particular kind of neural network model, the multilayer perceptron one, allows to account for these features. Indeed, through the multilayer perceptron neural network one is able to include include hidden layers and non-linear activation functions that may capture the non-linearity of the option market. An organic description of multilayer perceptron neural networks can be found, for example, in Haykin et al. (
In this subsection the the assessment criteria used to evaluate our models are presented. Performances of our pricing methods are judged according to three widely employed measures, i.e., the mean absolute error (MAE), mean squared error (MSE), and the mean absolute percentage error (MAPE). These criteria are defined by
where
An option market for cryptocurrencies—and Bitcoin—is gradually emerging. I analyze data from deribit.com, a platform offering trading of futures and European style options written on Bitcoin. In particular, the corresponding underlying on which the options are written consists of the deribit BTC index
Data are collected from 16 May 2018 to 15 July 2018, on a daily basis, every day at the same time (11:00 UTC). To be precise, the retrieved data are the deribit BTC index and all available option prices related to that day (European calls and puts).
Following Liang et al. (
Given the set of restrictions adopted above, the dataset ends up with a total number of 281 call and 695 put prices. In the current analysis, the first 10 weeks will be used for the estimation purposes, while the last 2 weeks will be used for out-of-sample performance assessment.
As far as the parameter specifications, a 15-days historical volatility for the deribit BTC index and the 2-months Libor interest rate as risk-free rate are used. Moreover, the finite difference method has a grid of size 3
The neural network involves several specifications, too. Firstly, the study relies on the widely spread backpropagation algorithm for the parameter estimation. Secondly, the most widely employed activation functions are tested in order to choose the one ensuring the best performance in terms of fitting
In this section results are presented distinguishing between call and put options.
Without loss of generality, a plot of a representative option price evolution against one of the parametric methods (the trinomial tree) prediction is shown in
Real option prices (black) against trinomial tree price predictions (red) for the option expiring on 29 June 2018,
Prediction errors associated with each category of options are illustrated in
In-sample performance of neural network and classical models.
MAPE | 0.0713 | 0.0713 | 0.0716 | 0.0670 |
MAE | 42.78 | 42.79 | 43.2 | 33.55 |
MSE | 5,362.41 | 5,362.65 | 5,401.13 | 1,926.66 |
MAPE | 0.0546 | 0.0547 | 0.0546 | 0.0506 |
MAE | 56.00 | 56.05 | 56.08 | 33.63 |
MSE | 4,764.71 | 4,764.81 | 4,765.29 | 2,299.11 |
In-sample performance of neural network and “best” classical model. The figure compares the in-sample performance of the neural network model (red) and “best” classical model (blue).
The obtained results are in accord with the existing literature on option pricing through non-parametric methods and, particularly, neural networks—see Hutchinson et al. (
With the aim of testing the robustness of our model, this section provides an out-of-sample performance analysis as well as a cross-validation analysis through repeated random sub-sampling.
The out-of-sample performance is tested on the options available on the deribit platform between 1 August 2018 and 15 August 2018. Options are selected according to the same criteria described in section 3. The final out-of-sample dataset consists of 29 call and 47 put option prices.
Results of the out-of-sample performance of the investigated models are illustrated in
Out-of-sample performance of neural network and classical models.
MAPE | 0.0429 | 0.0429 | 0.0425 | 0.0283 |
MAE | 26.64 | 26.65 | 26.77 | 17.93 |
MSE | 1,016.11 | 1,016.28 | 1,026.79 | 441.94 |
MAPE | 0.0642 | 0.0643 | 0.0642 | 0.035 |
MAE | 73.4 | 73.4 | 73.23 | 41.45 |
MSE | 6,668.17 | 6,667.56 | 6,646.12 | 2,978.26 |
As also depicted in
Out-of-sample performance of neural network and “best” classical model. The figure compares the out-of-sample performance of the neural network model (red) and “best” classical model (blue).
To further assess the robustness of our proposed model, the approach of repeated random sub-sampling for cross-validation purposes is adopted. In other words, the dataset is randomly split into training and validation set for 50 times and then the methodology and procedures described in this study are repeated. In this way one is able to determine whether the neural network performance achieved in the results section are stable, as well as to evaluate the model's relative performance after random sub-sampling with respect to the conventional option pricing methods.
Results linked to the random sub-sampling procedure are illustrated through the boxplots contained in
Model performance distribution (call).
Model performance distribution (put).
Furthermore, comparing the distributions of the assessment criteria with the results in
To conclude, there may be room for improvement in the modeling strategy, as well as this needs to be adapted to the specific case of interest. As an example, it can be argued that the neural network performances would benefit from increasing the number of observations and, specifically, by using high frequency data. In addition, as the market is highly volatile and the option market follows fast changing rules and patterns, different choices of the neural network specifications—different input layers, structure of the layers, activation functions, etc.,—may result more feasible in other contexts. Nevertheless, it may be claimed that the multilayer perceptron neural network model proposed is suitable for pricing options written on Bitcoin. Moreover, it may be argued that its application can be extended to the whole cryptocurrency framework, as well as to traditional markets.
This paper proposes an approach that relies on artificial neural network models for the purpose of Bitcoin option pricing. The methodology involves a first step in which options are priced according to some of the most widely employed parametric methodologies, i.e., tree models, Monte Carlo simulation, and finite difference method. The option prices obtained in this way are then used as input layers in a second step by the neural network, which is capable to refine the price predictions delivered by the parametric models in the first step. We believe that the proposed model can be extended, without loss of generality, to other cryptocurrency derivatives, as well as to traditional ones.
Empirical results show that the investigated conventional pricing methodologies yield to the conclusion that Bitcoin options are extensively overpriced. In contrast, by applying the proposed neural network model one is able to better represent the real market dynamics of Bitcoin option prices. Indeed, prediction errors consistently reduce when comparing the neural network pricing model to the classical parametric ones.
Further studies may benefit and improve prediction precision by using high frequency data as well as different model specifications. As an example, improvements could be achieved by the use of different models, such as stochastic volatility models, as input layers in the proposed neural network framework.
The author confirms being the sole contributor of this work and has approved it for publication.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
I heartily thank Matthias for the time he spent on the insightful discussions we had toward this common interesting topic. I also thank Dina for her great help in getting the dataset.
1“Altcoin” stands for “alternative coin.” The term is used to indicate all cryptocurrencies except for Bitcoin.
2Detailed information regarding the deribit BTC index can be found on
3In particular, the following activation functions are tested: sigmoid, taylor, identity, tanh, softplus, gauss.