Enhancing Options Pricing Techniques via Deep Learning

Abstract

This thesis develops a density first framework for option valuation that separates learning of the transition densities from contract specific pricing. Neural networks are trained to approximate the cumulative distribution functions of the terminal state over a parameter domain by solving the backward Kolmogorov problem. Numerical differentiation of the learned cumulative distribution function produces a transition probability density that is inserted into a high order quadrature engine. The same trained network then supports many payoffs without recalibration. This design keeps the model-specific work in the learning stage and reuses a single deterministic pricer for valuation. The approach is instantiated with an exponentially weighted neural network that introduces a logarithm and exponential pathway with bounded activations and soft step units aligned with the qualitative shape of cumulative distribution functions. The network promotes stable training and reliable recovery of the transition density by differentiation. The architecture and its properties are presented in detail, together with the precise block equations used in experiments.

The first empirical study benchmarks four neural network architectures for transition densities that are later used for pricing. Deep Galerkin, feed forward, convolutional, and long short term memory networks are trained on the backward cumulative distribution function formulation. Evaluation covers moment diagnostics, total variation, Kullback-Leibler divergence, Wasserstein distance, tail mass discrepancies, entropy differences, cumulative distribution function error, transition density accuracy via differentiation, loss convergence, and pricing error. A comparison under both geometric Brownian motion and Heston dynamics is reported, followed by a synthesis that relates price errors to the underlying distribution metrics. This chapter provides an implementation guide for resource constrained users and motivates the search for architectures that retain accuracy while improving efficiency.

The second study introduces the exponentially weighted network and evaluates it against the Deep Galerkin baseline across geometric Brownian motion, Heston, and SABR. For SABR the exponentially weighted network matches the benchmark implied volatility smile across all strikes while the Deep Galerkin curve sits below the benchmark through the entire strike range. For Heston the exponentially weighted network produces smoother tails in the marginal density, tight option prices across strikes, and implied volatilities close to the reference. The Deep Galerkin model captures the qualitative shape but exhibits tail ripples, a systematic underpricing in calls and puts, and an implied volatility surface that is uniformly low in the reported configuration. These diagnostics favour the exponentially weighted design in these setups.

The final study unifies the pieces into a universal pricing pipeline. A cumulative distribution function network is trained for geometric Brownian motion on two billion training points on a T4 GPU. The trained network is differentiated to obtain the transition density and is then coupled to Simpson quadrature with the usual backward induction where needed. The same pipeline is repeated for Heston in log variance coordinates with the corresponding loss, training set, and boundary conditions.

Across geometric Brownian motion, the pipeline reproduces benchmark European call and put prices with absolute errors O(10^-4) over wide strike, spot, and volatility ranges, and the grid preserves a no arbitrage structure. Up and out calls show absolute errors of O(10^-3) and percentage errors near one percent, with errors rising in the number of monitoring dates and falling in volatility because higher volatility prunes the effective integration domain by additional knock outs, not to mention difficulties with approximating low sigma densities. Bermudan puts show absolute errors that cluster near O(10^(-3) with percentage errors typically below one percent, with peaks in the vicinity of the early exercise frontier. American puts follow the same shapes with moderately larger magnitudes that are consistent with the continuous exercise limit and are computed through Richardson extrapolation from discretely monitored values.

Under Heston, European puts match reference prices closely with mean percentage error near 0.44% and largest absolute deviation near 6.3×(10^-4) in the at the money region. Absolute errors in spot slices peak near at the money and decline in the tails, with percentage errors ordered by moneyness. Varying the initial variance produces nearly linear growth of absolute error for in the money slices, a bowl shaped profile at the money, and a decline for out of the money slices after a small bump at very low variance. Down and out barrier puts lie between mid O(10^-4) and low O(10^-3) in absolute error. Heston Bermudan puts with five exercise dates are near O(10^-3) in absolute error and about one percent in percentage terms and show a mild underpricing that points to a shortfall in early exercise value. American puts reach several times O(10^-3) in absolute error and roughly 1.6 - 4.8 % in percentage terms with a uniform underpricing sign.


The thesis concludes that a cumulative distribution function network can act as a reusable transition engine and that quadrature provides a fast and accurate pricing stage that requires no intermediate times between observation dates. The exponentially weighted network attains competitive accuracy with cleaner tails and improved stability in the regimes that matter for pricing. The combined system provides a practical route to high quality valuation across models and features with a single training pass over parameters.

Date of Award	25 Mar 2026
Original language	English
Awarding Institution	University of Bath
Supervisor	David Newton (Supervisor) & Dimitrios Gounopoulos (Supervisor)