In April 2020, the Chicago Mercantile Exchange temporarily switched the pricing formula for West Texas Intermediate oil market options from the Black model to the Bachelier model. In this context, we introduce an additive Bachelier model that provides a simple closed-form solution and a good description of the implied volatility surface. This new additive model exhibits several notable mathematical and financial properties. It ensures the no-arbitrage condition, a critical requirement in highly volatile markets, while also enabling a parsimonious synthesis of the volatility surface. The model features only three parameters, each with a clear financial interpretation: the volatility term structure, the vol-of-vol, and a parameter for modelling skew. Model calibration can follow a cascade procedure: first, it accurately replicates the term structures of forwards and At-The-Money volatilities observed in the market; second, it fits the smile of the volatility surface. The proposed model also supports efficient pricing of path-dependent exotic options via Monte Carlo simulation, using a straightforward and computationally efficient approach. Overall, this model provides a robust and parsimonious description of the oil option market during the exceptionally volatile first period of the Covid-19 pandemic.

An elementary Recurrent Neural Network that operates on p time lags, called an RNN(p), is the natural generalisation of a linear autoregressive model ARX(p). It is a powerful forecasting tool for variables displaying inherent seasonal patterns across multiple time scales, as is often observed in energy, economic, and financial time series. The architecture of RNN(p) models, characterised by structured feedbacks across time lags, enables the design of efficient training strategies. We conduct a comparative study of learning algorithms for these models, providing a rigorous analysis of their computational complexity and training performance. We present two applications of RNN(p) models in power consumption forecasting, a key domain within the energy sector where accurate forecasts inform both operational and financial decisions. Experimental results show that RNN(p) models achieve excellent forecasting accuracy while maintaining a high degree of interpretability. These features make them well-suited for decision-making in energy markets and other fintech applications where reliable predictions play a significant economic role.

Modeling the dependence between multiple risk types is a central challenge in contemporary insurance risk management. The standard approaches, Lévy copulas and zero-mixed models, often face practical difficulties in simulation and parameter calibration. In this paper, we introduce the Comb-Bernoulli model, a novel framework for capturing dependence between sparse time series of insurance risks, bridging the benefits of the two standard approaches. The (traditional) copula structure of the proposed model enables tractable: i) simulation, ii) likelihood evaluation, and iii) estimation of dependence parameters. We present the general properties of the model and analyze in detail the Gaussian copula case with lognormal marginals. Moreover, we illustrate an application using the Danish fire insurance dataset, highlighting both the modeling strengths and numerical efficiency of our approach in real-world risk management.

L & eacute;vy-driven Ornstein-Uhlenbeck (OU) processes represent a versatile class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. However, in the current state of play, their Monte Carlo simulation is not straightforward for two main reasons: i) algorithms are only available for some specific processes within this class, and ii) they are often computationally expensive. In this paper, we introduce a new simulation technique designed to address both challenges. It relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all L & eacute;vy-driven OU processes. Moreover, leveraging FFT, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. Lastly, the algorithm allows explicit control of the numerical error. We apply the proposed technique to the pricing of energy derivatives, comparing the results with the existing benchmarks. Our findings indicate that the proposed methodology is at least one order of magnitude faster than the existing algorithms, while maintaining an equivalent level of accuracy.

A growing number of contributions in the literature have identified a puzzle in the European carbon allowance (EUA) market. Specifically, a persistent cost-of-carry spread (C-spread) over the risk-free rate has been observed. We are the first to explain the anomalous C-spread with the credit spread of the corporates involved in the emission trading scheme. We obtain statistical evidence that the C-spread is cointegrated with both this credit spread and the risk-free interest rate. This finding has a relevant policy implication: the most effective solution to solve the market anomaly is including the EUA in the list of European Central Bank eligible collateral for refinancing operations. This change in the ECB monetary policy operations would greatly benefit the carbon market and the EU green transition.

We investigate the asymptotic behaviour of the implied volatility in the Bachelier setting, extending the large-strike results established for the Black-Scholes framework. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier implied volatility in the wings of the smile, directly linking them to the tail decay of the underlying returns' distribution. Furthermore, we establish a rigorous connection between the analyticity strip of the characteristic function and the asymptotic slope of the volatility smile. Moreover, we show that if the implied variance grows linearly for large absolute moneyness degree, the underlying returns' distribution must exhibit exponential tail decay, and the corresponding characteristic function is analytic in a horizontal strip of the complex plane. These findings characterizes valid models based solely on the observable asymptotic behaviour of the smile.

Quantifying predictive uncertainty regarding future electricity demand is the main goal of probabilistic load forecasting. A good probabilistic model is often identified with forecasted densities that are as concentrated ("sharp") as possible. However, this goal is frequently achieved by sacrificing forecast reliability, i.e. the statistical compatibility between forecasted densities and observed frequencies. In real-world applications, reliability is the crucial measure of model quality, especially when predicting distribution tails. We propose a new methodology for probabilistic load forecasting, introducing a novel loss function which allows an excellent balance between forecast sharpness and reliability. We apply the proposed modelling approach for predicting the electricity load on a benchmark dataset. Experimental results show that the obtained density forecasts are extremely reliable and also close to optimal in terms of sharpness and point accuracy.

Monroe (1978) demonstrates that any local semimartingale can be represented as a time-changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time-change are independent? We prove that a local semimartingale is not equivalent to a BM with a time-change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to calibrate the equity volatility surface accurately. We notice that the sub-class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.

Probabilistic forecasting of power consumption in a middle-term horizon (few months to a year) is a main challenge in the energy sector. It plays a key role in planning future generation plants and transmission grid. This paper proposes a novel model that (i) incorporates seasonality and autoregressive features in a traditional time-series analysis and (ii) includes weather conditions in a parsimonious machine learning approach, known as Gaussian Process. Applying to a daily power consumption dataset in North East England, provided by one of the largest energy suppliers, we obtain promising results in Out-of-Sample density forecasts up to one year, even using a small dataset, with only a two-year calibration set. For the evaluation of the achieved probabilistic forecasts, we consider the pinball loss—a metric common in the energy sector—and we assess the coverage—a procedure standard in the banking sector after the introduction of Basel II Accords—also running the conditional and unconditional tests for probability intervals. Results show that the proposed model outperforms benchmarks in terms of both accuracy and reliability.

In this paper, we present a very fast Monte Carlo scheme for additive processes: the computational time is of the same order of magnitude of standard algorithms for simulating Brownian motions. We analyze in detail numerical error sources and propose a technique that reduces the two major sources of error. We also compare our results with a benchmark method: the jump simulation with Gaussian approximation. We show an application to additive normal tempered stable processes, a class of additive processes that calibrates “exactly” the implied volatility surface. Numerical results are relevant. This fast algorithm is also an accurate tool for pricing path-dependent discretely-monitoring options with errors of one basis point or below.

Monroe (1978) demonstrates that any local semimartingale can be represented as a time-changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time-change are independent? We prove that a local semimartingale is not equivalent to a BM with a time-change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to accurately calibrate the equity volatility surface. We notice that the sub-class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.

We introduce a simple additive process for equity index derivatives. The model generalizes Lévy Normal Tempered Stable processes (e.g. NIG and VG) with time-dependent parameters. It accurately fits the equity index volatility surfaces in the whole time range of quoted instruments, including options with small time-horizon (days) and long time-horizon (years). We introduce the model via its characteristic function. This allows using classical Fourier pricing techniques. We discuss the calibration issues in detail and we show that, in terms of mean squared error, calibration is on average two orders of magnitude better than both Lévy and Sato processes alternatives. We show that even if the model loses the classical stationarity property of Lévy processes, it presents interesting scaling properties for the calibrated parameters.

In Internal Rating Based approaches, the regulator indicates a model to determine bank’s credit capital requirements. The main concern is on model’s econometric usage and on the estimation of its key parameters: the probability of default and the loss given default. In this study, we point out that taking into account only parameters’ expectation leads to a significant underestimation of bank’s risk and its Regulatory Capital. In particular, we statistically test distributional assumptions on these two parameters and we underline the key role played by parameters’ dependency. We analyse two benchmark datasets: one with all corporations rated by Moody’s and another one that includes only speculative grade firms. Results are striking: we obtain that, considering parameters’ uncertainty, the Regulatory Capital should be increased by an amount in the range between 38% and 66%. A clear policy implication stems from this study: the scaling factor for model risk, removed by Basel III accord, should be reintroduced in the determination of credit Regulatory Capital.

A Recurrent Neural Network that operates on several time lags, called an RNN(p), is the natural generalization of an Autoregressive ARX(p) model. It is a powerful forecasting tool when different time scales can influence a given phenomenon, as it happens in the energy sector where hourly, daily, weekly and yearly interactions coexist. The cost-effective BPTT is the industry standard as learning algorithm for RNNs. We prove that, when training RNN(p) models, other learning algorithms turn out to be much more efficient in terms of both time and space complexity. We also introduce a new learning algorithm, the Tree Recombined Recurrent Learning, that leverages on a tree representation of the unrolled network and appears to be even more effective. We present an application of RNN(p) models for power consumption forecasting on the hourly scale: experimental results demonstrate the efficiency of the proposed algorithm and the excellent predictive accuracy achieved by the selected model both in point and in probabilistic forecasting of the energy consumption.

Empirical studies have emphasized that the equity implied volatility is characterized by a negative skew inversely proportional to the square root of the time-to-maturity. We examine the short-time-to-maturity behavior of the implied volatility smile for pure jump exponential additive processes. An excellent calibration of the equity volatility surfaces has been achieved by a class of these additive processes with power-law scaling. The two power-law scaling parameters are β , related to the variance of jumps, and δ , related to the smile asymmetry. It has been observed, in option market data, that β =1 and δ =-1/2 . In this paper, we prove that the implied volatility of these additive processes is consistent, in the short-time, with the equity market empirical characteristics if and only if β =1 and δ =-1/2 .

Credit capital requirements in Internal Rating Based approaches require the calibration of two key parameters: the probability of default and the loss-given-default. This letter considers the uncertainty about these two parameters and models this uncertainty in an elementary way: it shows how this estimation risk can be computed and properly taken into account in regulatory capital. We analyse two standard real datasets: one composed by all corporates rated by Moody's and one limited only to the speculative grade ones. We statistically test model hypotheses on both marginal distributions and parameter dependency. We compute the estimation risk impact and observe that parameter dependency raises substantially the tail risk in capital requirements. The results are striking with a required increase in regulatory capital in the range $38\%$-$66\%$.

We propose an option approach for pricing bond illiquidity that is reminiscent of the celebrated work of Longstaff (1995) on the non-marketability of some non-dividend-paying shares in IPOs. This approach describes a quite common situation in the fixed income market: it is rather usual to find issuers that, besides liquid benchmark bonds, issue some other bonds that either are placed to a small number of investors in private placements or have a limited issue size. We model interest rate and credit risks via a convenient reduced-form approach. We deduce a simple closed formula for illiquid corporate coupon bond prices when liquid bonds with similar characteristics (e.g. maturity) are present in the market for the same issuer. The key model parameter is the time-to-liquidate a position, i.e. the time that an experienced bond trader takes to liquidate a given position on a corporate coupon bond. We show that illiquid bonds present an additional liquidity spread that depends on the time-to-liquidate aside from bond volatility. We provide a detailed application for two issuers in the European market.

This study introduces a new technique to recover the implicit discount factor in the derivative market using only European put and call prices: this discount is grounded in actual transactions in active markets. Moreover, this study identifies the implied cost of funding, over OIS, of major market players. Does a liquid equity market allow arbitrage? The key idea is that the (unique) forward contract -- built using the put-call parity relation -- contains information about the market discount factor: by no-arbitrage conditions we identify the implicit interest rate such that the forward contract value does not depend on the strike. The procedure is applied to options on S&P 500 and EURO STOXX 50 indices. There is statistical evidence that, in the EURO STOXX 50 market, the implicit interest rate curve coincides with the EUR OIS one, while, in the S&P 500 market, a cost of funding of, on average, 34 basis points is added on top of the USD OIS curve.

Middle-term horizon (months to a year) power consumption prediction is a main challenge in the energy sector, in particular when probabilistic forecasting is considered. We propose a new modelling approach that incorporates trend, seasonality and weather conditions, as explicative variables in a shallow Neural Network with an autoregressive feature. We obtain excellent results for density forecast on the one-year test set applying it to the daily power consumption in New England U.S.A.. The quality of the achieved power consumption probabilistic forecasting has been verified, on the one hand, comparing the results to other standard models for density forecasting and, on the other hand, considering measures that are frequently used in the energy sector as pinball loss and CI backtesting.