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Formulation of a Rational Option Pricing Model Using Artificial Neural Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 December 2020 doi:10.20944/preprints202012.0426.v1 Formulation Of A Rational Option Pricing Model using Artificial Neural Networks Kaustubh Yadav School Of Computer Science and Engineering (Vellore Institute Of Technology) Vellore, Tamil Nadu, India [email protected] Abstract—This paper inquires on the options pricing modeling Subsequent approaches in improving on the Black-Scholes using Artificial Neural Networks to price Apple(AAPL) European Model, included modelling volatility as a stochastic process Call Options. Our model is based on the premise that Artificial [2], [4], [5], [6], [3]. Though these modes are an improvement Neural Networks can be used as functional approximators and can be used as an alternative to the numerical methods to some on the Black-Scholes Model in the sense that they use stochas- extent, for a faster and an efficient solution. This paper provides tic volatility to recreate practical conditions, these models a neural network solution for two financial models, the Black- still fail to capture deep out-of-money options. Also with the Scholes-Merton model, and the calibrated-Heston Stochastic introduction of stochastic volatility, there are two stochastic Volatility Model, we evaluate our predictions using the existing processes which increases the computational costs, and these numerical solutions for the same, the analytic solution for the Black-Scholes equation, COS-Model for Heston’s Stochastic models don’t perform exceptionally better than Black-Scholes Volatility Model and Standard Heston-Quasi analytic formula. hence Black-Scholes is still widely used. The research for The aim of this study is to find a viable time-efficient alternative finding a better solution brought non-parametric and semi- to existing quantitative models for option pricing. parametric models to light. These models are essentially Index Terms—Black Scholes Equation, Heston Model Cali- “model-free” and take a tangent from the theory of finance bration, Option Pricing, Stochastic Processes, Artificial Neural and rely explicitly on data. Hence an ANN approach can be Networks. considered as a non-parametric solution. As non-parametric models rely on historical data and statistical approaches they I. INTRODUCTION suffer from poor-generality and as they discard the theory of finance to some extent they cannot be considered as Pricing financial derivatives has always been a very centric an empirical. This paper aims to provide a semi-parametric problem in the field of quantitative finance. Option contracts solution that doesn’t solely rely on historical data but stays have seen tremendous growth after the crisis of 2008 mainly true to the theory of finance and option pricing. because options are considered as leverage and the party II. RELATED WORK possessing can avoid risks of financial volatility by exercising these options. Options contract can also be considered as a A. Quantitative Methods “zero-sum game”, where if the holder of an option is benefited, With the introduction of the Black-Scholes equation, the that directly correlates to the loss of the other party and vice academic research sought to find out new models that relax the versa, this can be considered as an incentive for investing in restrictive assumptions made by the Black-Scholes Model. The an options contract. Pricing of Options took its first leap with first was making the volatility in the model to be a stochastic the works of Black Scholes and Metron in 1973 [1], where process rather than a constant, this implied the replication both studies gave a model that could roughly estimate the of the volatility smile curve as seen in the practical market price of a European Option, yet the Black-Scholes formula conditions, which were introduced through the contributions still remains one of the most used tools for pricing options. of the Heston Model[6]. Other models, also known as Levy Even with the success of the Black-Scholes and Merton Models, used Levy Process to model the volatility instead of model, there were substantial flaws, flaws that make the model a Weiner Process, through which the volatility curve could more theoretical rather than practical. The volatility for the capture the rare jumps in the underlying, these are also called Black-Scholes model was assumed to be constant, which in jump-diffusion models [7], [8], [9]. practicality is impossible, the volatility can itself be considered As mentioned earlier with the increasing complexity of the as a stochastic process, the Black-Scholes model also assumed model, it becomes harder to model the underlying PDE. To that the returns on an asset are normally distributed, which model such PDE, a catalog of numerical methods have been implies that the stock prices follow a Geometrical Brownian used, eg. Spectral Method, Finite Differences, Monte Carlo Motion(GBM) ignoring the large changes in prices [2] which Simulations, Lapace-Carson transform[10], etc. Some models can also be attributed to the models’ inability to price deep including Heston Models have parameters that need to be out-of-money options [3]. tuned with the market data, these calibrations can be done © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 17 December 2020 doi:10.20944/preprints202012.0426.v1 either by historical data and or on options that are heavily European Option can only be exercised on the date of expiry. traded. Hence calibration of parameters can be considered Options further are of two types, Call and Put. A call option as a multidimensional optimization process, so as to reduce is a contract that allows the holder to buy the underlying the distance between the market price and the model output. asset at a fixed price point on(European) or before(American) But calibrating over a huge amount of data increases the the expiry. Whereas a put option is a contract that allows complexity and is computationally inefficient, hence many of the holder to sell the underlying asset at a fixed price point the stochastic volatility models are not as mainstream. on(European) or before(American) the expiry. The underlying in an option contract can be stock, bond, commodity, etc but B. Option Pricing And Neural Networks in a generic portfolio, it is usually a stock. Hence options are Artificial Neural Networks are the building blocks of Deep also called Financial Derivatives as their own value depends Learning, introduced in 1943 by McCullouch as a single on an underlying asset. A stock option can be compared to a neuron [11], since then ANN’s have been used in many stock warrant as well, but the key difference is, an option areas such as image classification, recommender systems, etc. is a contract between two investors whereas a warrant is One of the utilities of an ANN is functional approximation a contract between the company that issues stock and the for non-linear equations and PDEs [12], [13]. The premise investor. Another key difference is that the warrants are put behind a functional approximator is that if the model needs to out by the company hence the aggregate amount of warrants approximate a function y = F (x), the feed-forward requires in a market is positive, whereas options exist on an exchange the model inputs to be multiplied with the weights of the hence the aggregate amount of options remains zero [16]. neurons, then the weighted inputs pass through a transfer As the subject of this study is European Call Options, function and finally pass through an activation function, which we shall look into the mathematical formalization of the in turn yields the first approximate, the result is then compared aforementioned statements. Let the price of an underlying(S) with the ground truth, and simultaneously the weights of the be $100 we represent the present day as t = 0 and the expiry model are modified till the difference between the ground of the option be in 10 days(T = 10), hence time to maturity is truth and models’ prediction is minimal. Hence, a functional (τ = T − t = 10). Let the exercise price for the option or the approximation is itself an optimization problem, if the model Strike Price(K) be $120 if the price of the underlying becomes output is represented by y = FANN (x; w), that is as a function $150, the holder has the incentive to exercise the option as he fo weights and the input the optimization can be stated as, can buy the stock for $120 and has an opportunity to sell it at P argminw( i jjF (x) − FANN (xi)jj). a cost of $150, hence making a profit of $40. But if the price This can be applied to Option Pricing as well, by providing of the underlying drops to $80, the holder won’t exercise the appropriate model parameters. Although these methods are call option. Hence the payoff for a call option can be given an example of data-driven approaches in finding a solution, by: they do suffer from loss of generality if the training size isn’t diverse, also being completely data-driven they don’t conform max(0;ST − K) (1) to the empiricals of Option Pricing Theory itself. Even with these flaws, neural networks have on occasion outperformed To quantify the worth of an option, the goal is to find long-standing financial models [14], [15]. the price of the option at t = 0. This can be computed by The aim of this study is to provide an accurate yet time- discounting if we know the price of the underlying at T , Z 1 efficient solution for pricing a European Call Option under 0 −rτ both constant volatility (Black-Scholes) and stochastic volatil- E[C (S; K; τ)] = e max(0;ST − K)f(ST jSt;T )dST 0 ity(Heston’s Method) using ANN’s but enforcing the rational- (2) ity conditions laid out in [16] and [17] for a “rational” option Here C’(.) is the price of an option at t = 0, r is the risk-free pricing model which makes this study novel compared to rate or risk associated with a 3-month government treasury existing models.
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