Taming the Greeks: Option Portfolios with Inductive Biases

Wee Ling Tan; Stephen Roberts; Stefan Zohren

q-fin.PMSep 27, 2026

Taming the Greeks: Option Portfolios with Inductive Biases

Authors: Wee Ling Tan, Stephen Roberts, Stefan Zohren

Organizations: Department of Engineering Science Oxford-Man Institute of Quantitative Finance University of Oxford

Abstract

We present an end-to-end deep learning framework for systematic options trading that directly embeds hedging behavior through explicit control of portfolio-level risk exposures. While neural networks trained to optimize risk-adjusted performance have been shown to outperform traditional rules-based strategies, such approaches remain agnostic to the sensitivities of the resulting portfolios with respect to specific underlying risk factors. We propose a general training objective that combines a performance-driven loss with a differentiable risk-sensitivity penalty, enforcing neutrality to selected risk dimensions. Unlike reinforcement learning methods that approximate optimal hedging policies via simulated market dynamics, our framework operates entirely on historical data and jointly optimizes risk-adjusted returns and targeted risk constraints in a single learning problem. We instantiate the framework on static delta-neutral straddle portfolios with the penalty directed at first-order directional exposure, and evaluate two penalty variants -- an exposure-normalized penalty and a Greek-ratio drift penalty. Empirical results on Nasdaq 100 equity options demonstrate that appropriately calibrated regularization simultaneously improves out-of-sample risk-adjusted performance relative to an unregularized baseline while reducing realized directional exposure.

Figures & tables

Benchmarks
Model	E[Return]	Vol.	Downside Deviation	MDD	Sharpe	Sortino	Calmar	Hit Rate	$\mathbf{\frac{\text{Ave. P}}{\text{Ave. L}}}$
Long Only	0.100	0.161	0.088	0.340	0.621	1.129	0.294	0.447	1.390
TSMR	0.132	0.161	0.086	0.311	0.822	1.537	0.426	0.449	1.431
MACDMR	0.128	0.163	0.086	0.276	0.785	1.487	0.465	0.447	1.440
TSHestonMR	0.117	0.156	0.100	0.268	0.747	1.166	0.435	0.494	1.168
CSHestonMR	0.118	0.158	0.104	0.292	0.742	1.135	0.403	0.511	1.098

Table 1 : Performance Metrics - Rescaled to Target Volatility

Figure 1 : Distribution of Daily Net Position-Normalized Delta across the Out-of-Sample Period

Benchmarks
Model	Mean	Std	Min	Q1	Median	Q3	Max
Long Only	0.023	0.142	-0.465	-0.054	0.024	0.117	0.432
TSMR	0.011	0.076	-0.234	-0.033	0.002	0.053	0.452
MACDMR	0.015	0.084	-0.243	-0.033	0.009	0.062	0.464
TSHestonMR	0.004	0.061	-0.398	-0.023	0.003	0.035	0.206
CSHestonMR	0.001	0.067	-0.276	-0.038	-0.000	0.037	0.276

Table 2 : Summary Statistics: Net Position-Normalized Delta

Figure 2 : Distribution of Daily Gross Position-Normalized Delta across the Out-of-Sample Period

Benchmarks
Model	Mean	Std	Min	Q1	Median	Q3	Max
Long Only	0.218	0.110	0.000	0.133	0.221	0.306	0.484
TSMR	0.218	0.110	0.000	0.132	0.221	0.306	0.484
MACDMR	0.201	0.101	0.000	0.128	0.198	0.273	0.486
TSHestonMR	0.218	0.110	0.000	0.133	0.221	0.306	0.484
CSHestonMR	0.219	0.113	0.000	0.134	0.221	0.311	0.525

Table 3 : Summary Statistics: Gross Position-Normalized Delta

Figure 3 : Distribution of Daily Net Position-Normalized Delta across all Penalty Strengths

Figure 4 : Distribution of Daily Gross Position-Normalized Delta across all Penalty Strengths

Baseline
Model	E[Return]	Vol.	Downside Deviation	MDD	Sharpe	Sortino	Calmar	Hit Rate	$\mathbf{\frac{\text{Ave. P}}{\text{Ave. L}}}$
Baseline	0.399	0.187	0.131	0.222	2.131	3.043	1.797	0.656	1.053
Exposure-Normalized Penalty (L1)
ENP (L1) $\alpha=10^{0}$	0.365	0.199	0.144	0.283	1.833	2.527	1.290	0.655	1.020
ENP (L1) $\alpha=10^{1}$	0.415	0.196	0.141	0.268	2.113	2.941	1.549	0.671	1.034
ENP (L1) $\alpha=10^{2}$	0.417	0.187	0.128	0.255	2.231	3.260	1.632	0.661	1.125

Table 4 : Performance Metrics: Baseline vs. Regularized Models - Rescaled to Target Volatility

Figure 5 : Distribution of Mean Turnover for Strategies

Benchmarks
Model	0.0 bps	0.5 bps	1.0 bps	2.0 bps	3.0 bps	4.0 bps	5.0 bps	10.0 bps	20.0 bps	50.0 bps	100.0 bps
Long Only	0.621	0.620	0.618	0.616	0.613	0.611	0.608	0.596	0.571	0.496	0.372
TSMR	0.822	0.820	0.817	0.812	0.807	0.802	0.797	0.772	0.723	0.574	0.326
MACDMR	0.785	0.781	0.776	0.767	0.758	0.750	0.741	0.697	0.608	0.343	-0.100
TSHestonMR	0.747	0.745	0.742	0.737	0.732	0.727	0.722	0.697	0.647	0.497	0.247
CSHestonMR	0.742	0.741	0.739	0.736	0.733	0.730	0.727	0.711	0.680	0.587	0.431

Table 5 : Impact of Transaction Costs on Sharpe Ratio – Rescaled to Target Volatility

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameters	Search Grid
Minibatch Size	32, 64, 128, 256
Dropout Rate	0.1, 0.2, 0.3, 0.4, 0.5
Hidden Layer Size	5, 10, 20, 40, 80, 160
Learning Rate	$10^{-5},~10^{-4},~10^{-3},~10^{-2},~10^{-1},~10^{0}$
Max Gradient Norm	$10^{-4},~10^{-3},~10^{-2},~10^{-1},~10^{0},~10^{1}$
Risk-aversion Coefficient	$10^{0},~10^{1},~10^{2},~10^{3},~10^{4},~10^{5},~10^{6},~10^{7}$

Appendix

Table 6 : Hyperparameter Search Range

Figure 6 : Time-Series Plot of Daily Net Position-Normalized Delta - ENP (L1)

Figure 7 : Time-Series Plot of Daily Net Position-Normalized Delta - ENP (L2)

Figure 8 : Time-Series Plot of Daily Net Position-Normalized Delta - DP (L1)

Figure 9 : Time-Series of Daily Net Position-Normalized Delta - DP (L2)

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