Implementing Neural SDEs for Data-Driven Dynamics of the Bitcoin Option Surface

Abstract

This paper presents a full implementation of data-driven modelling of the dynamics of the options on Bitcoin, using high-frequency data from the Deribit exchange. To this end, we provide a synthesis of methods established in prior papers, namely, the works involving “neural SDE market models,” to build a pipeline to go from raw option quotes to a functioning non-parametric model. The option surface is decomposed into a low-dimensional latent space designed to minimise arbitrage in reconstruction, and the temporal evolution of these factors is modelled with a stochastic differential equation (SDE). The drift and diffusion of the SDE are learned from data using neural networks, thereby forming a “neural SDE”. These networks are constrained in order to guarantee the absence of static arbitrage and to minimise dynamic arbitrage in the resulting model. The networks are trained using a likelihood-based objective function in an SDE transition discretisation. The framework produces arbitrage-free simulations of option surfaces and enables risk management applications, such as Value-at-Risk estimation and hedging applications.

1. Introduction

1.1. Motivation

Accurately modelling the dynamics of options is a core function of modern quantitative finance, underpinning both valuation and risk management in numerous asset classes. Traditional approaches to option pricing, such as Black and Scholes (1973), Heston (1993) and SABR (Hagan et al. 2002), have been extensively developed and tested in the context of mature financial markets, such as equities or foreign exchange. As with any modelling approach, these frameworks have simplifying assumptions, in particular, in the form of parametric assumptions for the drift and diffusion processes.

Cryptocurrency markets, and by extension their derivatives markets, represent a markedly different environment than established financial markets. It is because of these altered dynamics that the assumptions in traditional option pricing models may be even more restrictive when applied to cryptocurrency as opposed to traditional asset classes. Ahmed et al. (2024) provide a review paper that highlights that high volatility seems to be a structural feature in cryptocurrency markets. They also suggest that machine learning methods, in particular, deep learning models, may better capture non-linear dynamics and handle high-frequency data. The other recent literature (e.g., Avordeh et al. 2026) similarly provides evidence that classical parametric approaches (such as the stochastic volatility model of Heston (1993)) struggle to reflect cryptocurrency stochastic dynamics, such as periods of very high volatility, and that enhancing models by adaptive machine learning techniques can result in better model performance. This point is also underscored in a wide-ranging review of mathematical models for cryptocurrency analytics by Dote-Pardo and Espinosa-Jaramillo (2026). That is to say, models incorporating neural networks for data-driven stochastic dynamics (as opposed to models based on a priori assumed parametric structures) are particularly important if one seeks to capture crypto-specific tail risks,1 heavy-tailed distributions, and sudden structural jumps. However, given that these issues are typically also present in more traditional assets, albeit in a less extreme form, one would expect the methodology in the present paper to perform similarly well (or possibly better) in those markets. Note that our focus here is on the dynamics of the cryptocurrency option surface over time, as opposed to cross-sectional (i.e., a single point in time) calibration to an observed option surface.2

Furthermore, there has been little work done in the space of cryptocurrency option pricing within the literature. While all option modelling techniques are domain-agnostic (i.e., the market itself does not matter, and the mathematics of the models do not change), as far as we are aware, there has not been any work done on the modelling of a multi-strike, multi-maturity options book (i.e., the so-called volatility surface) for cryptocurrencies.

1.2. Goals

The broad goal of this project is to use raw option quotes, obtained from the Deribit exchange, and create a data-driven model for the joint dynamics of the options observable in the market. This can be broken down into three points.

• Data-driven method to model the dynamics of the observable call option surface. We use non-parametric machine learning methods to extract the model dynamics from the observable call options on Deribit. Rather than assuming a parametric form for the drift and diffusion of the underlying, the model seeks to learn the evolution of option prices directly from historical time-series data across strikes and maturities. By focusing on the evolution of the entire surface rather than individual options, the model captures the co-movements and structural relationships inherent in a live options market, enabling a more realistic depiction of the dynamics observed in high-frequency cryptocurrency data.
• Ensure the model does not admit arbitrage. Absence of arbitrage is a basic consistency requirement for financial models. We want to ensure our model accounts for this by ensuring that outputs admitting arbitrage are either eliminated entirely or at least minimised. To ensure theoretical and practical validity, the model is explicitly designed to satisfy static, and minimise any violation of dynamic, no-arbitrage conditions. Static-arbitrage constraints impose shape restrictions on the option surface—such as monotonicity in strike and convexity, and calendar spread consistency—ensuring that prices at a single point in time form a feasible and economically consistent surface. This is the more fundamental requirement in the sense that violations of static-arbitrage constraints lead to immediate, model-free arbitrage opportunities. Dynamic arbitrage constraints, in contrast, govern the temporal evolution of option prices, ensuring that the model’s drift and diffusion dynamics are consistent with an equivalent martingale measure. What each of these constraints means definitionally will be discussed in more depth in Section 2.2 and Section 2.3 below. The static conditions are incorporated directly into the learning process through constrained optimisation and diffusion shrinkage methods derived from the arbitrage-free neural SDE framework of Cohen et al. (2023a).
• Use of the model for risk management and hedging. As application examples of the model fitted to the market in the present paper, we build on the work of Cohen et al. (see Cohen et al. 2022, 2023b) to hedge option portfolios and provide Value-at-Risk (VaR) estimates via market simulation.

1.3. Significance and Contributions

This work contributes to the emerging literature on data-driven financial modelling by applying the arbitrage-free neural SDE framework to the cryptocurrency derivatives market. It provides the first empirical application of an arbitrage-consistent neural SDE to intraday, multi-strike, multi-maturity crypto options data. Beyond methodological adaptation, the model demonstrates practical relevance by producing risk and hedging metrics that are derived from the same underlying stochastic dynamics, bridging the gap between theoretical consistency and applied market utility.

1.4. Structure of the Paper/Study Overview

From Deribit,3 we were able to obtain high-frequency data for all options on Bitcoin available on this exchange. It is on this data that we implement the neural SDE methodology.

This paper is organised as follows. After providing the background and rationale for our modelling approach in Section 2, in Section 3, we discuss the data selected for this study, as well as the pre-processing required before using this data to train the arbitrage-free neural SDE framework, which is detailed in Section 4. Example applications in hedging and risk management are presented in Section 5, and Section 6 concludes.

2. Background

2.1. Market Models

The present paper follows the approach of so-called market models, which model the joint dynamics of all liquidly tradable assets simultaneously and impose hard and/or soft constraints on the modelling process to ensure the absence of arbitrage. This idea is similar to the interest rate modelling framework of Heath–Jarrow–Morton (HJM) (Heath et al. 1992), though models for options are inherently more complex due to the constrained state space of option prices created through no-arbitrage relationships.

The literature for market models contains works that have been done in special cases. Schönbucher (1999) and Lyons (1995) have proposed a model for the implied volatility of single options. For example, denote by $S_t$ the price of the underlying asset at time t, where $C_t$ is the price of a call option, K is the strike price, and T is the maturity. If there is only one option to model, one has the static bounds ${(S_t-K)}^{+}\le C_t\le S_t$ and $X=C_T={(S_T-K)}^{+}$, where the latter corresponds to the terminal payoff condition. The authors of these papers do this by reparameterising the option price by its implied volatility. By contrast, modelling the joint dynamics of multiple options across strikes and maturities introduces a much richer set of static-arbitrage constraints that prices must satisfy (Carr and Madan 2005; Davis and Hobson 2007; Gatheral 2011). To our knowledge, only two substantial contributions address this setting: the arbitrage-consistent approach of Wissel (2008) and the data-driven neural SDE framework of Cohen et al. (2023a). The latter introduces the concept of a “liquid lattice”—a structured, yet arbitrage-consistent, state space for option prices—allowing dynamic modelling of an entire option surface rather than a single contract. This addresses the need for having a data-driven model for options that can take in a time series of data, so the present paper builds on the methods provided by Cohen et al. (2020, 2022, 2023b).

2.2. Static Arbitrage

Static arbitrage refers to the ability to make a risk-free profit by exploiting the prices of options at a specific point in time (Carr and Madan 2005; Davis and Hobson 2007; Gatheral 2011). Static-arbitrage strategies are immediate and model-free; thus, there is a strong requirement that option price surfaces should not allow static arbitrage. Effectively, these static-arbitrage constraints are a set of shape constraints on the price surface. Specifically, for call options, the conditions are:

• Monotonicity in maturity. Whenever two call prices share the same moneyness m but have different maturities ${\tau }_i<{\tau }_j$, the call price must satisfy

  $$C({\tau }_j,m)\ge C({\tau }_i,m).$$
• Monotonicity in moneyness. At any fixed maturity $\tau$, when two moneyness values satisfy $m_k<m_{\ell }$, the corresponding call prices must satisfy

  $$C(\tau ,m_k)\ge C(\tau ,m_{\ell }).$$
  These conditions are derived from the absence of “calendar spread” arbitrage formalised by Carr and Madan (2005) and further reviewed by Gatheral (2011).
• Convexity in moneyness. Also at a fixed $\tau$, for any three consecutive moneyness points $m_{k-1}<m_k<m_{k+1}$, convexity requires (see, e.g., Breeden and Litzenberger 1978; Roper and Rutkowski 2009)

  $$C(\tau ,m_{k-1})-2C(\tau ,m_k)+C(\tau ,m_{k+1})\ge 0.$$
• Non-negativity. For every node $({\tau }_i,m_i)$, the call price must be non-negative:

$$C({\tau }_i,m_i)\ge 0.$$

As in Cohen et al. (2020), we can collect all the linear inequalities arising from conditions (1)–(3) into a single matrix inequality

$$A\mathbf{c}\ge \mathbf{0},$$

giving a set of R linear constraints in ${\mathbb{R}}^n$. In particular, each row of A corresponds to exactly one of the following three types of finite-difference expressions:

$$\begin{array}{cc}& \left(\mathrm{a}\right)\mathrm{If}({\tau }_j,m)\mathrm{and}({\tau }_i,m)\mathrm{are}\mathrm{adjacent}\mathrm{in}\tau ,C({\tau }_j,m)-C({\tau }_i,m)\ge 0,\hfill \\ & \left(\mathrm{b}\right)\mathrm{If}(\tau ,m_k)\mathrm{and}(\tau ,m_{\ell })\mathrm{are}\mathrm{adjacent}\mathrm{in}m,C(\tau ,m_k)-C(\tau ,m_{\ell })\ge 0,\hfill \\ & \left(\mathrm{c}\right)\mathrm{If}(\tau ,m_{k-1}),(\tau ,m_k),(\tau ,m_{k+1})\mathrm{are}\mathrm{consecutive}\mathrm{in}m,\hfill \\ & C(\tau ,m_{k-1})-2C(\tau ,m_k)+C(\tau ,m_{k+1})\ge 0.\hfill \end{array}$$

Meanwhile, the non-negativity requirement is captured by the coordinate-wise condition $\mathbf{c}\ge \mathbf{0}$. We use this parameterisation in the data pre-processing step for detecting and removing arbitrage in our historical option prices. Similar arbitrage-cleaning procedures are applied in Cohen et al. (2020) and Wissel (2008) to enforce shape constraints before model calibration.

2.3. Dynamic Arbitrage

While static arbitrage concerns the shape of the option price surface at a single point in time, dynamic arbitrage refers to risk-free profits that can be generated by trading options (and the underlying asset) over time as their prices evolve. In other words, even if an option surface satisfies all static no-arbitrage conditions at each date, its temporal evolution may still violate arbitrage-free dynamics if there is an inconsistent drift or diffusion structure across maturities and strikes (Derman and Kani 1994; Dupire 1994; Fukasawa 2011). Specifically, in the context of driving the option price surface by a limited number of factors (as we do below), some options will be replicable by self-financing dynamic trading strategies in a Harrison and Pliska (1981, 1983) sense, giving rise to arbitrage opportunities if the option price dynamics do not match the value dynamics of these self-financing strategies.

The existence of an equivalent martingale measure is sufficient to preclude such arbitrage opportunities. Formally, a family of call price processes ${\left\{C_t(\tau ,m)\right\}}_{t\ge 0}$ is ensured to be dynamically arbitrage-free if there exists a risk-neutral probability measure $\mathbb{Q}$ under which the discounted price processes of the underlying and of all replicable portfolios of options are martingales. This implies that:

$$e^{-{\int }_0^t{r}_{s}ds}{S}_t\mathrm{and}{e}^{-{\int }_0^t{r}_{s}ds}{C}_t(\tau ,m)\mathrm{are}\mathbb{Q}-\mathrm{martingales}.$$

2.4. Neural Networks in Option Pricing and Modelling

The use of neural networks in option pricing was first pursued by Hutchinson et al. (1994), who trained feedforward networks to learn the pricing function of S&P 500 index options. The resurgence of neural networks in the 2010s brought new techniques for option pricing. Major advances include neural SDEs, generative models, and the use of physics-informed neural nets (PINNs).

2.4.1. Dynamic Hedging via Deep Learning

The concept of using deep learning for hedging, or “deep hedging”, was introduced by Buehler et al. (2019). Here, the hedging problem is treated as a reinforcement learning task (Sutton and Barto 1998), where an agent (the hedging strategy) learns to optimise a reward or risk objective over time. Buehler et al. (2019) used a synthetic dataset, with transaction costs, and found that the learned hedging strategy outperforms the traditional Black–Scholes delta hedging in terms of risk-adjusted returns. Similar work by Kolm and Ritter (2019) applied reinforcement learning (RL) to dynamic replication and hedging, training a hedging agent that learns how to optimally trade to manage option risk. These RL-based approaches mark a transition from using neural networks merely as function approximators to using them as decision-makers in a sequential setting.

2.4.2. Generative Models

Another use of neural networks is generative models, which seek to capture the joint distribution of asset prices and implied volatility surfaces, enabling realistic market simulation and data augmentation. Generative Adversarial Networks (GANs) (Goodfellow et al. 2020), and other related generative models, have been used to learn distributions of option prices across strikes, maturities and time, with the aim of producing synthetic data that is statistically the same as real market data. Work by Wiese et al. (2019), building on deep hedging, proposed a GAN-based simulator for equity options. The approach of Wiese et al. (2019) involved a generator network producing a sequence of market states (underlying price and entire implied volatility surface over time), which a discriminator network tries to distinguish from real historical market data. By training these networks against each other, the generator learns to create highly realistic option price dynamics. Similarly, the model introduced as “VolGAN” by Vuletić and Cont (2024) has a GAN-based architecture trained on historical sequences of implied volatility surfaces and underlying prices, capable of simulating realistic joint dynamics of the volatility surface and the underlying while respecting static-arbitrage constraints. Cont and Vuletić (2023) also introduce another, tractable, method capable of simulating arbitrage-free volatility surfaces using more conventional stochastic and factor dynamics with constraints and calibration steps.

2.4.3. Neural SDEs

Notable examples of neural SDE martingale models include Cuchiero et al. (2020), using GANs to match risk-neutral probability distributions, and Gierjatowicz et al. (2022), who train the neural SDE in a more traditional way. The thesis by Phytides (2023) demonstrates that neural SDEs, when properly constrained with no-arbitrage rules, can accurately replicate a benchmark process (like Black–Scholes and constant elasticity of variance (CEV)). Another approach to note is the work by DeLise (2021), who treats neural SDEs as universal Itô process approximators. Using standard martingale theory and Girsanov’s Theorem, these trained neural SDEs reliably reproduce the Black–Scholes option price (i.e., he trains the neural SDE on just the underlying geometric Brownian motion dynamics and has it reproduce Black–Scholes option prices). In contrast, Fan and Sirignano (2026) use real-world historical S&P 500 and S&P 100 index options to train neural SDE models for pricing and hedging. Cohen et al. (2023a) developed an arbitrage-free neural SDE market model for European options, incorporating explicit constraints to enforce static-arbitrage-free relationships among options. They tested their methodology on synthetic data generated from a Heston (1993) stochastic volatility model, demonstrating the ability to recover known dynamics. In their framework, a set of latent factors (such as implied volatility levels or slopes) evolves in time along with the underlying asset price, and these jointly drive the entire surface of option prices. The drift and volatility of these factors are given by neural networks, which are trained on time-series data of options and underlying prices. To ensure the learned model does not violate no-arbitrage conditions, the neural network outputs are passed through transformations that enforce the linear inequalities reflecting static-arbitrage constraints. This constrained neural network approach guarantees that every price surface produced by the model is free of calendar arbitrage and butterfly arbitrage by design. The result is a non-parametric yet arbitrage-consistent market model that can be calibrated to the market’s entire option book. Such a model can serve as a market simulator, generating realistic paths of future implied volatility surfaces for risk management use, or be used to price illiquid exotic derivatives by Monte Carlo simulation in the learned dynamics.

2.4.4. Physics-Informed Neural Networks (PINNs) for Option Pricing

PINNs offer a powerful way to reduce the “black-box” nature of deep learning by embedding known physical or theoretical laws directly into the learning process. In PINNs, the solution to a (partial) differential equation (PDE) is approximated by a neural network ${\mu }_{\theta }(x,t)$, and the governing PDE is enforced by penalising any deviation from its expected behavior.

There have been several attempts at applying PINNs to the option pricing problem. The thesis by Tanios (2021) demonstrates the feasibility of applying PINNs to the high-dimensional option pricing problem by solving both the forward and inverse formulations of the Black–Scholes and Heston partial differential equations (PDEs). In his work, PINNs are used not only to accurately approximate the solution to these PDEs, but also to compute sensitivities (Greeks) via automatic differentiation, thereby overcoming the limitations of traditional numerical methods in high-dimensional settings. Similarly, recent work by Dhiman and Hu (2023) extends the PINN framework to both European and American options, showing that embedding financial constraints—such as no-arbitrage conditions—directly into the neural network’s loss function can lead to improved accuracy and computational efficiency. While Tanios uses synthetically generated data, Dhiman and Hu (2023) include the application of PINNs to real market data.

2.5. Choice of Modelling Approach

Static constraints define a feasible state space for option prices, while dynamic constraints govern admissible drifts/diffusions over time. The “liquid lattice”, i.e., linear-inequality parameterisation, enables tractable enforcement of static constraints during learning and cleaning. For our use case, these constraints are not optional regularisation but necessary domain knowledge that stabilises learning and ensures economically meaningful outputs.

Deep hedging (i.e., RL) targets hedging strategy optimisation, not market option surface dynamics. Generative models (e.g., GANs) excel at sample realism but can struggle with enforcing the absence of arbitrage and with the interpretability of calibrated factor dynamics. PINNs embed the PDE structure but align best with parametric PDE models. Thus, we will focus on neural SDEs, which provide a stochastic, non-parametric, time-series view with continuous-time semantics. Combined with explicit no-arbitrage conditions, they can model the joint evolution of the entire option surface while remaining economically consistent. Building on the approach of Cohen et al. (2023a), we will construct a model with dynamics, which are learned directly from the data, for the temporal evolution of the option surface.

3. Data Pre-Processing

This section details the pre-processing steps required to take raw option quotes, obtained from Deribit, and clean and transform them so that they are ready to be fed into the neural SDE model. This involves the creation of a static lattice of options at high frequency, arbitrage repair of these option price surfaces and a transformation of the surfaces into a lower-dimensional representation.

3.1. Overview of Dataset Used

The dataset has been obtained from the Deribit exchange, which is a cryptocurrency options exchange, where at each time step, we see every available option on Bitcoin from the exchange. This effectively provides a time series of option prices, though we need to perform some pre-processing to create a static grid of strikes and maturities for the time series. This will be explored more in depth in Section 3. An important feature to note of the raw data is that it is high frequency, meaning that each consecutive snapshot is provided at the sub-second level; see Figure 1. There are $118,193$ unique timestamps in one dataset, which spans 24 h.

The date for the market data used in the initial experiment is 22 May 2025. For context, the price of BTCUSD on that day is plotted in Figure 2.

We also perform an extended analysis on a dataset following the same schema, i.e., over a week’s worth of data as opposed to a day. This extended analysis covers the dates 7–14 (inclusive) of September 2025. The distribution of time step lengths is seen in Figure 3, and the movement of the underlying asset is seen in Figure 4. This dataset has 842,940 unique time steps.

The analysis on this longer data period is conducted to complement the 24 h results of the main text and serves two purposes: to confirm that the pre-processing and decoding pipeline of Section 3 scales to a longer horizon, and to scrutinise the validity of the distributional approximations when more data are present.

The choice of data periods for which we present our empirical results, both the 24 h period for the initial analysis and the seven-day period for the extended analysis, is random and not motivated by particular market conditions. We would expect these data to be fairly representative of the market conditions prevailing in recent times.

3.2. Static Grid for Options

Over time, the time-to-maturity/moneyness coordinates of liquid options change stochastically. To facilitate training our model, it is hence required to create a static lattice of time-to-maturity/moneyness coordinates, to which market options are then matched. Let $\tau =T-t$ denote time to maturity. m is forward log-moneyness, i.e.,  $m=ln({\displaystyle \frac{K}{F_t\left(T\right)}})$, with K being the strike and $F_t$ being the forward price.

It is important to note that the entire option surface is never observable at once; it is only ever a subset of options for which we currently see active quotes within the order book. Because of this, for each time to maturity $\tau$, there is a different set of log-moneynesses m. In particular, options further from expiry have a larger range of traded moneyness values in comparison to options traded closer to maturity. This represents what Cohen et al. (2023a) call the ”liquid lattice” of options. When constructing our static grid, it is important to capture this aspect, as we are interested in modelling the movement of tradeable options that have sufficient liquidity. Conversely, we are not interested in modelling a very deep out-of-the-money option that has a one-day expiry, because we are very unlikely, given most market conditions, to observe this option ever being actively traded.

Constructinga Static Grid at High Frequency

Prior “market model” work in the literature has also parameterised the price of options in this fashion. Wissel (2008) constructs a square lattice (i.e., each maturity has the same distribution of moneyness), while Cohen et al. (2023a) use end-of-day closing data for STOXX 50 index and DAX options, which naturally provide a distribution of moneyness for each maturity, and hence the shape of the liquid lattice is chosen heuristically.

However, at high frequency/intraday frequency, each maturity has a continuous distribution of traded moneynesses effectively. Figure 5 highlights the observed distribution of available log-moneyness values for each maturity cluster in the Deribit data for a 24 h period on 22 May 2025.

This leads to a method that captures the distribution of m for each $\tau$, or the shape of the “liquid lattice”. We are primarily interested in ensuring that this liquid lattice reflects the liquidity profile within our observed dataset. We construct a lattice over the joint $(\tau ,m)$ space as follows. First, let ${\left\{{\overline{\tau }}_i\right\}}_{i=1}^p$ be the sorted unique maturities observed in the dataset. For each ${\overline{\tau }}_i$, compute empirical moneyness percentiles $m_{\mathrm{lo},i}=P_1(m\mid \tau ={\overline{\tau }}_i)$ and $m_{\mathrm{hi},i}=P_{99}(m\mid \tau ={\overline{\tau }}_i)$, and form an $N_m$-point linear grid

$$m_{ij}=m_{\mathrm{lo},i}+(j-1){\displaystyle \frac{m_{\mathrm{hi},i}-m_{\mathrm{lo},i}}{N_m-1}},j=1,\dots ,N_m.$$

The lattice nodes are the Cartesian set $\mathcal{N}=\left\{({\overline{\tau }}_i,m_{ij})\right\}$. We then fit a one-nearest-neighbor map in the 2D space (Euclidean metric) and assign each quote k with features $({\tau }_k,m_k)$ directly to its nearest node index ${\ell }_k=arg{min}_{({\overline{\tau }}_i,m_{ij})\in \mathcal{N}}\parallel ({\tau }_k,m_k)-({\overline{\tau }}_i,m_{ij})\parallel$. For each calendar time t and node in $\mathcal{N}$, if multiple quotes map to the same node, we keep the one with the highest liquidity (measured by trading volume). Finally, we pivot to a time-by-node panel and fill any missing entries by linear interpolation over t, followed by forward/backward-fill as needed.

This method is implemented by Algorithm 1, noting that within the 24 h of data, we take all unique maturities rather than performing clustering on $\tau$. Let $\mathcal{Q}={\left\{q_k\right\}}_{k=1}^M$ be quote observations, where each

$$q_k=(t_k,T_k,K_k,B_k,A_k,S_k,V_k,{\mathrm{type}}_k),$$

with timestamp $t_k$, expiry $T_k$, strike $K_k$, best bid $B_k$, best ask $A_k$, underlying spot $S_k$, liquidity proxy $V_k$ (USD traded volume), and the option type. Fix reference date $t_{\mathrm{ref}}$, rates $(r,q)$, and lattice hyperparameters $n_{\tau }\in \mathbb{N}$, $n_m\in \mathbb{N}$, and $K_{max}\in \mathbb{N}$ (default 50).

For each retained quote k, define

$${\tau }_k:={\displaystyle \frac{T_k-t_{\mathrm{ref}}}{365.25}},F_k:=S_k{e}^{(r-q){\tau }_k},m_k:=log\left({\displaystyle \frac{K_k}{F_k}}\right),$$

$${\mathrm{mid}}_k:={\displaystyle \frac{B_k+A_k}{2}},c_k:={\mathrm{mid}}_k{e}^{-(r-q){\tau }_k}(=C_k/F_k).$$

Retain calls only (${\mathrm{type}}_k=\mathrm{C}$) with $V_k>0$.

If train/test splitting is used, build $(\mathcal{L},\mathrm{NN})$ on the training quotes only, apply the snapping-and-panel step separately to train and test, then keep the intersection of node indices:

$${\mathcal{L}}_{\cap }={\mathcal{L}}_{\mathrm{sub}}^{\mathrm{train}}\cap {\mathcal{L}}_{\mathrm{sub}}^{\mathrm{test}},$$

and reindex both panels to ${\mathcal{L}}_{\cap }$ with the same linear + forward/backward fill.

The resulting liquid lattice from this process is displayed in Figure 6, which is the lattice from 24 h of data from the 22nd of May 2025. Here we can see that, while the lattice is irregular and asymmetric, it does contain the general shape of an options lattice we would expect, with options closer to maturity having a smaller distribution support of traded moneyness values. Additionally, one can speculate that when running this process over a longer time period, say one month, the lattice would look more symmetric, as one would observe in more traded combinations of $(\tau ,m)$.

Algorithm 1 Static lattice construction

Require:  Cleaned quotes ${\left\{(t_k,{\tau }_k,m_k,c_k,V_k)\right\}}_{k=1}^M$, $n_{\tau }$, $n_m$, $K_{max}$, fill method (linear) Ensure:  Reduced node set ${\mathcal{L}}_{\mathrm{sub}}$, dense panel $C(t,\ell )$ on ${\mathcal{L}}_{\mathrm{sub}}$   1: $\mathcal{U}\leftarrow \mathrm{sort}\left(\mathrm{unique}{\left\{{\tau }_k\right\}}_{k=1}^M\right)$   2: Fit 1D K-means with $n_{\tau }$ clusters on $\mathcal{U}$; obtain centroids ${\left\{{\tilde{\tau }}_i\right\}}_{i=1}^{n_{\tau }}$   3: Sort centroids increasingly: ${\overline{\tau }}_1<\cdots <{\overline{\tau }}_{n_{\tau }}$ and relabel quote cluster indices accordingly   4: for $i=1,\dots ,n_{\tau }$ do   5:     ${\mathcal{M}}_i\leftarrow \{m_k:\mathrm{quote}k\mathrm{assigned}\mathrm{to}\mathrm{cluster}i\}$   6:     if ${\mathcal{M}}_i=⌀$ then   7:         $({\underline{m}}_i,{\overline{m}}_i)\leftarrow \left(P_1\left(\left\{m_k\right\}\right),P_{99}\left(\left\{m_k\right\}\right)\right)$   8:     else   9:         $({\underline{m}}_i,{\overline{m}}_i)\leftarrow \left(P_1\left({\mathcal{M}}_i\right),P_{99}\left({\mathcal{M}}_i\right)\right)$ 10:     end if 11:     for $j=1,\dots ,n_m$ do 12:         $m_{ij}\leftarrow {\underline{m}}_i+{\displaystyle \frac{j-1}{n_m-1}}({\overline{m}}_i-{\underline{m}}_i)$ 13:         Create node $\ell =(i,j)$ with coordinates $x_{\ell }:=({\overline{\tau }}_i,m_{ij})$ 14:     end for 15: end for 16: $\mathcal{L}\leftarrow \left\{x_{\ell }\right\}$, $N:=\left|\mathcal{L}\right|=n_{\tau }{n}_m$ 17: Fit 1-nearest-neighbor model on $\mathcal{L}$ in ${\mathbb{R}}^2$ (Euclidean distance) 18: for $k=1,\dots ,M$ do 19:     ${\ell }_k\leftarrow arg{min}_{\ell \in \{1,\dots ,N\}}{\parallel ({\tau }_k,m_k)-x_{\ell }\parallel }_2$ 20: end for 21: for each pair $(t,\ell )$ observed among $\left\{(t_k,{\ell }_k)\right\}$ do 22:     ${\mathcal{I}}_{t,\ell }\leftarrow \{k:t_k=t,{\ell }_k=\ell \}$ 23:     $k^{★}(t,\ell )\leftarrow arg{max}_{k\in {\mathcal{I}}_{t,\ell }}{V}_k$ 24:     $C^{\mathrm{sparse}}(t,\ell )\leftarrow c_{k^{★}(t,\ell )}$ 25: end for 26: Set all unfilled entries of $C^{\mathrm{sparse}}$ to NaN 27: Node coverage count: $n_{\ell }\leftarrow \#\{t:C^{\mathrm{sparse}}(t,\ell )\mathrm{is}\mathrm{observed}\}$ 28: Keep ${\mathcal{L}}_K$ = indices of the $K_{max}$ largest $n_{\ell }$ 29: Restrict $C^{\mathrm{sparse}}$ to columns in ${\mathcal{L}}_K$; drop columns entirely NaN 30: Let remaining column indices be ${\mathcal{L}}_{\mathrm{sub}}$ and corresponding nodes be ${\left\{x_{\ell }\right\}}_{\ell \in {\mathcal{L}}_{\mathrm{sub}}}$ 31: for each $\ell \in {\mathcal{L}}_{\mathrm{sub}}$ do 32:     Interpolate $C^{\mathrm{sparse}}(·,\ell )$ linearly in time 33:     Apply forward-fill, then backward-fill at time boundaries 34: end for 35: Denote resulting dense panel by $C(t,\ell )$ 36: return ${\mathcal{L}}_{\mathrm{sub}}$, $C(t,\ell )$

3.3. Arbitrage Repair of Historical Option Prices

When training an options model, we want to ensure that the input data is arbitrage-free. Thus cleaning input data of arbitrage violations is a logical next step in the data processing. We may encounter seeming arbitrage violations in the input data for two reasons:

• Using the mid-price to construct our model. Within our data, and in the microstructure of markets, we do not see a single price for any asset, but rather two quotes: the bid, which is the highest price a buyer is willing to pay, and the ask, which is the lowest price a seller is willing to accept. The information contained within the spread of these relates to the liquidity of the option’s price and is captured within the lattice creation, via volume weighting, and the arbitrage-cleaning process. The existence of an arbitrage-free surface between the spread is discussed in Appendix A.
• Data artefacts of our lattice creation method. The lattice of $(\tau ,m)$ pairs that are created using Algorithm 1 are not guaranteed to satisfy our static-arbitrage constraints. Even if the mid-price captured a static-arbitrage-free surface, we have interpolated these prices to common nodes, which could potentially break convexity and/or monotonicity conditions.

Thus it is important to note that what may seem like arbitrage violations in the data are unlikely to be opportunities to make actual arbitrage profits in the market.

From Algorithm 1, we have a time series of option surfaces. We can use methods from Cohen et al. (2020) to “repair” these constructed surfaces of static-arbitrage violations. Note that Cohen et al. (2020) applied this method to end-of-day closing data, while in our context, it is applied to intraday, high-frequency cryptocurrency options data.

Recall the representation of static-arbitrage constraints reproduced from Cohen et al. (2020) in Section 2.2 above:

$$Ac\ge 0,$$

where A represents a matrix encoding each of the option shape constraints, and c is a stacked vector of call option prices, across moneyness/time-to-maturity pairs. Let $A\in {\mathbb{R}}^{m\times N}$ and $b\in {\mathbb{R}}^m$ encode the shape constraints (following Cohen et al. 2020):

(i)

Monotone in maturity:

$$\begin{array}{c}\hfill c({\tau }_{i+1},m)-c({\tau }_i,m)\ge 0.\end{array}$$

(1)

(ii)

Monotone in strike:

$$\begin{array}{c}\hfill c(\tau ,m_j)-c(\tau ,m_{j+1})\ge 0.\end{array}$$

(2)

(iii)

Convex in strike:

$$\begin{array}{c}\hfill {\displaystyle \frac{c(\tau ,m_j)-c(\tau ,m_{j-1})}{x_j-x_{j-1}}}\le {\displaystyle \frac{c(\tau ,m_{j+1})-c(\tau ,m_j)}{x_{j+1}-x_j}},x_j:=e^{m_j}.\end{array}$$

(3)

(iv)

Bounds and intrinsic:

$$\begin{array}{cc}\hfill \mathrm{Lower}\mathrm{envelope}:& 0\le c(\tau ,m)\le 1,c(\tau ,m)\ge {\left(1-e^m\right)}_{+}.\hfill \end{array}$$

(4)

On a non-uniform $x=e^m$ grid, (3) is implemented as a linear inequality

$${\displaystyle \frac{1}{x_j-x_{j-1}}}{c}_{j-1}-\left({\displaystyle \frac{1}{x_j-x_{j-1}}}+{\displaystyle \frac{1}{x_{j+1}-x_j}}\right)c_j+{\displaystyle \frac{1}{x_{j+1}-x_j}}{c}_{j+1}\ge 0.$$

When adjacent $\tau$ rows have misaligned m grids, (1) evaluates $c({\tau }_{i+1},m)$ by linear interpolation on the ${\tau }_{i+1}$ row.

To speed up repeated projections, redundant rows of $Ac\ge b$ are eliminated by LP tests, a method which comes from Caron et al. (1989). For each row i, solve

$$\underset{c}{max}{A}_{i}c\mathrm{s}.\mathrm{t}.A_{-i}c\ge b_{-i},$$

where $A_{-i}$ and $b_{-b}$ refer to the matrix A with its ith row removed and the vector b with its ith element removed. We are trying to test if the ith constraint is redundant, so if the optimum $\le b_i+\mathrm{tol}$, row i is redundant and removed.

Subsequently, each time slice is repaired by the nearest feasible point

$$\begin{array}{c}\hfill {\hat{c}}_t=arg\underset{c\in {\mathbb{R}}^N}{min}\frac{1}{2}{\parallel c-c_t^{\mathrm{raw}}\parallel }_2^2\mathrm{s}.\mathrm{t}.Ac\ge b,0\le c\le 1.\end{array}$$

(5)

This strictly enforces all static constraints while minimally perturbing the raw panel. When bid/ask data is available, we perform a spread-aware L¹ projection, biasing adjustments to remain within the spread by introducing non-negative up/down slack ${ϵ}^{+},{ϵ}^{-}$ and weights $w^{+},w^{-}$ proportional to inverse half-spreads:

$$\begin{array}{cc}\hfill \underset{c,{ϵ}^{\pm }\ge 0}{min}& {\displaystyle \sum _{n=1}^N}\left(w_n^{+}{ϵ}_n^{+}+w_n^{-}{ϵ}_n^{-}\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& c=c_t^{\mathrm{mid}}+{ϵ}^{+}-{ϵ}^{-},Ac\ge b,0\le c\le 1,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & w_n^{+}={\displaystyle \frac{1}{max\{c_{t,n}^{\mathrm{ask}}-c_{t,n}^{\mathrm{mid}},\delta \}}},w_n^{-}={\displaystyle \frac{1}{max\{c_{t,n}^{\mathrm{mid}}-c_{t,n}^{\mathrm{bid}},\delta \}}}.\hfill \end{array}$$

(8)

The spread-aware L¹ adjustment matches the method of Cohen et al. (2020) exactly.

After these steps, the repaired surfaces ${\left\{{\hat{c}}_t\right\}}_t$ are static-arbitrage-free and feed all downstream steps. Because each time slice solves a convex program with warm starts, the method scales to high-frequency data while remaining faithful to the observed bid–ask data. Figure 7 highlights an example of the adjustments made to create an arbitrage-free surface. We can see that the deep out-of-the-money options are the ones that have the largest corrections. This can make sense, as these options are typically the ones with the largest bid–ask spread and are also the ones with the most unreliable (and potentially stale) quotes.

3.4. Linear Factor Representation

Now that we have a series of arbitrage-free option surfaces, we notice that it is high-dimensional and that there are likely redundancies in information contained within the high-dimensional representation, in the sense that we can expect prices on the surface to be correlated in some way. As such it is useful and efficient to decompose the surface into a low-dimensional representation.

We follow the method presented in Cohen et al. (2023a), where it is labelled Algorithm 1. This algorithm is designed to decompose the surface into latent factors that are designed to minimise arbitrage in the reconstruction.

Suppose that the surface can be described by d latent factors ${\xi }_{1t},\dots ,{\xi }_{dt}$, collected into the factor vector ${\Xi }_t={({\xi }_{1t},\dots ,{\xi }_{dt})}^{\top }\in {\mathbb{R}}^d$. We link these factors to the vector of call option prices via a linear decomposition:

$$\begin{array}{c}\hfill \tilde{c}=G_0(\tau ,m)+{\displaystyle \sum _{i=1}^d}{G}_i(\tau ,m){\xi }_{it}.\end{array}$$

(9)

Here $G_0(\tau ,m)\in {\mathbb{R}}^N$ is a baseline price vector (the temporal mean of the arbitrage-repaired surfaces) and each $G_i(\tau ,m)\in {\mathbb{R}}^N$, $i=1,\dots ,d$, is a vector of linear loading coefficients indexed over the N lattice nodes $(\tau ,m)$, mapping the scalar factor ${\xi }_{it}$ to price perturbations across the surface. Stacking the loading vectors column-wise gives the $N\times d$ loading matrix $M=[G_1\mid G_2\mid \dots \mid G_d]$, so that (9) can be written compactly as $\tilde{c}=G_0+M{\Xi }_t$.

The loading vectors $G_i$ and associated factor scores ${\xi }_{it}$ are obtained via a modified principal component analysis (PCA) procedure, following Algorithm 1 of Cohen et al. (2023a). The decomposition is performed sequentially, with each successive factor chosen to optimise a different objective:

• Minimal Dynamic Arbitrage: The first loading vector $G_1$ (denoted $G_{\mathrm{dyn}}$) is chosen to align with the leading principal component of the empirical z-factor, i.e., the deterministic drift operator applied to the reconstructed surface (see Appendix B). This ensures the dominant direction of surface evolution is representable by the factor model, thereby minimising the violation of the HJM-type drift restriction. It is worth noting that this approach to minimising dynamic arbitrage in the model is by the construction of an approximation, justified by the fact that dynamic arbitrage opportunities are only practicable if they are relatively large (given that truly continuous-time trading is not possible and any such arbitrage strategies would be subject to substantial model risk).
• Statistical Accuracy: The second loading vector $G_2$ (denoted $G_{\mathrm{stat}}$) is the leading principal component of the residual price variation after projecting out $G_1$. This captures the direction of the maximum remaining variance, ensuring accurate reconstruction.
• Minimal Static Arbitrage: The third loading vector $G_3$ (denoted $G_{\mathrm{sa}}$) is chosen from the residual subspace to minimise the static-arbitrage violations (measured via the constraint matrix A) in the reconstructed surfaces. A hinge-penalty objective penalises violations of $A\tilde{c}\ge 0$ during the optimisation of this direction.

The loading matrix is therefore partitioned as $M=[G_{\mathrm{dyn}}\mid G_{\mathrm{stat}}\mid G_{\mathrm{sa}}]$ under a three-factor model ($d=3$).

Figure 8 illustrates the three loading vectors and the baseline $G_0$ under a three-factor model. While the stochastic loading vectors are difficult to interpret individually, the baseline $G_0$ exhibits the standard call surface shape one would expect. Similar to the approach by Cohen et al. (2023a), all loading vectors $G_i$ ($i\ge 1$) are hard-coded to zero at maturity $\tau =0$ to ensure that the terminal payoff condition is satisfied.

4. Neural SDE Modelling

This section builds on the method of Cohen et al. (2023a), applying it to high-frequency cryptocurrency options data to obtain a data-driven, arbitrage-free model taking in the time series of available cryptocurrency option market data.

4.1. Stochastic Dynamics of Market Factors

We first posit an SDE for the evolution of our market factors under the statistical (a.k.a. “objective”, “physical” or “real-world”) probability measure, commonly denoted as the $\mathbb{P}$ measure. Specifically, the dynamics of the d-dimensional factor vector ${\Xi }_t={({\xi }_{1t},\dots ,{\xi }_{dt})}^{\top }$ introduced in Section 3.4 are assumed to be governed by the SDE,

$$d{\Xi }_t={\mu }_{\theta }({\Xi }_t,t)dt+{\sigma }_{\varphi }({\Xi }_t,t)d{W}_t,$$

(10)

where ${\mu }_{\theta }$ is the drift vector, ${\sigma }_{\varphi }$ is the diagonal diffusion matrix and $W_t$ is a standard Brownian motion. Each of ${\mu }_{\theta }$, ${\sigma }_{\varphi }$ and $W_t$ is d-dimensional, where d refers to the number of factors chosen from our factor decoding, as described in Section 3.4. Our diffusion matrix is assumed to be diagonal, meaning we assume that the noise terms between our $\Xi$ factors are uncorrelated, and with the 24 h of data, this assumption seems to hold relatively well when analysing the residual factor plots; see Section 4.4.2.

In our data-driven approach, we represent the drift and diffusion components of this SDE with neural networks with weights $\theta$ and $\varphi$ respectively. The specific architecture of the neural networks is detailed in Appendix E.

We can formulate the loss function for training the neural networks in this model by considering the Euler–Maruyama discretisation for a stochastic differential equation. Under this expansion, we have a defined transition. Let $\Delta t_k:=t_{k+1}-t_k$ and $\Delta W_k:=W_{t_{k+1}}-W_{t_k}\sim \mathcal{N}(0,\Delta t_k{I}_d)$. The Euler–Maruyama step for the Itô SDE is

$$\begin{array}{cc}\hfill {\Xi }_{k+1}& ={\Xi }_k+{\mu }_{\theta }({\Xi }_k,t_k)\Delta t_k+{\sigma }_{\varphi }({\Xi }_k,t_k)\Delta W_k,\hfill \\ \hfill & ={\Xi }_k+{\mu }_{\theta }({\Xi }_k,t_k)\Delta t_k+{\sigma }_{\varphi }({\Xi }_k,t_k)\sqrt{\Delta t_k}{\epsilon }_k,{\epsilon }_k\sim \mathcal{N}(0,I_d).\hfill \end{array}$$

(11)

Conditionally on ${\Xi }_k$, the transition density is Gaussian:

$$\begin{array}{c}{\Xi }_{k+1}\mid {\Xi }_k\sim \mathcal{N}\left(m_k,\Delta t_k{\Gamma }_k\right),\hfill \\ \hfill m_k:={\Xi }_k+{\mu }_{\theta }({\Xi }_k,t_k)\Delta t_k,{\Gamma }_k:={\sigma }_{\varphi }({\Xi }_k,t_k){\sigma }_{\varphi }{({\Xi }_k,t_k)}^{\top }.\end{array}$$

(12)

We can then formulate an expectation maximisation (EM) loss function. Define the observed increment $\Delta {\Xi }_k:={\Xi }_{k+1}-{\Xi }_k$. The one-step negative log-likelihood (NLL) is

$$\begin{array}{c}{\ell }_k(\theta ,\varphi )=\frac{1}{2}[{\left(\Delta {\Xi }_k-{\mu }_{\theta }({\Xi }_k,t_k)\Delta t_k\right)}^{\top }{\left(\Delta t_k{\Gamma }_k\right)}^{-1}\left(\Delta {\Xi }_k-{\mu }_{\theta }({\Xi }_k,t_k)\Delta t_k\right)\hfill \\ \hfill +logdet\left(\Delta t_k{\Gamma }_k\right)+dlog\left(2\pi \right)],\end{array}$$

(13)

so that for a path ${\left\{{\Xi }_k\right\}}_{k=0}^K$, the total loss function value is

$$\begin{array}{c}\hfill {\mathcal{L}}_{\mathrm{NLL}}(\theta ,\varphi )={\displaystyle \sum _{k=0}^{K-1}}{\ell }_k(\theta ,\varphi ).\end{array}$$

(14)

This NLL function now formulates our test loss, where minimising the NLL is equivalent to maximising the likelihood of seeing the next step given the current data. This follows the classical EM framework from Dempster et al. (1977) and is the neural SDE learning strategy from Dridi et al. (2021). This is structurally analogous to the maximisation/inference loop approach from Greff et al. (2017).

This training scheme does make the assumption that the transition/discretisation of the process is Gaussian, and at ultra-high frequency (sub-second level intervals), this assumption does not hold. When using 24 h of high-frequency cryptocurrency options data, the prices were aggregated, by taking the median price, to minutely increments in order for goodness-of-fit tests to look reasonable. The training results at high frequency and the associated discussion can be found in Appendix C.

One can also consider using different expansions or quasi-likelihoods as the loss term for the SDE. Aït-Sahalia (2002) develops a closed-form approximation for the likelihood of discretely observed diffusion processes, addressing the fact that most continuous-time SDEs lack an analytically tractable transition density. The approach constructs a sequence of explicit Hermite polynomial expansions that converge uniformly to the true but unknown transition density. By successively correcting the leading Gaussian term through higher-order terms in the sampling interval $\Delta t$, this method yields a quasi-maximum likelihood estimator that remains asymptotically equivalent to the true MLE. Conceptually, it extends the Euler–Maruyama Gaussian transition by incorporating higher-order corrections that depend on the local drift, diffusion, and their spatial derivatives, thereby capturing the non-linearity and skewness inherent in finite-time transitions. We explored using this approximation as a higher-order likelihood correction to the Gaussian transition in (12), attempting to capture the non-Gaussianity of the high-frequency intervals, and the training results in this case are presented and discussed in Appendix D.

4.2. Arbitrage-Free Factor Representation

In order to constrain the outputs of our model to be free of static arbitrage, we must first consider what these constraints mean in our factor space. As our constraints are linear inequalities and our factor representation of option prices is also linear, these constraints have a representation in the factor space. Recall the linear constraints in Section 2.2, $Ac\ge 0$ and the linear factor decomposition, $\tilde{c}=G_0(\tau ,m)+{\sum }_{i=1}^d{G}_i(\tau ,m){\xi }_{it}$. Substituting this decomposition into the constraint system gives

$$A\tilde{c}=A{G}_0+{\displaystyle \sum _{i=1}^d}A{G}_i{\xi }_{it}\ge 0.$$

Hence, in the factor space, the arbitrage-free region is characterised by all factor vectors ${\Xi }_t={({\xi }_{1t},\dots ,{\xi }_{dt})}^{\top }$, satisfying

$$A_G{\Xi }_t\ge b_G,A_G:=[A{G}_1,A{G}_2,\dots ,A{G}_d],b_G:=-A{G}_0.$$

This defines a convex polytope, or region of space, in the factor space, within which any combination of factors yields option prices that respect all static-arbitrage constraints. During training or simulation, we therefore enforce that the decoded factors ${\Xi }_t$ remain inside this feasible region

$${\mathcal{P}}_{\mathrm{arb}}=\left\{{\Xi }_t|A_G{\Xi }_t\ge b_G\right\},$$

ensuring that the reconstructed surface is statically arbitrage-free.

Removing redundant linear constraints, as described in Section 3.3, we can plot our static-arbitrage-free convex polytope and the associated $\xi$ factors to visually inspect if our factor decoding process in Section 3.4 has been done correctly. Figure 9 presents scores in a two-factor model and the corresponding static no-arbitrage polygon. Inspecting this image we can see that the vast majority of our latent $\xi$ factors lie within the acceptable polygon. Those few scores, which lie either on the boundary or just outside the polygon, are there due to the hinge-penalty-based optimisation done to reduce static arbitrage in option price reconstruction in the factor decoding process given in Section 3.4.

4.3. Constrained Networks and No-Static-Arbitrage Guarantees

Given the specified region of space in which the $\Xi$ factors can evolve without option prices allowing static arbitrage, we can enforce constraints on the neural SDE to ensure that the process remains within the required polytope. Cohen et al. (2023a) provide the method to achieve this using Friedman and Pinsky conditions for the non-attainability of a boundary for an SDE (Friedman and Pinsky 1973).

Definition 1 

(Friedman–Pinsky non-attainability conditions). Consider an Itô diffusion in ${\mathbb{R}}^d$,

$$d{\Xi }_t=\mu \left({\Xi }_t\right)dt+\sigma \left({\Xi }_t\right)d{W}_t,$$

and let $\mathcal{P}=\{\Xi \in {\mathbb{R}}^d:a_i^{\top }\Xi \ge b_i,i=1,\dots ,n_f\}$ be a convex polytope defined by $n_f$ linear constraints (i.e., $n_f$ faces). For each face i with the boundary function

$$q_i(\Xi ):=a_i^{\top }\Xi -b_i=0,$$

define the normal drift and diffusion components near the boundary as

$${\mu }_i(\Xi ):=a_i^{\top }\mu (\Xi ),{\sigma }_i^2(\Xi ):=a_i^{\top }\Gamma (\Xi )a_i,\Gamma (\Xi ):=\sigma (\Xi )\sigma {(\Xi )}^{\top }.$$

Then, a sufficient condition for the boundary $\partial \mathcal{P}$ to be non-attainable (i.e., the process remains in $\mathcal{P}$ almost surely if started inside) is that, for each i,

$$\begin{array}{cc}\hfill & \left(i\right){\mu }_i(\Xi )\ge 0\mathit{whenever}{q}_i(\Xi )=0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & \left(\mathit{ii}\right){\sigma }_i^2(\Xi )=0\mathit{whenever}{q}_i(\Xi )=0.\hfill \end{array}$$

(16)

That is to say, the drift of the SDE points inward at the boundary, and the diffusion vanishes in the direction normal to it.

Cohen et al. (2023a) enforce these conditions via a smooth Lipschitz function. This function will shrink the diffusion as it gets closer to the boundary and slowly add drift in the opposite direction. Choosing such a function to be smooth and Lipschitz is important so as not to break backpropagation for training the neural networks. Recall the linear price map $\tilde{c}=G_0+M{\Xi }_t$ from (9), where ${\Xi }_t\in {\mathbb{R}}^d$ is the factor vector and $M=[G_{\mathrm{dyn}}\mid G_{\mathrm{stat}}\mid G_{\mathrm{sa}}]\in {\mathbb{R}}^{N\times d}$ is the loading matrix. Given the lattice constraints $A\tilde{c}\ge b$, we map them into factor space as

$$H{\Xi }_t\ge h,H:=AM,h:=b-A{G}_0,$$

and augment with a data-driven box $L\le {\Xi }_t\le U$. The box bounds $L,U\in {\mathbb{R}}^d$ are obtained component-wise from the training factor scores as $L_k=Q_{\alpha }\left({\left\{{\xi }_{k,t}\right\}}_t\right)$ and $U_k=Q_{1-\alpha }\left({\left\{{\xi }_{k,t}\right\}}_t\right)$ for a small quantile level $\alpha$ (defaulting to component-wise min/max when quantile trimming is not applied). These box constraints are converted into additional half-space rows appended to $H{\Xi }_t\ge h$, so that the diffusion shrinkage and inward drift correction treat empirical range boundaries identically to the static-arbitrage faces. Let $n_f$ denote the total number of half-space constraints (arbitrage faces plus box faces) after this augmentation. Rows of H are row-normalised when building the shrink geometry, so the signed distance of ${\Xi }_t$ to face i is

$${\rho }_i\left({\Xi }_t\right):={\displaystyle \frac{H_i{\Xi }_t-h_i}{\parallel H_i\parallel }}=H_i{\Xi }_t-h_i(\mathrm{since}\parallel H_i\parallel = 1).$$

Following Cohen et al. (2023a) the Lipschitz shrink profile is given by

$$\varphi \left(x\right)={\displaystyle \frac{x_{+}}{1+x_{+}}},x_{+}:=max\{x,0\}.$$

Near the boundary ${\rho }_i\left({\Xi }_i\right)\downarrow 0$, we have $\varphi \left({\rho }_i\right)\downarrow 0$, far inside the polytope $\varphi \left({\rho }_i\right)\uparrow 1$.

At a given state ${\Xi }_t$ we select the $k=min\{d,n_f\}$ most relevant faces (smallest ${\rho }_i$), build an orthonormal frame $Q\left({\Xi }_t\right)\in {\mathbb{R}}^{d\times d}$ whose first k columns point along the corresponding unit normals, and compute shrink coefficients $\epsilon \left({\Xi }_t\right)\in {[0,1]}^k$ with components ${\epsilon }_j\left({\Xi }_t\right)=\varphi \left({\rho }_{i_j}\left({\Xi }_t\right)\right)$. This yields two equivalent shrink operators (chosen by mode):

-

Matrix shrink:

$$P\left({\Xi }_t\right)=Q\left({\Xi }_t\right)\left[\begin{array}{cc}\mathrm{diag}\left(\epsilon \left({\Xi }_t\right)\right)& 0\\ 0& I_{d-k}\end{array}\right]Q{\left({\Xi }_t\right)}^{\top }\in {\mathbb{R}}^{d\times d},$$

so the effective diffusion is

$${\Gamma }_{\mathrm{eff}}\left({\Xi }_t\right)=P\left({\Xi }_t\right)\Gamma \left({\Xi }_t\right)P{\left({\Xi }_t\right)}^{\top }.$$

-

Diagonal shrink (default, diagonal noise): Compute elementwise multipliers $s\left({\Xi }_t\right)\in {[0,1]}^d$ so that $\mathrm{diag}\left(s\left({\Xi }_t\right)\right)\sigma \left({\Xi }_t\right)$ best matches $P\left({\Xi }_t\right)\mathrm{diag}\left(\sigma \left({\Xi }_t\right)\right)$ in a covariance sense. The effective diffusion used inside the model is

$${\sigma }_{\mathrm{eff}}\left({\Xi }_t\right)=s\left({\Xi }_t\right)\odot \sigma \left({\Xi }_t\right).$$

In both cases, along any active face normal $a_i$, one has $a_i^{\top }{\Gamma }_{\mathrm{eff}}\left({\Xi }_t\right)a_i\to 0$ as ${\rho }_i\left({\Xi }_t\right)\downarrow 0$, satisfying (16). The same shrink can also be applied at the loss level: rather than modifying the diffusion $\sigma$ directly, the constraint geometry is absorbed into a state-dependent preconditioner $\mathsf{\Omega }\left({\Xi }_t\right)\in {\mathbb{R}}^{d\times d}$ (derived from $P\left({\Xi }_t\right)$) that transforms the observed increments and likelihood. Specifically, the tuple $(\mathsf{\Omega },det\mathsf{\Omega },{\mathsf{\Omega }}^{-\top }\Delta \Xi )$ is passed to the negative log-likelihood (13), so that the Gaussian transition is evaluated in the preconditioned coordinate system where the boundary constraints are automatically respected.

To enforce (15) via an inward drift correction, we precompute, along the observed factor path $X_t$, per-time inward directions and normalised boundary distances using

$$({\mathrm{dirs}}_t,{\epsilon }_t^{\mu })=(W,b,X_{·};{\rho }_{★},{\epsilon }_{★}^{\mu }),$$

where $(W,b)$ are the row-normalised half-spaces from the polytope output. At the training time, for a batch indexed by t, the raw drift $\mu (·)$ is corrected by

$${\mu }_{\mathrm{corr}}\left(X_t\right)=\mu \left(X_t\right)+\Delta {\mu }_t,\Delta {\mu }_t=\sum _k{w}_{t,k}{\mathrm{dirs}}_{t,k},w_{t,k}={\displaystyle \frac{{\lambda }_{\mathrm{bc}}}{{\epsilon }_{t,k}^{\mu }+{\epsilon }_0}}\mathbf{1}\{{\epsilon }_{t,k}^{\mu }\le {\tau }_{\mathrm{cut}}\},$$

with user-selectable parameters $({\lambda }_{\mathrm{bc}},{\epsilon }_0,{\tau }_{\mathrm{cut}})$ and an optional norm cap $\parallel \Delta {\mu }_t\parallel \le \mathrm{cap}$.

Together, the smooth diffusion shrink guarantees $a_i^{\top }{\Gamma }_{\mathrm{eff}}\left({\Xi }_t\right)a_i\to 0$ as the boundary is approached, and the inward drift correction ensures $a_i^{\top }{\mu }_{\mathrm{corr}}\left({\Xi }_t\right)\ge 0$ there; hence the Friedman–Pinsky non-attainability conditions are satisfied for the learned SDE on the static-arbitrage polytope.

4.4. Training Results on Single-Day Data

This section will discuss and review training results and goodness-of-fit tests of this neural SDE model on our Deribit cryptocurrency options data from 22 May 2025.

4.4.1. Loss Curve

Illustrated in Figure 10, each loss curve, with and without the drift/diffusion constraints, suggests convergence. However, the loss with the constraints is far lower than the loss without them. Additionally, the gap between the train and test loss is smaller. This is interpreted as suggesting that the constraints are indeed working as expected, reducing the state space of our $\Xi$ factor values.

Here, the train/test split is 80/20. Meaning, the first 80% of the 24 h period is used as training, and the remaining 20% is the test, or out-of-sample, region.

4.4.2. Residual Plots

Following the estimation of the neural SDE, we assess the adequacy of the fitted drift and diffusion functions by examining the one-step standardised residuals. Under the Euler–Maruyama discretisation of the stochastic differential equation

$$d{Y}_t={\mu }_{\theta }\left(Y_t\right)dt+{\sigma }_{\varphi }\left(Y_t\right)d{W}_t,$$

the discrete-time transition between observations can be approximated as

$$Y_{t+\Delta t}\approx Y_t+{\mu }_{\theta }\left(Y_t\right)\Delta t+{\sigma }_{\varphi }\left(Y_t\right)\sqrt{\Delta t}{\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,I_d),$$

where ${\epsilon }_t$ are the model innovations implied by the fitted dynamics. Given observed data ${\left\{Y_t\right\}}_{t=1}^T$, we therefore compute the empirical residuals as

$${\hat{\epsilon }}_t={\displaystyle \frac{Y_{t+\Delta t}-Y_t-{\mu }_{\theta }\left(Y_t\right)\Delta t}{{\sigma }_{\varphi }\left(Y_t\right)\sqrt{\Delta t}}}.$$

If the estimated SDE correctly captures the conditional mean and variance of the data, then ${\hat{\epsilon }}_t$ should be approximately independent and standard normal across time and dimensions.

These residuals thus serve as a key diagnostic measure: quantile–quantile (QQ) plots of ${\hat{\epsilon }}_t$ against a standard normal distribution test for deviations from Gaussianity (such as skewness or excess kurtosis), while pairwise scatter plots of ${\hat{\epsilon }}_t^{\left(i\right)}$ and ${\hat{\epsilon }}_t^{\left(j\right)}$ assess the cross-sectional independence across latent factors.

From the QQ plots in Figure 11, we can see that the assumption of Gaussian transition increments from our Euler–Maruyama training step is approximately correct, though not perfect. There is still some overdispersion at the tails and underdispersion at the centrality. This suggests that deviations from our model predictions are still possible, which makes sense considering the cryptocurrency market’s higher volatility.

In addition to marginal normality checks, we also compute the pairwise correlations between residual components to evaluate the cross-sectional independence of the model’s innovations. For each pair of factors $(i,j)$, the sample correlation coefficient is given by

$${\rho }_{ij}={\displaystyle \frac{\mathrm{Cov}({\hat{\epsilon }}^{\left(i\right)},{\hat{\epsilon }}^{\left(j\right)})}{\sqrt{\mathrm{Var}\left({\hat{\epsilon }}^{\left(i\right)}\right)\mathrm{Var}\left({\hat{\epsilon }}^{\left(j\right)}\right)}}}.$$

Under a correctly specified diffusion, the residuals should be uncorrelated across factors, reflecting that each dimension of the learned SDE captures an independent source of risk or market variation.

Analysing the residuals as plotted in Figure 12, we can see that there is some slight correlation between the “dynamic arbitrage” factor and the “statistical accuracy” factor. In a sense, this is not unexpected, as in testing, we noticed that the vast majority of variance is explained with a two-factor model. When training with two factors (one statistical-accuracy factor and one static-arbitrage factor), we observe an $R^2$ value of 99.5–99.8%, indicating that almost all variation in the observed option prices is captured by the model’s reconstructed surfaces. In this context, $R^2$ measures the proportion of the total price variance explained by the factor representation, reflecting both the completeness and goodness-of-fit of the decomposition. In a way this is unsurprising, as other martingale stochastic volatility models are also two-factor models (see, e.g., Hagan et al. 2002; Heston 1993). Aside from that, the correlations are low, and the residuals are centre-aligned.

4.5. Training Results of a Week of Data

This section reports the calibration of the arbitrage-free neural SDE on the extended dataset introduced in Section 3.1, comprising the trading week of 7–14 September 2025, whose timestamp-spacing and underlying-price profiles are shown in Figure 3 and Figure 4. It complements the 24 h results above and serves two purposes: to confirm that the pre-processing and decoding pipeline of Section 3 scales to a longer horizon, and to scrutinise the distributional assumptions underlying the Gaussian likelihood and the diagonal diffusion when more data, spanning more market regimes, are present.

4.5.1. Liquid Lattice for the Week

Figure 13 shows the liquid lattice $\mathcal{N}$ obtained for the week through the construction of Section 3.2. It exhibits the characteristic shape of an options lattice: a dense concentration of nodes at short maturities ($\tau <0.1$) and near the money, with the traded moneyness range widening at longer maturities, where nodes extend from roughly $m=-0.35$ to $m=1.2$. The maturities cluster into a small number of discrete buckets ($\tau \approx 0.02$–$0.08$, $0.15,0.22,0.30,0.55,0.80$ years), reflecting the maturity-clustering step of Algorithm 1, and the lattice contains 39 nodes in total. Relative to the single-day lattice of the main text, the week’s lattice is denser and spans a wider moneyness range, because more $(\tau ,m)$ combinations become liquid once aggregated over five trading days; as shown in Appendix F.3, this is precisely the regime in which the factor representation is best identified.

As Figure 14 shows, on this data set the training and test negative log-likelihood is also well-behaved: The objective function converges within ≈130 epochs and then flattens out, leaving a small and stable train/test gap with no deterioration of the test loss.

4.5.2. Residual Diagnostics: Marginal Normality

The likelihood treats each one-step factor increment as conditionally Gaussian. To assess this, Figure 15 presents normal quantile–quantile plots of the standardised one-step residuals of the three factors ($G_{\mathrm{dyn}}$, $G_{\mathrm{stat}}$ and $G_{\mathrm{sa}}$).

We can see that while all three are still approximately linear, there are fatter tails in the QQ plot in comparison to when training over only a single day. The dynamic factor is the most peaked, with its near-flat central segment indicating a concentration of small increments punctuated by occasional large moves.

4.5.3. Residual Diagnostics: Cross-Factor Dependence

The diffusion matrix is specified to be diagonal (Section 3.4), which presumes that the driving noises of the factors are mutually uncorrelated. Figure 16 shows the pairwise scatter of the factor residuals together with their Pearson correlations. The dynamic and static-arbitrage residuals are essentially uncorrelated ($\rho =0.02$), and the dynamic and statistical residuals are only weakly correlated ($\rho =-0.16$), whereas the statistical and static-arbitrage residuals retain a moderate correlation ($\rho =0.27$). The cross- or star-shaped point clouds, with mass concentrated along the axes and heavy tails, are consistent with weakly dependent, leptokurtic marginals. The diagonal specification is thus broadly supported but not exact over the week: the residual correlation is larger than in the 24 h analysis of Section 4.4.2, suggesting that as the sample spans more regimes, a small amount of cross-factor dependence—chiefly between the statistical and static-arbitrage factors—is left unmodelled by the diagonal diffusion.

5. Applications

This section considers two fundamental downstream applications: hedging and risk management, in the form of Value-at-Risk (VaR). These represent the most immediate and practically meaningful uses of a market pricing model. Both tasks provide direct validation of the model’s ability to capture market dynamics: hedging evaluates how well the learned dynamics can replicate real-world sensitivities and risk exposures, while VaR assesses the model’s capacity to simulate realistic loss distributions under stochastic evolution. However, this is not to say that this is an exhaustive list of the model’s capabilities, as it provides a complete stochastic representation of how the implied volatility and option prices evolve through time. In principle, this framework can be extended to a wide range of quantitative finance tasks—including scenario generation, risk-neutral valuation, stress testing, derivative portfolio simulation, and even data-driven model calibration or the discovery of latent market factors.

5.1. Market Simulation and VaR

Cohen et al. (2023b) detail the methodology on how to create VaR estimates for option portfolios using neural SDE market models. In essence, the neural SDE is a generative model, meaning that it can simulate the learned $\Xi$ market factors. Via a factor decoding process (i.e., moving from the low-dimensional factor space back to a higher-dimensional option surface), simulated prices for various option portfolios can be obtained. As we now have a distribution of potential returns for the options on our defined liquid lattice, we can naturally compute VaR estimates (and hence other risk metrics, such as expected shortfall) using this distribution.

5.1.1. Market Simulation Methodology

Let ${\Xi }_t\in {\mathbb{R}}^d$ denote the vector of latent market factors, evolved according to the trained neural SDE:

$$d{\Xi }_t={\mu }_{\theta }\left({\Xi }_t\right)dt+{\sigma }_{\varphi }\left({\Xi }_t\right)d{W}_t,{\sigma }_{\varphi }\left({\Xi }_t\right):=diag\left(\sigma \left({\Xi }_t\right)\right)\ge 0,$$

where ${\mu }_{\theta }$ and ${\sigma }_{\varphi }$ represent the learned drift and diffusion networks, and $W_t$ is a standard Brownian motion. During simulation, we again ensure that the factor process ${\Xi }_t$ remains within the static-arbitrage polytope defined by the half-space constraints:

$${\mathcal{P}}_{\mathrm{arb}}=\{{\Xi }_t\mid A_G{\Xi }_t\ge b_G\}.$$

For each face i of the polytope, define the signed distance

$${\rho }_i\left({\Xi }_t\right)=A_{G,i}{\Xi }_t-b_{G,i},i=1,\dots ,m,$$

which measures proximity to the ith constraint boundary.

As before, we apply a smooth Lipschitz function ${\phi }_{\sigma }:{\mathbb{R}}_{+}\to [0,1]$ to modulate the effective diffusion near the boundary:

$${\epsilon }_i\left({\Xi }_t\right)={\phi }_{\sigma }\left({\rho }_i\left({\Xi }_t\right)\right)={\displaystyle \frac{{\rho }_i\left({\Xi }_t\right)}{1+{\rho }_i\left({\Xi }_t\right)}}.$$

Let $Q\left({\Xi }_t\right)$ be an orthonormal basis spanning the k most active faces (smallest ${\rho }_i$), and define

$$P\left({\Xi }_t\right)=Q\left({\Xi }_t\right)\left[\begin{array}{cc}diag\left(\epsilon \left({\Xi }_t\right)\right)& 0\\ 0& I_{d-k}\end{array}\right]Q{\left({\Xi }_t\right)}^{\top }.$$

The effective diffusion is then given by

$${\sigma }_{\mathrm{eff}}\left({\Xi }_t\right)=diag\left(s\left({\Xi }_t\right)\odot \sigma \left({\Xi }_t\right)\right),$$

where $s\left({\Xi }_t\right)\in {[0,1]}^d$ are elementwise shrink coefficients matching $P\left({\Xi }_t\right)\sigma \left({\Xi }_t\right)$ in covariance. By construction, the normal component $A_{G,i}^{\top }{\Gamma }_{\mathrm{eff}}\left({\Xi }_t\right)A_{G,i}\to 0$ as ${\rho }_i\left({\Xi }_t\right)\downarrow 0$, satisfying the diffusion non-attainability condition.

Again following our previous procedure, a smooth correction is added to ensure the drift points inward at the boundary. Define a Lipschitz function ${\phi }_{\mu }\left(\rho \right)={\displaystyle \frac{1}{1+\rho }}$ and inward directions $d_i$ associated with each active face normal $A_{G,i}$. The corrected drift is:

$${\mu }_{\mathrm{corr}}\left({\Xi }_t\right)={\mu }_{\theta }\left({\Xi }_t\right)+\sum _{i\in \mathcal{I}\left({\Xi }_t\right)}{\omega }_i\left({\Xi }_t\right)d_i,{\omega }_i\left({\Xi }_t\right)={\displaystyle \frac{{\left[-A_{G,i}^{\top }{\mu }_{\theta }\left({\Xi }_t\right)-{\phi }_{\mu }\left({\rho }_i\left({\Xi }_t\right)\right)\right]}_{+}}{\langle d_i,A_{G,i}\rangle }},$$

ensuring $A_{G,i}^{\top }{\mu }_{\mathrm{corr}}\left({\Xi }_t\right)\ge 0$ along all active faces.

Given a discretisation step $\Delta t$, each Euler–Maruyama step evolves as:

$${\Xi }_{t+\Delta t}={\Xi }_t+{\mu }_{\mathrm{corr}}\left({\Xi }_t\right)\Delta t+{\sigma }_{\mathrm{eff}}\left({\Xi }_t\right)\sqrt{\Delta t}{\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,I_d).$$

Running B Monte Carlo paths of ${\Xi }_t$ generates a sample distribution of factor evolutions ${\left\{{\Xi }_t^{\left(b\right)}\right\}}_{b=1}^B$.

5.1.2. VaR of Option Portfolios

Table 1. Value-at-Risk (VaR) comparison: neural SDE vs. historical simulation.

Portfolio	VaR NSDE 99	VaR HS 99	VaR NSDE 95	VaR HS 95	Description
Short 10% strangle ∼1 d	−0.018899	−0.003865	−0.013648	−0.002575	Short 10% OTM call and 10% OTM put, $\tau \approx 1$ d.
Short 10% strangle ∼7 d	−0.014940	−0.003988	−0.010646	−0.002645	Same short strangle with maturity ∼7d.
Long ATM straddle ∼1 d	−0.014211	−0.002680	−0.013370	−0.001680	Long ATM call + long ATM put, $\tau \approx 1$ d.
Risk-reversal ±10% ∼1 d	−0.006106	−0.003822	−0.006106	−0.002410	Long 10% OTM call, short 10% OTM put (directional skew), $\tau \approx 1$ d.
Put spread 5–10% OTM ∼1 d	−0.004825	−0.002017	−0.003476	−0.001376	Long 10% OTM put, short 5% OTM put; same expiry (debit put spread).
Iron condor 10/20% ∼30 d	−0.004094	−0.003429	−0.003197	−0.002309	Short 10%/20% OTM call and put wings (short condor), $\tau \approx 30$ d.
Butterfly 10% wings ∼7 d	−0.003645	−0.002377	−0.002677	−0.001515	Same as above (ATM long body, OTM short wings).
Fly wings = 15% $\tau$ = 1 d	−0.002920	−0.004423	−0.001855	−0.002849	Wider ±15% wing butterfly, 1 d.
Fly wings = 10% $\tau$ = 30 d	−0.002821	−0.004889	−0.002049	−0.003254	±10% wing butterfly, 30 d.
Fly wings = 15% $\tau$ = 30 d	−0.002810	−0.001905	−0.001703	−0.001362	±15% wing butterfly, 30 d.
Iron condor 10/20% ∼1 d	−0.002717	−0.001909	−0.001937	−0.001197	Short condor with inner 10% and outer 20% strikes, $\tau \approx 1$ d.
Fly wings = 10% $\tau$ = 1 d	−0.002506	−0.004034	−0.001470	−0.002632	±10% wing butterfly, 1 d.
Butterfly 10% wings ∼1 d	−0.002506	−0.004034	−0.001470	−0.002632	Equivalent short-dated butterfly.
Fly wings = 15% $\tau$ = 7 d	−0.002457	−0.001571	−0.002000	−0.001165	±15% wing butterfly, 7 d.
Calendar ATM 7 d/30 d	−0.001962	−0.000974	−0.001381	−0.000652	Long 30 d ATM, short 7 d ATM (calendar spread).
Fly wings = 5% $\tau$ = 1 d	−0.001693	−0.003198	−0.001181	−0.002135	Tight ±5% wing butterfly, 1 d.
Call spread ±5% ∼1 d	−0.001653	−0.002181	−0.001653	−0.001498	Long 5% ITM call, short 5% OTM call; $\tau \approx 1$ d.
Fly wings = 5% $\tau$ = 7 d	−0.001541	−0.003935	−0.001112	−0.002596	±5% wing butterfly, 7 d.
Call spread ±5% ∼30 d	−0.001495	−0.001580	−0.001054	−0.001022	5% call spread, 30 d.
Fly wings = 5% $\tau$ = 30 d	−0.001460	−0.001538	−0.000985	−0.000962	Tight ±5% wing butterfly, 30 d.

presents VaR results for various option portfolios, for both the neural SDE simulation and a standard historical simulation approach.

The results in Table 1 show that across all portfolios, the neural SDE simulated Value-at-Risk (VaR) estimates are consistently higher in magnitude than those obtained via historical simulation. This indicates that the neural SDE captures fatter tails and greater short-term variability in the factor dynamics, reflecting more realistic risk in high-frequency markets, such as cryptocurrency options. The largest VaR differences appear in short-maturity, short-volatility positions (e.g., short strangles and short condors), where non-linear payoffs are most sensitive to volatility clustering and jump-like behaviour. Conversely, for longer-dated or more balanced portfolios (e.g., butterflies or calendar spreads), the neural SDE and historical methods yield more comparable results, suggesting that the learned diffusion smooths risk over longer horizons.

Plotting a few examples of loss distributions in Figure 17 and comparing against a distribution from historical simulation, we can see that the neural SDE does indeed provide a different distribution. We interpret these as reflecting data-driven and arbitrage-consistent dynamics of the underlying surface. Noting that traditional historical simulation approaches assume that options, or market factors, move individually, however, we know that options move as a surface with constraints to their state, as discussed in Section 2.2. The historical simulation approach taken here, in comparison, is a simulation of the market factors; that is to say, we look at the historical distribution of returns we have (in the test period) and simulate returns by sampling from this distribution. Option prices/portfolios are simulated by a factor decoding process.

5.2. Hedging of Option Portfolios

Cohen et al. (2022) highlight how we can use the learned model to hedge a portfolio of options. While their work presents both sensitivity-based hedging and minimum-variance (MV) hedging, in our case, the applications to cryptocurrency options will be focused on the MV hedge, as it directly uses the learned neural SDE model, whereas the sensitivity-based hedge only uses the factor representation of option prices.

5.2.1. Minimum-Variance Hedging

Let ${\Xi }_t\in {\mathbb{R}}^d$ denote the latent market factors driving the option surface, and recall the linear decoder from Section 3.4,

$$\tilde{c}\left({\Xi }_t\right)=G_0+{\displaystyle \sum _{i=1}^d}{G}_i{\xi }_{it}=G_0+M{\Xi }_t,M\in {\mathbb{R}}^{N\times d},$$

(17)

where $M=[G_1\mid \cdots \mid G_d]$ is the loading matrix defined in Section 3.4. For a portfolio defined by node weights $w\in {\mathbb{R}}^N$, the portfolio’s factor exposure vector is given by

$$s=M^{\top }w\in {\mathbb{R}}^d.$$

(18)

To first order, the one-step change in portfolio value is given by

$$\Delta V_t\approx s^{\top }\Delta {\Xi }_t,\Delta {\Xi }_t:={\Xi }_{t+1}-{\Xi }_t.$$

(19)

From the learned neural SDE model, the drift and diffusion functions ${\mu }_{\theta }\left({\Xi }_t\right)$ and ${\sigma }_{\varphi }\left({\Xi }_t\right)$ define the instantaneous covariance of the latent factors as

$${\Sigma }_t^{\mathrm{base}}={\sigma }_{\varphi }\left({\Xi }_t\right){\sigma }_{\varphi }{\left({\Xi }_t\right)}^{\top }.$$

(20)

To ensure that this covariance structure respects the static-arbitrage geometry defined by the polytope constraints $A_G{\Xi }_t\ge b_G$, we apply a state-dependent linear map ${\mathsf{\Omega }}_t$ that shrinks the diffusion along the active boundary directions, as detailed in Equation (16). The resulting adjusted covariance is

$${\Sigma }_t={\mathsf{\Omega }}_t{\Sigma }_t^{\mathrm{base}}{\mathsf{\Omega }}_t^{\top }.$$

(21)

This procedure ensures that factor dynamics remain within the feasible arbitrage-free region during simulation and hedging.

Now, consider a set of k hedging instruments indexed by $\mathcal{H}=\{j_1,\dots ,j_k\}$. Let $B\in {\mathbb{R}}^{d\times k}$ denote their factor exposures,

$$B:=M_{:,\mathcal{H}}^{\top }.$$

(22)

The static minimum-variance (MV) hedge solves

$$u^{★}=arg\underset{u\in {\mathbb{R}}^k}{min}Var\left({(s-Bu)}^{\top }\Delta {\Xi }_t\right)={\left(B^{\top }\overline{\Sigma }B\right)}^{-1}{B}^{\top }\overline{\Sigma }s,$$

(23)

where $\overline{\Sigma }={\displaystyle \frac{1}{T-1}}{\sum }_t{\Sigma }_t$ is the time-averaged model-implied factor covariance. A dynamic version of this hedge recomputes $u_t^{★}$ at each time step, according to

$$u_t^{★}={\left(B^{\top }{\Sigma }_{t}B\right)}^{-1}{B}^{\top }{\Sigma }_{t}s,$$

(24)

yielding time-varying hedge weights that evolve consistently with the latent factor covariance implied by the neural SDE. In practice, the ${\Sigma }_t$ matrices are computed per step using the endogenous shrinkage ${\mathsf{\Omega }}_t$ and the trained model parameters $(\theta ,\varphi )$.

5.2.2. Hedging an Example Portfolio (a 30-Day ATM Option)

We present hedging results for a portfolio consisting of an at-the-money option with a 30-day expiry. Figure 18 showcases the root mean squared error of the neural SDE MV hedge compared with standard Black–Scholes delta and delta-vega hedging.

While we cannot say that the MV hedging procedure from the learned neural SDE model outperforms the delta or delta-vega hedge, we can state that it is comparable to these traditional hedging methods. These hedging results on cryptocurrency are similar to the results from Cohen et al. (2022), who present hedging results from traditional equities and forex markets.

6. Limitations and Conclusions

6.1. Limitations of Modelling Methods

There are several limitations that one should consider with this neural SDE modelling choice. First, the neural SDE model assumes continuous-diffusion dynamics and therefore will not reflect jumps. The data-driven nature of the neural SDEs will, to some extent, allow it to adapt to volatility clustering and heavy-tailed distributions over longer time periods, but at the scale of each single time step, the approximation by Euler/Maruyama discretisation introduces an implicit assumption of Gaussianity. A full Lévy or jump-diffusion extension of the approach presented in the present paper may be a fruitful avenue of further investigation, as it may yield a more realistic specification. Second, training was limited to a single day and a single week of data due to computational and storage constraints; extending the analysis across multiple weeks could provide stronger statistical validation. Third, although static-arbitrage constraints are enforced strictly, dynamic arbitrage is only minimised, not eliminated, and a theoretical guarantee of global consistency across time remains an open challenge. Finally, the current modelling requires a fixed-factor dimension and lattice shape. Future research should aim to address these limitations.

6.2. Conclusions

This paper provides a synthesis of several methods in the literature and presents a pipeline that takes in raw option quotes and produces a non-parametric model for the entire option surface. The presented neural SDE model adheres to static-arbitrage constraints, and the applications to risk management and hedging have been showcased. One can conclude that combining machine learning with known financial laws, e.g., arbitrage constraints, can provide non-parametric alternatives to traditional models.

Our results demonstrate the feasibility of applying the arbitrage-free neural SDE market model of Cohen et al. (2023a) to an entirely new asset class—high-frequency cryptocurrency options—where the microstructure and volatility regimes differ substantially from traditional markets. In particular, this extends the model’s applicability to ultra-high-frequency data, developing pre-processing methods for constructing static liquid lattices and repairing arbitrage in sparse intraday panels. The elements brought together here form a complete pipeline from raw data to an operational, data-driven, arbitrage-free simulator of the Bitcoin options market.