SANOS -- Smooth strictly Arbitrage-free Non-parametric Option Surfaces

TL;DR

The paper addresses the challenge of fitting observed European option bids and asks into a smooth, arbitrage-free surface across strikes and maturities. It develops a non-parametric framework that represents prices as $\hat{C}(T_j,K) = \sum_{i=0}^N q^i_j \,\mathrm{Call}(K^i, K, V_j)$ and calibrates $q_j$ via linear programming, guaranteeing absence of arbitrage in time and strike. A key contribution is the Production Model, which introduces a smooth surface with $\hat{C}_j(K) = \sum_{i=1}^N q_j^i \mathrm{Call}(K^i, K, \eta V_j)$ and a parameter $\eta$ controlling smoothness, plus an efficient second contribution: a Discrete Local Volatility (DLV) parametrization that yields positive, tractable surface dynamics through martingale transition operators. The framework accommodates bid-ask spreads, allows strikes to be added outside the observed grid, and supports generalized, per-strike dynamics, enabling fast, arbitrage-free interpolation suitable for pricing, risk-management, and data-driven generative modelling. Overall, the method combines numerical efficiency with strict arbitrage constraints to produce high-fidelity, smooth option surfaces on SPX-like data.

Abstract

We present a simple, numerically efficient but highly flexible non-parametric method to construct representations of option price surfaces which are both smooth and strictly arbitrage-free across time and strike. The method can be viewed as a smooth generalization of the widely-known linear interpolation scheme, and retains the simplicity and transparency of that baseline. Calibration of the model to observed market quotes is formulated as a linear program, allowing bid-ask spreads to be incorporated directly via linear penalties or inequalities, and delivering materially lower computational cost than most of the currently available implied-volatility surface fitting routines. As a further contribution, we derive an equivalent parameterization of the proposed surface in terms of strictly positive "discrete local volatility" variables. This yields, to our knowledge, the first construction of smooth, strictly arbitrage-free option price surfaces while requiring only trivial parameter constraints (positivity). We illustrate the approach using S&P 500 index options

Figures (6)

• Figure 1: Fit of our model to the market, illustrated for a few DTE's on several days in 2023 using IvyDB OptionMetrics data. The thin bars in the call price and volatility graphs illustrate fit errors, expressed relative to the relevant spread on the right hand axis. (We do see some fitting errors in these charts because of the smoothness we impose; see section \ref{['sec:prod_model']}.) The red dots below the zero line in the density graphs represent cases where the market-mid prices have arbitrage.
• Figure 2: Fit of our model with different smoothness parameters $\eta$ to 1 day to expiry options on SPX. The right hand side shows the density which is much smoother for the case $\eta=0.25$ while actually retaining a decent fit: the biggest fitting error is 40% of vol spread.
• Figure 3: Fit of our model with different smoothness parameters $\eta$ to SPX options with 35 days to expiry. In this case $\eta=0.06$ fits the market within bid/ask while imposing sufficient smoothness on the model.
• Figure 4: Fit of our model with different smoothness parameters $\eta$ to SPX options with 1249 days to expiry. This expiry has sparse strikes which leads to the unnatural bump on the right wing for the linear model. This is rectified by the smoother model with $\eta=0.25$.
• Figure 5: Example of the idea \ref{['eq:inhC']} where the call prices at two expiries are increasing in the market strikes $K=(0.5,1,1.5)$ but not at the extrapolated strikes $K\geq 1.6$ (the error is plotted at the right hand axis). The example uses constant log-normal volatilities $(0.05,1.2,0.05)$ for the three strikes for both expiries. The densities are $q_1=(0,1,0)$ and $q_2=(0.5,0,0.5)$.

Theorems & Definitions (39)

• Definition 2.1: Arbitrage Free Call Price Surfaces
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Definition 2.5: Smooth call price function
• Theorem 2.6
• Remark 2.7
• proof
• Remark 2.8
• Definition 2.9: Martingale transition operator