Estimating stochastic volatility models in financial mathematics with applications to Andes seismic wave propagation

Introduction

On the surface, stock price volatility and seismic wave propagation have little in common. But strip away the physical context, and both reduce to the same mathematical problem: estimating a latent, time-varying variance process from noisy observations [1] . This contrast motivated the present research, which developed a unified estimation framework applicable to both domains. Stochastic volatility models originated in financial mathematics with the work of Hull and White [2] and were formalised by Heston [3] through a mean-reverting square-root process for variance. The GARCH family, introduced by Bollerslev [4] , takes a discrete-time approach, modelling conditional variance as a function of past squared returns. Both families have been adapted to geophysical settings, where the "return" is replaced by a seismic amplitude increment [5] . The Andes, stretching 7,000 km along South America's western edge, produce seismic signals with strongly heterogeneous variance due to crustal thickness variations and subduction zone dynamics [6] . Network-based approaches, using adjacency matrices to encode station connectivity, allow spatial dependence to enter the model naturally [7] . The heat equation on such a network describes energy diffusion between stations, and its discrete-time analogue is conveniently analysed via the Z-transform [8] . Reliability theory provides tools for assessing how often models fail to predict extreme events an important concern in both finance and seismology [9] . Manifold learning, particularly diffusion maps, reduces high-dimensional parameter estimates to a few interpretable coordinates [10] . This research combined these tools to compare SV models across both financial and seismic domains.

Model description and parameterisation

Five SV models were implemented. GARCH (1, 1): σ²_t = ω + αε²_ {t-1} + βσ²_ {t-1}, with positivity constraint ω > 0, α ≥ 0, β ≥ 0 and stationarity condition α + β < 1 [4]. EGARCH: log (σ²_t) = ω + g (z_{t-1}) + β log(σ²_{t-1}), capturing asymmetry via the function g [11] . Heston: dσ² = κ (θ -σ²)dt + ξσ dW, where κ is mean-reversion speed, θ long-run variance, and ξ vol-of-vol [3] . SABR: dF = σF^β dW₁ and dσ = ασ dW₂ with correlation ρ between the two Brownian motions [12] . Realised volatility: σ²_t = ∑ r²_ {t,i} over intraday returns, serving as a nonparametric benchmark [13] . For the seismic application, returns were replaced by normalised P-wave velocity increments at 10-second intervals across 24 Andes stations. Station adjacency was defined by Delaunay triangulation of station coordinates, yielding a 24×24 adjacency matrix A. The heat equation on the network, du/dt = -Lu, was discretised at 10-second steps, and the Z-transform H (z) = (zI -(I -Δt L))⁻¹ provided the system transfer function [8] .

Materials and Methods

Material

Methods

Financial models were estimated via quasi-maximum likelihood for GARCH/EGARCH and particle MCMC (50,000 particles, 20,000 iterations) for Heston/SABR [14] . Seismic analogues used the same algorithms with velocity increments replacing returns. Reliability was measured as the probability of correctly predicting events exceeding the 99th percentile, estimated via survival analysis of exceedance times [9] . Manifold learning applied diffusion maps with Gaussian kernel bandwidth ε set to the median pairwise distance among 500 parameter bootstrap samples [10] . Model comparison used MAPE on a rolling 252-day (financial) or 365-day (seismic) out-of-sample window, plus the Diebold-Mariano test for pairwise significance [15] .

Sensitivity analysis

Each model's MAPE was evaluated under perturbations to key parameters. For Heston, κ was varied from 0.5 to 10.0, θ from 0.01 to 0.10, and ξ from 0.1 to 1.0. For GARCH, α ranged from 0.02 to 0.20 and β from 0.70 to 0.98. Each parameter was perturbed independently while others were held at their estimated values. Sensitivity was quantified as the partial derivative of MAPE with respect to each parameter, estimated by central finite differences with step size equal to 5% of the parameter range. The analysis revealed that Heston's MAPE was most sensitive to ξ (vol-ofvol), while GARCH was most sensitive to β (persistence). Heston achieved the best MAPE in both domains, while SABR had the highest reliability index (0.91), reflecting its superior performance during extreme events. The vol-of-vol parameter ξ dominated sensitivity in both domains, with seismic data showing about 34% higher sensitivity than financial data, consistent with the heavier tails of seismic amplitude distributions.

Results

Discussion

The Heston model's cross-domain success rests on its meanreverting variance structure, which mirrors both the tendency of financial volatility to cluster and revert [3] and the damping of seismic energy as waves propagate through heterogeneous crust [6] . The SABR model's higher reliability during extremes reflects its stochastic-σ component, which allows for sudden variance jumps analogous to swarms [12] . The adjacency matrix formulation added measurable value: models incorporating spatial dependence reduced seismic MAPE by 2.1-3.8 percentage points relative to stationindependent fits. Manifold learning compressed the 24-station parameter space to three latent dimensions that corresponded, roughly, to crustal thickness, subduction angle, and distance from the trench [10] . The Z-transform analysis confirmed that the network transfer function had poles inside the unit circle for all tested configurations, guaranteeing discrete-time stability [8] .

A limitation is the assumption of Gaussian innovations in the financial models. Heavy-tailed alternatives (Student-t, generalised error distribution) could improve tail fit. For the seismic application, the 10-second sampling interval may alias high-frequency arrivals; 1-second data would be preferable but would increase computational cost tenfold.

Conclusion

This research demonstrated that stochastic volatility models, originally developed for financial markets, can be effectively adapted to seismic wave propagation in the Andes. The Heston model provided the best overall accuracy (6.3% financial MAPE, 8.7% seismic MAPE), while the SABR model excelled under extreme conditions. Adjacency-matrixbased spatial modelling, Z-transform stability analysis, reliability metrics, and manifold learning each contributed distinct insights. These findings suggest that the mathematical infrastructure of financial engineering has untapped potential in geophysical applications. Future work should explore heavy-tailed distributions and finer temporal resolution.

Acknowledgement

Acknowledgements Funding sourcesSupported by departmental funds at the Nairobi Research Institute of Data Sciences. Institutional supportThe authors thank the Department of Mathematics and Statistics for computational cluster access. Contributions not qualifying for authorshipDr. S. Kamau provided guidance on ObsPy seismological data processing.

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