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A neural network-based framework for financial model calibration
Shuaiqiang Liu, Anastasia Borovykh, Lech A. Grzelak, Cornelis W. Oosterlee
TL;DR
Financial model calibration must recover model parameters from market data despite nonconvex objectives that can have multiple local minima. The paper proposes CaNN, combining ANN-based calibration with parallel global optimization, and reports that optimal values can be found within a second. Its sensitivity analysis is preliminary, with more than two parameters requiring advanced tools for study.
Problem
Calibration must recover model parameters from market data, while nonconvex objectives can have multiple local minima.
Source: §1 · pp. 2–3Method
CaNN uses an ANN calibration framework with parallel global optimization, including Differential Evolution, to avoid reliance on specific initial values.
Source: §1 · p. 3, §3.4.2 · p. 13Results
Optimal values can be found within a second even when using a global optimization algorithm.
Source: §5 · p. 30Takeaways & Limitations
The framework is reported as robust for high-dimensional Heston and Bates calibration, with Differential Evolution preventing calibration from getting stuck in a local minimum without specific initial values.
Source: §5 · p. 30The sensitivity study is preliminary, and studying effects of more than two parameters requires advanced tools.
Source: §4.1.2 · p. 19
Abstract
from arXiv · showhide
A data-driven approach called CaNN (Calibration Neural Network) is proposed to calibrate financial asset price models using an Artificial Neural Network (ANN). Determining optimal values of the model parameters is formulated as training hidden neurons within a machine learning framework, based on available financial option prices. The framework consists of two parts: a forward pass in which we train the weights of the ANN off-line, valuing options under many different asset model parameter settings; and a backward pass, in which we evaluate the trained ANN-solver on-line, aiming to find the weights of the neurons in the input layer. The rapid on-line learning of implied volatility by ANNs, in combination with the use of an adapted parallel global optimization method, tackles the computation bottleneck and provides a fast and reliable technique for calibrating model parameters while avoiding, as much as possible, getting stuck in local minima. Numerical experiments confirm that this machine-learning framework can be employed to calibrate parameters of high-dimensional stochastic volatility models efficiently and accurately.
1 Introduction
Financial calibration infers model parameters from market data, but repeated pricing for high-dimensional models is computationally burdensome and non-convex objectives can have multiple local minima. The paper proposes CaNN, an ANN-based framework with parallel global search to address these challenges.
1 Introduction
- Calibration recovers stochastic differential equation parameters from liquid option data, supporting OTC derivative pricing, hedging, and risk management.The task may recur numerous times daily, and repeated model valuation is especially burdensome for multidimensional asset models.Source: §1 · p. 2
- CaNN trains hidden-layer parameters offline, freezes them to predict outputs, then learns the input layer from observed market quantities during calibration.The authors call the training and prediction stages the forward pass and the input-layer calibration stage the backward pass.Source: §1 · p. 2
- ANNs are presented as a way to accelerate pricing, while global optimization addresses calibration objectives that may have multiple local minima.The paper situates its ANN approach among work using neural networks to learn financial SDE solutions.Source: §1 · pp. 2–3
- The framework treats training, prediction, and calibration as one ANN system whose learnable units change by phase, and it accepts a flexible number of market inputs.The authors contrast this input flexibility with CNN approaches that require a fixed number of inputs.Source: §1 · p. 3
- CaNN combines an ANN calibration framework with parallel Differential Evolution, aiming for robust global search without specific initial values.The framework uses a unified computing platform for its components and tasks, apart from data generators.Source: §1 · p. 3
2 Financial Model Calibration
This section introduces Heston and Bates stochastic-volatility models and formulates calibration as minimizing discrepancies between observed market quantities and model outputs. It also describes numerical valuation methods and the nonlinear, potentially nonconvex nature of the optimization.
2 Financial Model Calibration
- Financial model calibration estimates model parameters from market samples by minimizing the discrepancy between observed and model-based quantities.The section introduces stochastic models, their option-valuation equations, and standard calibration requirements.Source: §2 · p. 4, §2.2 · pp. 5–6
2.1.1 The Heston model
- The Heston model uses a mean-reverting square-root variance process, with parameters for long-run variance, reversion speed, variance volatility, and initial variance.Its variance process is modeled to preclude negative variance values.Source: §2.1.1 · p. 4
2.1.2 The Bates model
- The Bates model extends Heston by adding jumps to the stock-price process, parameterized by jump intensity and normally distributed jump sizes.The jump process is assumed independent of the Brownian motions and jump sizes; option valuation uses a partial integro-differential equation.Source: §2.1.2 · p. 5
2.1 Asset pricing models
- Because Heston and Bates lack analytic option values, the paper uses the COS Fourier method for pricing and Brent’s method to recover implied volatilities.Fourier pricing requires the asset-price characteristic function.Source: §2.1.2 · p. 5
2.2 The calibration procedure
- Calibration minimizes weighted errors across available instruments, optionally with regularization, and the procedure may require thousands of option valuations.Equal weights, no regularization, and squared errors give the mean squared error; ANNs can value an array of options in parallel.Source: §2.2 · pp. 5–6
2.3 Choices within calibration
- Calibration objectives can be nonlinear and nonconvex, while correlated parameter effects can produce multiple local minima in high-dimensional problems.The section contrasts price-based and implied-volatility-based objectives; implied-volatility calibration requires option pricing and numerical root finding but is preferred by practitioners for specifying vanilla-option prices.Source: §2.3 · pp. 6–7
3 An ANNs-based approach to calibration
CaNN frames calibration as optimization and uses an ANN to approximate financial-model outputs, then fits model inputs to observed market data. Its forward and backward passes separate offline solver learning from online calibration, with parallel Differential Evolution used to address multiple local minima.
3 An ANNs-based approach to calibration
- CaNN treats calibration and ANN training as optimization problems, respectively fitting model parameters to market quantities and network weights to a user-defined loss.The paper identifies this correspondence as the inspiration for its machine-learning approach to calibration.Source: §3 · p. 8, §3.3 · p. 11
3.1 Artificial Neural Networks
- An ANN maps input-output pairs through neurons and layers, and its weights and biases are trained to minimize a loss function.The paper notes that its stated approximation assumption requires the training data to define the true function over the domain.Source: §3.1 · p. 8
3.2 The forward pass: learning the solution with ANNs
- For Heston implied volatility, the forward pass merges COS option pricing and Brent root-finding into one ANN mapping model parameters to implied volatility.A similar ANN is used for Bates model parameters.Source: §3.2 · p. 9
3.3 The backward pass: calibration using ANNs
- In the backward pass, the trained hidden layers stay fixed while the input layer becomes learnable, allowing market prices and implied volatilities to determine model parameters.The strike price is an input but is not estimated; training is offline and performed once, whereas calibration is online and computationally cheap.Source: §3.3 · pp. 11–12
3.4 Optimization
- The framework combines gradient-based and gradient-free calibration algorithms and permits any number of market quotes without requiring a fixed grid of inputs.Gradients of the calibration objective with respect to input parameters can provide search directions without external numerical techniques.Source: §3.3 · p. 12
3.4.1 Stochastic Gradient Descent
- SGD trains the ANN, but it is faster and may fail during calibration without a good initial guess.The ANN training loss is based on MSE, and mini-batches approximate gradients when memory is limited.Source: §3.4.1 · pp. 12–13
3.4.2 Differential Evolution
- Differential Evolution is a population-based, derivative-free optimizer that requires no specific initialization and is used to address nonconvex calibration objectives.Its described procedure comprises initialization, mutation, crossover, and selection.Source: §3.4 · p. 12, §3.4.2 · p. 13
3.4.3 Acceleration of calibration
- With a population around 50, parallel CaNN is at least 10 times faster than conventional CaNN on either a CPU or GPU.The ANN evaluates market samples in batches while parallel DE processes a generation of candidates together.Source: §3.4.3 · p. 15
4 Numerical results
The numerical-results section outlines experiments on Heston and Bates calibration with CaNN. It examines parameter effects, forward-pass sensitivity analysis, and backward-pass calibration performance.
Heston calibration
- The section begins with Heston calibration and discusses how Heston parameters affect implied volatility despite no explicit mapping between them.It then presents an ANN trained to map model parameters to implied volatilities.Source: §4 · p. 15
Forward pass
- The trained forward pass is also used as a tool for sensitivity analysis of model parameters.Source: §4 · p. 15
Backward pass and Bates calibration
- The section evaluates backward-pass Heston calibration and concludes with Bates-CaNN calibration for a model with more parameters than Heston.Source: §4 · p. 15
4.1 Parameter sensitivities for Heston model
The Heston sensitivity analysis examines how individual parameters shape implied volatility and how parameter interactions affect the calibration objective. It also identifies a limit of the two-parameter study: higher-dimensional objective surfaces are more complex and require more advanced sensitivity tools.
4.1.1 Effect of individual parameters
- Increasing γ makes the implied-volatility smile more pronounced when ρ = 0%, while increasingly negative ρ steepens the implied-volatility skew.κ has a limited effect on the smile or skew, up to 1%–2%, and determines the speed of convergence to long-term volatility.Source: §4.1.1 · p. 16, Figure 3 · p. 17
- κ and γ have similar effects on implied-volatility shape, so practitioners often fix κ = 0.5 and optimize γ, reducing calibration to four parameters.The analysis also notes that the number of Heston parameters calibrated depends on the scenario and that CaNN can handle varying numbers of parameters.Source: §4.1.1 · pp. 16, 18
- Short-maturity ATM implied volatility squared can provide an initial guess for ν0 as maturity approaches zero.The passage also describes projecting the variance process onto its expectation to relate Heston calibration to a time-dependent Black–Scholes model.Source: §4.1.1 · p. 17
4.1.2 Effect of two combined parameters
- The objective surface has a narrow valley in ν0 and a flat direction in κ, while κ and long-term variance also show flat minima with similar MSE values.For κ and γ, small κ with large γ can yield essentially the same option prices as large κ with small γ, producing multiple local minima.Source: §4.1.2 · pp. 18–19, Figure 4 · p. 18, Figure 5 · p. 19
- Higher-dimensional objective functions become more complex, and this preliminary two-parameter study calls for advanced tools to examine more than two parameters.The paper proposes extracting the Hessian of all five Heston parameters with the ANN to show broader dependencies and sensitivities.Source: §4.1.2 · p. 19
4.2 The forward pass
The forward pass trains an ANN to map Heston model and option-contract inputs to Black–Scholes implied volatility. It is trained on numerically priced option data, and the reported test performance is similar to training performance.
4.2 The forward pass
- The ANN uses four hidden layers of 200 neurons each and a linear regression output, balancing calibration speed and accuracy.The selected network has 122,601 trainable parameters and uses MSE as its training loss.Source: §4.2 · p. 19, Table 2 · p. 20, Figure 6 · p. 20
- Training data contain around one million randomly generated samples across six Heston parameters and two option-contract inputs, split into 80% training, 10% validation, and 10% testing.The samples are European put-option prices with fixed strike K = 1.Source: §4.2 · p. 20, Table 3 · p. 21
- Heston option prices are computed with a robust COS method, followed by Black–Scholes implied-volatility calculation using Brent’s method.The COS integration interval is enlarged automatically in extreme parameter cases when the default setting lacks precision.Source: §4.2 · p. 20
- The forward pass can handle both call and put implied volatilities without additional effort because identical Heston parameters yield the same implied volatilities.Option prices are intermediate variables during training.Source: §4.2 · pp. 20–21
- The trained forward-pass ANN learns the model-parameter-to-implied-volatility mapping, with test performance very similar to training performance.The paper reports that this indicates good generalization.Source: §4.2 · p. 21, Table 4 · p. 21
4.3 The backward pass
The backward pass uses a trained ANN to recover model parameters from implied-volatility quotes, then evaluates Heston calibration, parameter sensitivity, and calibration to Bates-generated quotes.
4.3.1 Calibration to Heston option quotes
- With κ and ν0 fixed, the three-parameter tests use a DE population of 45, and GPU runtime is around half the CPU runtime.Results average five runs per case; increasing the population ratio further showed no significant benefit.Source: §4.3.1 · p. 23, Table 7 · p. 24
- Five-parameter Heston-CaNN calibration finds parameter values in approximately 0.5 seconds on a GPU, using around 20,000 function evaluations.The DE population size is 50; the reported tests span 3,125 parameter combinations and five random seeds.Source: Table 8 · p. 24, §4.3.1 · p. 24
4.3.2 Sensitivity analysis based on ANNs
- The ANN-derived Hessian measures local loss curvature and parameter sensitivities, using the MSE on 35 market data points without regularization.The loss is for computing Black-Scholes implied volatility from Heston parameters.Source: §4.3.2 · p. 25
- At the optimum, the ν0-to-κ Hessian sensitivity ratio is around 323; κ’s flat direction permits multiple values with similar loss.The authors report a ν̄-to-κ ratio around 45, compared with ratios sometimes of order 10^6 when fitting option prices; implied-volatility calibration appears to reduce the sensitivity disparity.Source: §4.3.2 · p. 25
- ν0 is the most sensitive parameter and is predicted most precisely, while less-sensitive parameters allow a wide convergence area that regularization or averaging top DE candidates may address.The five-parameter error measure cannot fall significantly below 7.18 × 10^-8, close to the ANN testing MSE of 1.23 × 10^-7.Source: §4.3.2 · p. 26
4.3.3 Calibration to Bates quotes
- When Bates jumps are small, Heston calibration can mimic Bates implied volatilities, but deviations grow with jump intensity, especially for short maturities.Figure 8 compares Bates-generated and calibrated Heston implied volatilities.Source: §4.3.3 · p. 27, Figure 8 · p. 28
- Weighting ATM options improves their fit, but increases total error; the authors note Heston’s difficulty fitting short-maturity market implied volatility.The third Figure 8 plot and Table 10 compare ATM-weighted and equally weighted calibration.Source: §4.3.3 · p. 28
4.4 Calibrating the Bates model
The Bates-CaNN calibrates eight Bates parameters from synthetic quotes, including more complex implied-volatility surfaces that the Heston model cannot capture.
4.4 Calibrating the Bates model
- The Bates-IV-ANN forward pass is trained on one million Bates-generated samples, varying three more model parameters than the Heston setup.All calibration samples have equal weight, with λ̄ = 1.0 × 10^-6.Source: §4.4 · p. 29
- Bates-CaNN calibrates eight parameters, with DE population size 80, on 35 synthetic Bates market samples.The example uses random seed 2 and addresses a challenging case with high jump intensity and large variance jumps.Source: Table 11 · p. 29, §4.4 · p. 29
- The calibrated Bates implied volatilities closely resemble the synthetic market volatilities, including high curvature at short maturities.Figure 9 shows the comparison for equal weights and λ̄ = 1.0 × 10^-6.Source: §4.4 · p. 29, Figure 9 · p. 30
5 Conclusion
The authors conclude that ANN-based calibration combined with differential evolution (DE) is efficient and accurate for financial models, including high-dimensional Heston and Bates cases. They attribute speed to parallel computation and use the trained ANN’s Hessian for calibration insight.
5 Conclusion
- DE global optimization avoids getting stuck in a local minimum without requiring specific initial values.Source: §5 · p. 30
- Numerical experiments find optimal values within one second even with a global optimizer, with ANN evaluation and parallel DE population processing contributing to speed.The framework computes market data simultaneously and evaluates a whole DE population in parallel during selection.Source: §5 · p. 30
- The ANN-derived Hessian supports sensitivity analysis for many model parameters and explains the optimization preference for implied volatility over option prices.Source: §5 · p. 30
- The framework does not require characteristic functions or explicit model gradients, and the amount of market data and parameters to calibrate is flexible.Source: §5 · pp. 30–31
- Future work could use recent calibrated values as starting points when markets change little over short periods, or combine gradient-based optimization with DE.The authors also suggest narrowing DE’s search space or using a smaller neural network to reduce computation time.Source: §5 · p. 31
A COS Pricing Method
The COS method approximates European option values by representing the underlying density with a truncated Fourier cosine expansion based on its characteristic function. Its stated convergence condition concerns smooth conditional densities.
A COS Pricing Method
- The risk-neutral valuation formula gives the option value, and truncating the integration range enables the COS approximation.The density is approximated using the characteristic function and a truncated Fourier cosine expansion.Source: A COS Pricing Method · p. 34
- The COS algorithm approximates European option values by substituting the cosine-series density into the valuation formula and interchanging integration and summation.The payoff’s Fourier cosine coefficient is available in closed form for several European option payoff functions.Source: A COS Pricing Method · p. 34
- The COS algorithm has exponential convergence when the conditional density is infinitely differentiable on the stated interval, whose size can be determined using cumulants.Formula (23) also forms the basis for pricing Bermudan options.Source: A COS Pricing Method · p. 34