Deep learning of transition probability densities for stochastic asset models with applications in option pricing

TL;DR

The paper tackles the challenge of rapidly computing transition probability densities for stochastic asset models across a broad parameter space. It introduces neural TPDF generators that solve parametric backward Kolmogorov equations for the cumulative distribution function, from which the TPDF is obtained by differentiation. The Deep Galerkin Method is adapted to parametric PDE problems, delivering a single offline training that enables fast online option pricing across GBM, Heston, SABR, and jump-diffusion dynamics, with substantial speedups in QUAD-based pricing. The approach demonstrates high accuracy in densities and prices, including non-affine and PIDEs, while highlighting limitations in tail accuracy and suggesting avenues for architecture improvements and further calibration work.

Abstract

Transition probability density functions (TPDFs) are fundamental to computational finance, including option pricing and hedging. Advancing recent work in deep learning, we develop novel neural TPDF generators through solving backward Kolmogorov equations in parametric space for cumulative probability functions. The generators are ultra-fast, very accurate and can be trained for any asset model described by stochastic differential equations. These are "single solve", so they do not require retraining when parameters of the stochastic model are changed (e.g. recalibration of volatility). Once trained, the neural TDPF generators can be transferred to less powerful computers where they can be used for e.g. option pricing at speeds as fast as if the TPDF were known in a closed form. We illustrate the computational efficiency of the proposed neural approximations of TPDFs by inserting them into numerical option pricing methods. We demonstrate a wide range of applications including the Black-Scholes-Merton model, the standard Heston model, the SABR model, and jump-diffusion models. These numerical experiments confirm the ultra-fast speed and high accuracy of the developed neural TPDF generators.

Figures (17)

• Figure 1: Illustration of the deep Galerkin method NN architecture used in this paper.
• Figure 2: The root mean squared errors (RMSE) are calculated using the NN approximate TPDF benchmarked against the exact TPDF Eq. \ref{['eq: BSM closed form density']}. This figure plots epoch against RMSE for different $\lambda_{L}=0.1,1,10,100,1000$ across various times $t=0.25$, $0.5$, $0.75$, $1.0$ set up under the Black-Scholes-Merton model. The epoch checkpoints are $5000$, $25000$, $50000$, $100000$, $250000$, $500000$, $1000000$, $1500000$, $2000000$. Each checkpoint represents the NN with the smallest training loss up to that epoch checkpoint. The dashed line in the graph means no better NN was found and we use the best NN in the previous checkpoint. The domain $Q$ used to train the NN is $x,y\in[-2.3,2.3]$, $\sigma\in[0,0.6]$, $t\in[0,1.2].$ In terms of validation, for each $t$, $\sigma=0.1$, $0.15$, $0.2$, $0.25$, $0.3$, $0.35$, $0.4$, $0.45$,$0.5$, $0.55$, $0.6$. The initial log price $x_{0}=0$, and the range of $y$ for each $(t,\sigma)$ set is $[-2.3,2.3]$.
• Figure 3: Gaussian TPDF, closed form vs. NN approximation is shown in the first row. The second row shows the same results in log scale of density. The figures shown correspond to the parameters $x=0$, $\sigma=0.2$, the time to maturity $t=0.25,0.5,0.75,1.0$, and $\lambda_{L}=100$. The domain used to train the network is $x,y\in[-2.3,2.3]$, $\sigma\in[0,0.6]$, $t\in[0,1.2].$
• Figure 4: RMSE are calculated using the NN approximate TPDF benchmarked against the true TPDF from Eq. \ref{['eq: BSM closed form density']}. This figure plots epoch against RMSE for different $\lambda_{L}=0.1$, $1$, $10$, $100$, $1000$ across various times $t=0.025$, $0.05$, $0.075$, $0.1$. The epoch checkpoints are $5000$, $25000$, $50000$, $100000$, $250000$, $500000$, $1000000$, $1500000$, $2000000$. Each checkpoint represents the NN with the smallest training loss up to that epoch checkpoint. The dashed line in the graph means no better NN can be found and we use the best NN in the previous checkpoint. The dotted line for $\lambda_L = 100$ represented the fine tuned model after $500$ epochs. The domain $Q$ used to train the network is $x,y\in[-2.3,2.3]$, $\sigma\in[0,0.6]$, $t\in[0,1.2].$ In terms of validation, for each $t$, $\sigma=0.1$, $0.15$, $0.2$, $0.25$, $0.3$, $0.35$, $0.4$, $0.45$, $0.5$, $0.55$, $0.6$. The initial log price $x_{0}=0$, and the range of $y$ for each $(t,\sigma)$ set is $[-2.3,2.3]$.
• Figure 5: The first row shows the exact Gaussian TPDF and the neural TPDF before and after tuning for small time to maturity. The second row shows the same results in log scale of density. $x=0$, $\sigma=0.2$, the time to maturity $T=0.025$, $0.05$, $0.075$, $0.1$. The NN results after 500 epochs tuning (and we select the best trained NN).

Theorems & Definitions (2)

• Remark 1
• Remark 2