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Pricing options and computing implied volatilities using neural networks

Shuaiqiang Liu, Cornelis W. Oosterlee, Sander M. Bohte

arXiv:1901.08943v2q-fin.CPcs.LGcs.NAmath.NA

TL;DR

Complex option-pricing models can require costly numerical computation, including for calibration and risk management. This paper trains ANNs on solver-generated data to approximate option prices and implied volatilities. The authors report that the ANN computes both efficiently, accurately, and robustly, with offline training followed by fast online prediction.

  • Problem

    Complex asset models often lack closed-form option values, while calibration can require thousands of prices and efficient computation is important for risk management.

    Source: §1 · p. 1

  • Method

    The paper trains ANNs on data generated by conventional financial models and numerical solvers to approximate option prices and implied volatilities.

    Source: §2.2 · p. 4, §5 · p. 19

  • Results

    The authors report that the ANN computes option prices and implied volatilities efficiently, accurately, and robustly.

    Source: §5 · p. 19

  • Takeaways & Limitations

    The approach separates longer offline training from fast online prediction, offering an approximation technique for more time-consuming asset-price computations.

    Source: §5 · p. 19

    The study focuses on European call options, while extension to American, Bermuda, or exotic options is presented as possible future work.

    Source: §5 · p. 19

Abstract

from arXiv · show

This paper proposes a data-driven approach, by means of an Artificial Neural Network (ANN), to value financial options and to calculate implied volatilities with the aim of accelerating the corresponding numerical methods. With ANNs being universal function approximators, this method trains an optimized ANN on a data set generated by a sophisticated financial model, and runs the trained ANN as an agent of the original solver in a fast and efficient way. We test this approach on three different types of solvers, including the analytic solution for the Black-Scholes equation, the COS method for the Heston stochastic volatility model and Brent's iterative root-finding method for the calculation of implied volatilities. The numerical results show that the ANN solver can reduce the computing time significantly.

1. Introduction

Complex asset models can make option pricing computationally demanding, especially when calibration requires thousands of prices. The paper trains ANNs on conventional solver outputs to approximate prices and implied volatility for faster online evaluation.

1. Introduction

  • Model calibration may require thousands of option prices, while real-time risk management makes efficient computation important.These demands can force a trade-off between computation time and accuracy.Source: §1 · p. 1
  • ANNs can approximate nonlinear functions,, and PDE solutions [4], [5], motivating their use as data-driven solver approximators.The paper also notes their potential for parallel evaluation, particularly on GPUs.Source: §1 · pp. 1–2
  • The proposed ANN learns option-pricing solver outputs from generated data, then approximates solutions online using parallelizable matrix operations.Training can be time-consuming but is performed offline.Source: §1 · p. 2
  • The approach targets both option-pricing PDE solutions and the implicit relation between variables and implied volatility.The paper distinguishes this model-based solver-learning direction from market-price regression approaches.Source: §1 · p. 2
  • The study evaluates Black-Scholes and Heston pricing and implied-volatility robustness, then reports numerical results and errors.The paper presents its ANN and hyper-parameters before reporting the numerical evaluation.Source: §1 · p. 2

2. Option pricing and asset models

The paper introduces Black-Scholes and Heston option models, defines implied volatility, and reviews numerical methods for pricing and root-finding. It then describes Brent’s method as a robust approach and uses it to compute Black-Scholes implied volatility for Heston prices.

2. Option pricing and asset models

  • European option pricing is considered under geometric Brownian motion and Heston stochastic volatility, with other option types noted as possible extensions.The Heston model can reproduce implied-volatility smiles or skews, but its additional stochastic process makes valuation more expensive than for a scalar asset-price process.Source: §2 · p. 2, §2.2 · p. 4

2.1. The Black-Scholes PDE

  • For European plain-vanilla options, the Black-Scholes PDE has an analytic solution, denoted BS(·).The PDE is accompanied by a final condition representing the option payoff.Source: §2.1 · p. 3

2.1.1. Implied volatility

  • Implied volatility is the volatility that makes the Black-Scholes value match an observed market option price, and the paper expresses it as a root-finding problem.The paper notes that an analytic solution for implied volatility does not exist.Source: §2.1.1 · p. 3

2.2. The Heston Model

  • Heston stochastic volatility can reproduce market volatility smiles or skews, but the model has no analytic solution and is solved numerically.The paper uses the COS method for Heston training data; it relies on the characteristic function, which may be unavailable for other modern asset models.Source: §2.2 · p. 4

2.3. Numerical methods for implied volatility

  • Newton-Raphson may fail when Vega is very small, particularly for deep in-the-money or out-of-the-money options.Vega is relatively large near at-the-money options; the paper identifies hybrid bracketing approaches as a robust alternative.Source: §2.3 · p. 5, Figure 1 · p. 5

2.3.1. Brent’s method for implied volatility

  • Brent’s method combines bisection, inverse quadratic interpolation, and the secant method, and is used here to compute Black-Scholes implied volatility for Heston prices.When two consecutive approximations coincide, the interpolation step switches to a secant-based approximation.Source: §2.3.1 · p. 6

3. Methodology

The methodology builds an ANN surrogate from solver-generated financial data, then trains and evaluates it before replacing the original solver. It specifies an MLP, tunes its hyperparameters, and selects learning-rate schedules.

3. Methodology

  • The workflow generates input-output data from a financial solver, splits it into training and test sets, trains and evaluates the ANN, then substitutes it for the original solver.The data generator supplies option prices or implied volatilities as outputs.Source: §3 · p. 6

3.1. Artificial neural network

  • An MLP maps inputs through connected layers of weighted neurons and biases; training minimizes approximation loss, with p=2 corresponding to mean squared error.The input and output layers represent x and y, respectively.Source: §3.1 · pp. 6–8

3.2. Hyper-parameters optimization

  • Random search with 3-fold validation selects initial configurations on a small data set, then the top five configurations are averaged and the final ANN is trained on combined training and validation data.Each initial model is trained for 200 epochs using MSE; the chosen architecture has four hidden layers, and batch normalization and dropout do not improve accuracy.Source: §3.2 · pp. 8–9

3.3. Learning rates

  • The loss sweep identifies 10^-5 to 10^-3 as the steepest reduction range; on Heston, DecayLR outperforms CLR at equal bounds, while CLR yields smaller training-validation loss differences.The authors use CLR to find the learning-rate range, then apply it in DecayLR training; the reported schedules show no over-fitting.Source: §3.3 · pp. 10–11, Figure 3 · p. 10, Figure 4 · p. 11

4. Numerical Results

The numerical evaluation covers ANN learning of Black–Scholes prices, Heston prices, and implied volatility from Brent’s method. It uses MSE to train the ANN and several metrics to evaluate it.

4. Numerical Results

  • The accuracy metrics provide the basis for ANN training and performance reporting.The section introduces the metrics used to assess the financial-model solvers.Source: §4 · pp. 11–12
  • MSE updates the ANN weights, while all listed accuracy metrics evaluate the selected network.The paper also reports other well-known metrics for completeness.Source: §4 · p. 12
  • The ANN learns Black–Scholes option prices, implied volatility from Brent’s method, and Heston prices from the COS method.These three targets correspond to a closed-form pricing solution, an iterative root-finding method, and a numerical method for the Heston model.Source: §4 · p. 12

4.1. Details of the data set

The data set pairs parameter samples with numerical option-price or implied-volatility results. Increasing training-set size improves prediction accuracy and reduces its variance, motivating a small-data search followed by training on more samples.

4.1. Details of the data set

  • Latin hypercube sampling generates parameter samples with better representation of the parameter space, and 90% of the data is assigned to training and 10% to testing.Black–Scholes prices come from the closed-form formula; Heston prices use the COS method, with Brent’s method then finding corresponding implied volatility.Source: §4.1 · p. 12
  • Table 3 lists the different training-data-set sizes used in the ANN experiments.The supplied caption identifies the table’s scope but does not specify its individual values.Source: Table 3 · p. 12
  • Increasing the training-set size improves prediction accuracy and reduces the variance across repeated runs.The implied-volatility example repeats training five times with different random seeds for each data size.Source: §4.1 · p. 12
  • The selected approach searches hyperparameters on a small data set, then trains the ANN on a larger one.Training and validation losses remain close as the data size increases, indicating no overfitting in this experiment.Source: §4.1 · p. 12

4.2. Black-Scholes model

For Black–Scholes pricing, the ANN takes scaled market parameters as input and predicts scaled option prices. Its errors are small on the wide test set and slightly improve on a narrower parameter range; the paper recommends training on a wider domain than the range of interest.

4.2. Black-Scholes model

  • The ANN receives {S0/K, τ, r, σ} and predicts the scaled option price V/K.The data set contains 1,000,000 random samples for European call options.Source: §4.2 · pp. 12–13
  • On the wide test set, RMSE is around 9 · 10−5, corresponding to an average pricing error of 0.009% of the strike price.The wide test set uses the same parameter ranges as the training set; the maximum absolute error is around 0.06%.Source: §4.2 · p. 13, Table 5 · p. 14
  • On the narrower test range, ANN test performance slightly improves and the largest deviation is less than 0.04%.There is no significant difference in R2 between the wide and narrow test cases.Source: §4.2 · pp. 13–14, Table 5 · p. 14
  • The paper recommends training on a slightly wider data set when the parameter range of interest is somewhat smaller.The wide and narrow tests distinguish same-range evaluation from evaluation on a narrower parameter range.Source: §4.2 · pp. 13–14, Figure 7 · p. 14

4.3. Implied volatility

This section develops an ANN approach to implied volatility where small Vega can create steep gradients, then evaluates prediction accuracy and runtime against iterative root-finding methods.

4.3. Implied volatility

  • The gradient-squash approach subtracts intrinsic value for ITM options and log-transforms the resulting time value to reduce gradient steepness.The transformed input is {log(Ṽ/K), S0/K, r, τ}, where Ṽ is option time value.Source: §4.3 · p. 14

4.3.1. Model performance

  • Scaling improved ANN performance significantly, and out-of-sample implied-volatility errors had a maximum deviation around 6·10^-4.Training data were generated with the Black–Scholes formula; volatility was the target, with option price and contract/model variables as inputs after a log transformation of time value.Source: Table 7 · p. 15, Figure 8 · p. 15, §4.3.1 · p. 15

4.3.2. Comparison of root-finding methods

  • Brent’s method was fastest among robust iterative methods requiring no domain knowledge to select an initial value; the ANN solver had averaged MAE ≈10^-4 and ran 100× faster on GPU and 10× faster on CPU than Newton–Raphson.Source: §4.3.2 · p. 16

4.4. The Heston stochastic volatility model

The section evaluates ANN approximations for Heston option prices and implied volatilities, including out-of-sample accuracy and generation of an implied-volatility surface.

4.4.1. Heston model for option prices

  • The Heston-ANN approximated option prices accurately despite having twice as many inputs as the Black–Scholes model.The model was trained for 3,000 epochs on around one million Latin-hypercube samples, split into 80% training, 10% validation, and 10% testing data.Source: §4.4.1 · p. 16, Table 10 · p. 17, Figure 9 · p. 17

4.4.2. Heston model and implied volatility

  • RMSE was ≈7·10^-4 for Heston-ANN plus IV-ANN over m∈[0.7,1.3] and τ∈[0.3,1.1]; shrinking parameter ranges improved fit, while domain boundaries were weaker.The IV-ANN takes Heston-ANN prices as input, so accumulated error slightly reduces overall accuracy.Source: Table 11 · p. 18, Figure 11 · p. 18, §4.4.2 · p. 18
  • For one Heston parameter setting, IV-ANN produced a volatility surface over m∈[0.7,1.3] and τ∈[0.5,1.0] with maximum deviation no more than 4·10^-4.Source: §4.4.2 · p. 18, Figure 12 · p. 19
  • At test time, the ANN approximates Heston option prices and implied volatilities without requiring the model’s characteristic function.Source: §4.4.2 · p. 19

5. Conclusions and discussion

The conclusion reports efficient, accurate ANN approximations for option prices and implied volatilities, while noting parameter-range effects and the European-call scope of the demonstrations.

5. Conclusions and discussion

  • Across the tested solvers, ANN approximations were efficient and accurate; in this case, batch processing delivered around 100× GPU speed over the original solver.The tests covered the Black–Scholes solution, the Heston COS method, and Brent’s implied-volatility root finder.Source: §5 · p. 19
  • Parameter boundaries affected ANN performance, and the authors recommend training on ranges somewhat wider than the values of interest.Source: §5 · p. 19
  • The demonstrations focus on European calls; pricing American, Bermuda, or exotic options is described as a possible extension.The authors also identify deeper or more complex networks as possible ways to improve accuracy, and shallower networks as a possible way to improve speed.Source: §5 · p. 19
  • The authors describe recovering option Greeks from the trained ANN, either from its gradient or with Auto-Differentiation.Source: §5 · pp. 19–20
  • Combining Heston-ANN and IV-ANN into one network is proposed as a way to make Heston implied-volatility-surface computation more efficient.Source: §5 · p. 20
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