Search Results (11)

The One-X property, introduced by Zetocha in a 2023 paper, provides a novel stochastic order with direct implications for constructing arbitrage-free implied volatility surfaces. The current work revisits its theoretical foundations and explores its connections with classical stochastic orders, thereby offering a deeper understanding of its mathematical structure and practical significance in calendar-arbitrage-free modeling. We first present an explicit counterexample to a conjecture raised in Zetocha’s previous paper, and then provide a natural and valid enhancement of this conjecture. After discussing the inherent relations between the One-X property and properties such as $T{P}_2$, $R{P}_2$, and unimodality of density ratio (introcuded by Glasserman and Pirjol in their 2024 papers), we further explore some sufficient conditions to achieve the One-X property for random variables of certain mixture types that are frequently seen in applications. Full article

Variational autoencoders (VAEs) have emerged as a promising tool for modeling volatility surfaces, with particular significance for generating synthetic implied volatility scenarios that enhance risk management capabilities. This study evaluates VAE performance using synthetic volatility surfaces, chosen specifically for their arbitrage-free properties and clean data characteristics. Through a comprehensive comparison with traditional methods including thin-plate spline interpolation, parametric models (SABR and SVI), and deterministic autoencoders, we demonstrate that our VAE approach with latent space optimization consistently outperforms existing methods, particularly in scenarios with extreme data sparsity. Our findings show that accurate, arbitrage-free surface reconstruction is achievable using only 5% of the original data points, with errors 7–12 times lower than competing approaches in high-sparsity scenarios. We rigorously validate the preservation of critical no-arbitrage conditions through probability distribution analysis and total variance strip non-intersection tests. The framework we develop overcomes traditional barriers of limited market data by generating over 13,500 synthetic surfaces for training, compared to typical market availability of fewer than 100. These capabilities have important implications for market risk analysis, derivatives pricing, and the development of more robust risk management frameworks, particularly in emerging markets or for newly introduced derivatives where historical data are scarce. Our integration of machine learning with financial theory constraints represents a significant advancement in volatility surface modeling that balances statistical accuracy with financial relevance. Full article

We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time driving, bitcoin returns and gives rise to an arbitrage-free option pricing model. In this framework, we derive two bitcoin volatility measures. The first combines NDIG option pricing with the Chicago Board Options Exchange VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both volatility measures are compared to the volatility based on the historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures the observed in-sample volatility. Full article

We implement the VIX methodology on intraday data of a large set of individual equity options. We thereby consider approaches based on monthly option contracts, weekly option contracts, and a cubic spline interpolation approach. Relying on 1 min, 10 min, and 60 min model-free implied volatility measures, we empirically examine the individual equity return–volatility relationship on the intraday level using quantile regressions. The results confirm a negative contemporaneous link between stock returns and volatility, which is more pronounced in the tails of the distributions. Our findings hint at behavioral biases causing the asymmetric return–volatility link rather than the leverage and volatility-feedback effects. Full article

We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an $L_1$-regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases. Full article

Since the collapse of the Metallgesellschaft AG due to hedging losses in 1993, energy practitioners have been concerned with the ability to hedge long-dated linear and non-linear oil liabilities with short-dated futures and options. This paper identifies a model-free non-parametric approach to extrapolating futures prices and implied volatilities. When we expand the analysis to implementing hedge portfolios for long-dated futures or option contracts over the time period 2007–2017, we utilize the useful benchmark of hedge ratios arising from Schwartz and Smith. With respect to the empirical consequences of hedging long-dated futures and options with their short-dated counterparts, we find that the long-term tracking errors are, on average, quite close to zero, but there is increasing risk entailed in attempting to do so, as the hedge-tracking errors for both futures and option contracts increase with time-to-maturity. Full article

While the main conceptual issue related to deposit insurances is the moral hazard risk, the main technical issue is inaccurate calibration of the implied volatility. This issue can raise the risk of generating an arbitrage. In this paper, first, we discuss that by imposing the no-moral-hazard risk, the removal of arbitrage is equivalent to removing the static arbitrage. Then, we propose a simple quadratic model to parameterize implied volatility and remove the static arbitrage. The process of removing the static risk is as follows: Using a machine learning approach with a regularized cost function, we update the parameters in such a way that butterfly arbitrage is ruled out and also implementing a calibration method, we make some conditions on the parameters of each time slice to rule out calendar spread arbitrage. Therefore, eliminating the effects of both butterfly and calendar spread arbitrage make the implied volatility surface free of static arbitrage. Full article

This paper explores the stochastic collocation technique, applied on a monotonic spline, as an arbitrage-free and model-free interpolation of implied volatilities. We explore various spline formulations, including B-spline representations. We explain how to calibrate the different representations against market option prices, detail how to smooth out the market quotes, and choose a proper initial guess. The technique is then applied to concrete market options and the stability of the different approaches is analyzed. Finally, we consider a challenging example where convex spline interpolations lead to oscillations in the implied volatility and compare the spline collocation results with those obtained through arbitrage-free interpolation technique of Andreasen and Huge. Full article

These days we are witnessing a deep change in the characteristics of the type of energy that our economies are supplied with. A clear trend is that sustainable and green energies are decisively replacing traditional fossil fuel-based sources of energy. For various reasons, this fundamental change implies an increasing risk in investments on portfolios heavily based on traditional energy industries. What is less known, is that these industries have returns that show a very low correlation with sustainable fossil fuel-free stock portfolios making them an appealing tool for portfolio managers to design properly diversified investments. In this study we examine this and related phenomena proposing statistical methods to implement the expected shortfall (ES), the challenging risk measure recently adopted by the financial regulator. We obtain evidence that a newly proposed backtesting procedure for the ES based on multinomial tests is an adequate and simple method to validate these risk measures when applied to a highly volatile stock index. Backtesting results of the ES show that flexible heavy-tailed distribution α–stable performs well for modelling the loss distribution. These results are even improved when the variances of fossil fuel price returns are included as external regressors in the GARCH model variance equation. In this case, the ES computed from the four considered loss distributions perform properly. Full article

Insurance companies issue guarantees that need to be valued according to the market expectations. By calibrating option pricing models to the available implied volatility surfaces, one deals with the so-called risk-neutral measure $\mathbb{Q}$ , which can be used to generate market consistent values for these guarantees. For asset liability management, insurers also need future values of these guarantees. Next to that, new regulations require insurance companies to value their positions on a one-year horizon. As the option prices at $t=1$ are unknown, it is common practice to assume that the parameters of these option pricing models are constant, i.e., the calibrated parameters from time $t=0$ are also used to value the guarantees at $t=1$ . However, it is well-known that the parameters are not constant and may depend on the state of the market which evolves under the real-world measure $\mathbb{P}$ . In this paper, we propose improved regression models that, given a set of market variables such as the VIX index and risk-free interest rates, estimate the calibrated parameters. When the market variables are included in a real-world simulation, one is able to assess the calibrated parameters (and consequently the implied volatility surface) in line with the simulated state of the market. By performing a regression, we are able to predict out-of-sample implied volatility surfaces accurately. Moreover, the impact on the Solvency Capital Requirement has been evaluated for different points in time. The impact depends on the initial state of the market and may vary between −46% and +52%. Full article

Certain exotic options cannot be valued using closed-form solutions or even by numerical methods assuming constant volatility. Many exotics are priced in a local volatility framework. Pricing under local volatility has become a field of extensive research in finance, and various models are proposed in order to overcome the shortcomings of the Black-Scholes model that assumes a constant volatility. The Johannesburg Stock Exchange (JSE) lists exotic options on its Can-Do platform. Most exotic options listed on the JSE’s derivative exchanges are valued by local volatility models. These models needs a local volatility surface. Dupire derived a mapping from implied volatilities to local volatilities. The JSE uses this mapping in generating the relevant local volatility surfaces and further uses Monte Carlo and Finite Difference methods when pricing exotic options. In this document we discuss various practical issues that influence the successful construction of implied and local volatility surfaces such that pricing engines can be implemented successfully. We focus on arbitrage-free conditions and the choice of calibrating functionals. We illustrate our methodologies by studying the implied and local volatility surfaces of South African equity index and foreign exchange options. Full article