CyberSec Research

We study the pricing of derivative securities in financial markets modeled by a sub-mixed fractional Brownian motion with jumps (smfBm-J), a non-Markovian process that captures both long-range dependence and jump discontinuities. Under this model, we derive a fractional integro-partial differential equation (PIDE) governing the option price dynamics. Using semigroup theory, we establish the existence and uniqueness of mild solutions to this PIDE. For European options, we obtain a closed-form pricing formula via Mellin-Laplace transform techniques. Furthermore, we propose a Grunwald-Letnikov finite-difference scheme for solving the PIDE numerically and provide a stability and convergence analysis. Empirical experiments demonstrate the accuracy and flexibility of the model in capturing market phenomena such as memory and heavy-tailed jumps, particularly for barrier options. These results underline the potential of fractional-jump models in financial engineering and derivative pricing.

We derive an explicit analytical approximation for the local volatility function in the Cheyette interest rate model, extending the classical Dupire framework to fixed-income markets. The result expresses local volatility in terms of time and strike derivatives of the Bachelier implied variance, naturally generalizes to multi-factor Cheyette models, and provides a practical tool for model calibration.

We consider islamic Profit and Loss (PL) sharing contract, possibly combined with an agency contract, and introduce the notion of {\em $c$-fair} profit sharing ratios ($c = (c_1, \ldots,c_d) \in (\mathbb R^{\star})^d$, where $d$ is the number of partners) which aims to determining both the profit sharing ratios and the induced expected maturity payoffs of each partner $\ell$ according to its contribution, determined by the rate component $c_{\ell}$ of the vector $c$, to the global success of the project. We show several new results that elucidate the relation between these profit sharing ratios and various important economic factors as the investment risk, the labor and the capital, giving accordingly a way of choosing them in connection with the real economy. The design of our approach allows the use of all the range of econometrics models or more general stochastic diffusion models to compute or approximate the quantities of interest.

We apply vector quantisation within mixed one- and two-factor Bergomi models to implement a fast and efficient approach for option pricing in these models. This allows us to calibrate such models to market data of VIX futures and options. Our numerical tests confirm the efficacy of vector quantisation, making calibration feasible over daily data covering several months. This permits us to evaluate the calibration accuracy and the stability of the calibrated parameters, and we provide a comprehensive assessment of the two models. Both models show excellent performance in fitting VIX derivatives, and their parameters show satisfactory stability over time.

In this paper we study the pricing and hedging of nonreplicable contingent claims, such as long-term insurance contracts like variable annuities. Our approach is based on the benchmark-neutral pricing framework of Platen (2024), which differs from the classical benchmark approach by using the stock growth optimal portfolio as the num\'eraire. In typical settings, this choice leads to an equivalent martingale measure, the benchmark-neutral measure. The resulting prices can be significantly lower than the respective risk-neutral ones, making this approach attractive for long-term risk-management. We derive the associated risk-minimizing hedging strategy under the assumption that the contingent claim possesses a martingale decomposition. For a set of nonreplicable contingent claims, these strategies allow monitoring the working capital required to generate their payoffs and enable an assessment of the resulting diversification effects. Furthermore, an algorithmic refinancing strategy is proposed that allows modeling the working capital. Finally, insurance-finance arbitrages of the first kind are introduced and it is demonstrated that benchmark-neutral pricing effectively avoids such arbitrages.

Despite significant advancements in machine learning for derivative pricing, the efficient and accurate valuation of American options remains a persistent challenge due to complex exercise boundaries, near-expiry behavior, and intricate contractual features. This paper extends a semi-analytical approach for pricing American options in time-inhomogeneous models, including pure diffusions, jump-diffusions, and Levy processes. Building on prior work, we derive and solve Volterra integral equations of the second kind to determine the exercise boundary explicitly, offering a computationally superior alternative to traditional finite-difference and Monte Carlo methods. We address key open problems: (1) extending the decomposition method, i.e. splitting the American option price into its European counterpart and an early exercise premium, to general jump-diffusion and Levy models; (2) handling cases where closed-form transition densities are unavailable by leveraging characteristic functions via, e.g., the COS method; and (3) generalizing the framework to multidimensional diffusions. Numerical examples demonstrate the method's efficiency and robustness. Our results underscore the advantages of the integral equation approach for large-scale industrial applications, while resolving some limitations of existing techniques.

The key objective of this paper is to develop an empirical model for pricing SPX options that can be simulated over future paths of the SPX. To accomplish this, we formulate and rigorously evaluate several statistical models, including neural network, random forest, and linear regression. These models use the observed characteristics of the options as inputs -- their price, moneyness and time-to-maturity, as well as a small set of external inputs, such as the SPX and its past history, dividend yield, and the risk-free rate. Model evaluation is performed on historical options data, spanning 30 years of daily observations. Significant effort is given to understanding the data and ensuring explainability for the neural network. A neural network model with two hidden layers and four neurons per layer, trained with minimal hyperparameter tuning, performs well against the theoretical Black-Scholes-Merton model for European options, as well as two other empirical models based on the random forest and the linear regression. It delivers arbitrage-free option prices without requiring these conditions to be imposed.

Here we demonstrate how we can use Small Volatility Approximation in calibration of Multi-Factor HJM model with deterministic correlations, factor volatilities and mean reversals. It is noticed that quality of this calibration is very good and it does not depend on number of factors.

In cryptocurrency markets, a key challenge for perpetual future issuers is maintaining alignment between the perpetual future price and target value. This study addresses this challenge by exploring the relationship between funding rates and perpetual future prices. Our results demonstrate that by appropriately designing funding rates, the perpetual future price can remain aligned with the target value. We develop replicating portfolios for perpetual futures, offering issuers an effective method to hedge their positions. Additionally, we provide path-dependent funding rates as a practical alternative and investigate the difference between the original and path-dependent funding rates. To achieve these results, our study employs path-dependent infinite-horizon BSDEs in conjunction with arbitrage pricing theory. Our main results are obtained by establishing the existence and uniqueness of solutions to these BSDEs and analyzing the large-time behavior of these solutions.

We investigate the asymptotic behaviour of the Implied Volatility in the Bachelier setting, extending the framework introduced by Benaim and Friz for the Black-Scholes setting. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier Implied Volatility in the wings of the smile, linking these to the tail behaviour of the underlying's returns' distribution. Furthermore, we establish a direct connection between the analyticity strip of the returns' characteristic function and the asymptotic formula for the Implied Volatility smile at extreme moneyness.

A common assumption in cryptocurrency markets is a positive relationship between total-value-locked (TVL) and cryptocurrency returns. To test this hypothesis we examine whether the returns of TVL-sorted portfolios can be explained by common cryptocurrency factors. We find evidence that portfolios formed on TVL exhibit returns that are linear functions of aggregate crypto market returns, that is they can be replicated with appropriate weights on the crypto market portfolio. Thus, strategies based on TVL can be priced with standard asset pricing tools. This result holds true both for total TVL and a simple TVL measure that removes a number of ways TVL may be overstated.

The increasing vulnerability of power systems has heightened the need for operating reserves to manage contingencies such as generator outages, line failures, and sudden load variations. Unlike energy costs, driven by consumer demand, operating reserve costs arise from addressing the most critical credible contingencies - prompting the question: how should these costs be allocated through efficient pricing mechanisms? As an alternative to previously reported schemes, this paper presents a new causation-based pricing framework for electricity markets based on contingency-constrained energy and reserve scheduling models. Major salient features include a novel security charge mechanism along with the explicit definition of prices for up-spinning reserves, down-spinning reserves, and transmission services. These features ensure more comprehensive and efficient cost-reflective market operations. Moreover, the proposed nodal pricing scheme yields revenue adequacy and neutrality while promoting reliability incentives for generators based on the cost-causation principle. An additional salient aspect of the proposed framework is the economic incentive for transmission assets, which are remunerated based on their use to deliver energy and reserves across all contingency states. Numerical results from two case studies illustrate the performance of the proposed pricing scheme.

We introduce a fast and flexible Machine Learning (ML) framework for pricing derivative products whose valuation depends on volatility surfaces. By parameterizing volatility surfaces with the 5-parameter stochastic volatility inspired (SVI) model augmented by a one-factor term structure adjustment, we first generate numerous volatility surfaces over realistic ranges for these parameters. From these synthetic market scenarios, we then compute high-accuracy valuations using conventional methodologies for two representative products: the fair strike of a variance swap and the price and Greeks of an American put. We then train the Gaussian Process Regressor (GPR) to learn the nonlinear mapping from the input risk factors, which are the volatility surface parameters, strike and interest rate, to the valuation outputs. Once trained, We use the GPR to perform out-of-sample valuations and compare the results against valuations using conventional methodologies. Our ML model achieves very accurate results of $0.5\%$ relative error for the fair strike of variance swap and $1.7\% \sim 3.5\%$ relative error for American put prices and first-order Greeks. More importantly, after training, the model computes valuations almost instantly, yielding a three to four orders of magnitude speedup over Crank-Nicolson finite-difference method for American puts, enabling real-time risk analytics, dynamic hedging and large-scale scenario analysis. Our approach is general and can be extended to other path-dependent derivative products with early-exercise features, paving the way for hybrid quantitative engines for modern financial systems.

Using an intangible intensity factor that is orthogonal to the Fama--French factors, we compare the role of intangible investment in predicting stock returns over the periods 1963--1992 and 1993--2022. For 1963--1992, intangible investment is weak in predicting stock returns, but for 1993--2022, the predictive power of intangible investment becomes very strong. Intangible investment has a significant impact not only on the MTB ratio (Fama--French high minus low [HML] factor) but also on operating profitability (OP) (Fama--French robust minus weak [RMW] factor) when forecasting stock returns from 1993 to 2022. For intangible asset-intensive firms, intangible investment is the main predictor of stock returns, rather than MTB ratio and profitability. Our evidence suggests that intangible investment has become an important factor in explaining stock returns over time, independent of other factors such as profitability and MTB ratio.

This paper develops a continuous-time filtering framework for estimating a hazard rate subject to an unobservable change-point. This framework arises naturally in both financial and insurance applications, where the default intensity of a firm or the mortality rate of an individual may experience a sudden jump at an unobservable time, representing, for instance, a shift in the firm's risk profile or a deterioration in an individual's health status. By employing a progressive enlargement of filtration, we integrate noisy observations of the hazard rate with default-related information. We characterise the filter, i.e. the conditional probability of the change-point given the information flow, as the unique strong solution to a stochastic differential equation driven by the innovation process enriched with the discontinuous component. A sensitivity analysis and a comparison of the filter's behaviour under various information structures are provided. Our framework further allows for the derivation of an explicit formula for the survival probability conditional on partial information. This result applies to the pricing of credit-sensitive financial instruments such as defaultable bonds, credit default swaps, and life insurance contracts. Finally, a numerical analysis illustrates how partial information leads to delayed adjustments in the estimation of the hazard rate and consequently to mispricing of credit-sensitive instruments when compared to a full-information setting.

Stablecoins represent a critical bridge between cryptocurrency and traditional finance, with Tether (USDT) dominating the sector as the largest stablecoin by market capitalization. By Q1 2025, Tether directly held approximately $98.5 billion in U.S. Treasury bills, representing 1.6% of all outstanding Treasury bills, making it one of the largest non-sovereign buyers in this crucial asset class, on par with nation-state-level investors. This paper investigates how Tether's market share of U.S. Treasury bills influences corresponding yields. The baseline semi-log time trend model finds that a 1% increase in Tether's market share is associated with a 1-month yield reduction of 3.8%, corresponding to 14-16 basis points. However, threshold regression analysis reveals a critical market share threshold of 0.973%, above which the yield impact intensifies significantly. In this high regime, a 1% market share increase reduces 1-month yields by 6.3%. At the end of Q1 2025, Tether's market share placed it firmly within this high-impact regime, reducing 1-month yields by around 24 basis points relative to a counterfactual. In absolute terms, Tether's demand for Treasury Bills equates to roughly $15 billion in annual interest savings for the U.S. government. Aligning with theories of liquidity saturation and nonlinear price impact, these results highlight that stablecoin demand can reduce sovereign funding costs and provide a potential buffer against market shocks.

We investigate the Gatheral model of double mean-reverting stochastic volatility, in which the drift term itself follows a mean-reverting process, and the overall model exhibits mean-reverting behavior. We demonstrate that such processes can attain values arbitrarily close to zero and remain near zero for extended periods, making them practically and statistically indistinguishable from zero. To address this issue, we propose a modified model incorporating Skorokhod reflection, which preserves the model's flexibility while preventing volatility from approaching zero.

We propose a tractable extension of the rough Bergomi model, replacing the fractional Brownian motion with a generalised grey Brownian motion, which we show to be reminiscent of models with stochastic volatility of volatility. This extension breaks away from the log-Normal assumption of rough Bergomi, thereby making it a viable suggestion for the Equity Holy Grail -- the joint SPX/VIX options calibration. For this new (class of) model(s), we provide semi-closed and asymptotic formulae for SPX and VIX options and show numerically its potential advantages as well as calibration results.