Martingale Optimal Transport (MOT) provides a framework for robust pricing and hedging of illiquid derivatives. Classical MOT enforces exact calibration of model marginals to the mid-prices of vanilla options. Motivated by the industry practice of fitting bid and ask marginals to vanilla prices, we introduce a relaxation of MOT in which model-implied volatilities are only required to lie within observed bid--ask spreads; equivalently, model marginals lie between the bid and ask marginals in convex order. The resulting Bid--Ask MOT (BAMOT) yields realistic price bounds for illiquid derivatives and, via strong duality, can be interpreted as the superhedging price when short and long positions in vanilla options are priced at the bid and ask, respectively. We further establish convergence of BAMOT to classical MOT as bid--ask spreads vanish, and quantify the convergence rate using a novel distance intrinsically linked to bid--ask spreads. Finally, we support our findings with several synthetic and real-data examples.
We consider the pricing of derivatives written on accumulated marks, such as weather derivatives or aggregate loss claims, using a self-exciting marked point process. The jump intensity mean-reverts between events and increases at jump times by an amount proportional to the mark. The resulting state process, where the variable $U_t$ accumulates jump magnitudes, is a piecewise deterministic Markov process (PDMP). We derive the discounted pricing equation as a backward partial integro-differential equation (PIDE) in two spatial dimensions. To overcome the dimensionality, we propose an exponential (Laplace/Fourier) transform in the accumulated mark variable, which diagonalizes the translation operator and reduces the pricing problem to a family of one-dimensional PIDEs in the intensity variable along a Bromwich contour. For Gamma-mixture mark laws (under actuarial or Esscher-tilted measures), the nonlocal jump term is efficiently approximated by generalized Gauss--Laguerre quadrature. We solve the reduced PIDEs backward in time using a monotone IMEX finite difference scheme (implicit upwind drift and discounting, explicit jump operator) and recover option prices via numerical inversion. We provide a rigorous, term-by-term global error bound covering time and space discretization, quadrature, interpolation, and boundary effects, supported by numerical experiments and Monte Carlo benchmarks.
In response to growing demand for resilient and transparent financial instruments, we introduce a novel framework for replicating private equity (PE) performance using liquid, AI-enhanced strategies. Despite historically delivering robust returns, private equity's inherent illiquidity and lack of transparency raise significant concerns regarding investor trust and systemic stability, particularly in periods of heightened market volatility. Our method uses advanced graphical models to decode liquid PE proxies and incorporates asymmetric risk adjustments that emulate private equity's unique performance dynamics. The result is a liquid, scalable solution that aligns closely with traditional quarterly PE benchmarks like Cambridge Associates and Preqin. This approach enhances portfolio resilience and contributes to the ongoing discourse on safe asset innovation, supporting market stability and investor confidence.
We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged filtrations. More specifically, the information available to the insider is modelled by Brownian filtrations progressively enlarged with the times of either the global maximum or minimum of the underlying risky asset price over the infinite time interval, which is not a stopping time in the filtration generated by the underlying risky asset. We show that the optimal exercise times are the first times at which the asset price process reaches either lower or upper stochastic boundaries depending on the current values of its running maximum or minimum given the occurrence of times of either the global maximum or minimum, respectively. The proof is based on the reduction of the original problems into the necessarily three-dimensional optimal stopping problems and the equivalent free-boundary problems. We apply either the normal-reflection or the normal-entrance conditions as well as the smooth-fit conditions for the value functions to characterise the candidate boundaries as either the maximal or minimal solutions to the associated first-order nonlinear ordinary differential equations and the transcendental arithmetic equations, respectively.
We propose a new financial model, the stochastic volatility model with sticky drawdown and drawup processes (SVSDU model), which enables us to capture the features of winning and losing streaks that are common across financial markets but can not be captured simultaneously by the existing financial models. Moreover, the SVSDU model retains the advantages of the stochastic volatility models. Since there are not closed-form option pricing formulas under the SVSDU model and the existing simulation methods for the sticky diffusion processes are really time-consuming, we develop a deep neural network to solve the corresponding high-dimensional parametric partial differential equation (PDE), where the solution to the PDE is the pricing function of a European option according to the Feynman-Kac Theorem, and validate the accuracy and efficiency of our deep learning approach. We also propose a novel calibration framework for our model, and demonstrate the calibration performances of our models on both simulated data and historical data. The calibration results on SPX option data show that the SVSDU model is a good representation of the asset value dynamic, and both winning and losing streaks are accounted for in option values. Our model opens new horizons for modeling and predicting the dynamics of asset prices in financial markets.
En italien, ci et vi sont des pronoms atones de première et deuxième personne du pluriel, mais aussi des pronoms adverbiaux équivalant au y français, là où ne correspond au en français. Or, à date ancienne, ces deux formes ont été respectivement concurrencées par no et vo puis ne et ve. De leur côté, les pronoms COI de troisième personne sont occupés par les formes atones gli (masculin) et le (féminin) pour le singulier, et par la forme tonique loro au pluriel. Cependant, en italien néo-standard, le et loro sont remplacés par gli, lui-même remplaçable par ci. En diachronie, ci et vi ont donc maintenu leurs capacités référentielles, ne les a rétrécies et gli les a augmentées. Quant à no, vo et ve, ils ont disparu. Notre objectif est donc d’analyser la fin de vie et les nouvelles vies de ces clitiques en examinant les corrélations possibles entre leurs gestes articulatoires et leurs capacités référentielles.
We propose a new risk sensitive reinforcement learning approach for the dynamic hedging of options. The approach focuses on the minimization of the tail risk of the final P&L of the seller of an option. Different from most existing reinforcement learning approaches that require a parametric model of the underlying asset, our approach can learn the optimal hedging strategy directly from the historical market data without specifying a parametric model; in addition, the learned optimal hedging strategy is contract-unified, i.e., it applies to different options contracts with different initial underlying prices, strike prices, and maturities. Our approach extends existing reinforcement learning methods by learning the tail risk measures of the final hedging P&L and the optimal hedging strategy at the same time. We carry out comprehensive empirical study to show that, in the out-of-sample tests, the proposed reinforcement learning hedging strategy can obtain statistically significantly lower tail risk and higher mean of the final P&L than delta hedging methods.
Sophia Zamudio-Haas, Kim Koester, Luz Venegas
et al.
Abstract Background Uptake of HIV pre-exposure prophylaxis (PrEP) remains low among transgender people as compared to other subgroups, despite high rates of HIV acquisition. In California, Latinx people comprise 40% of the population and Latina transgender women experience some of the highest burden of HIV of any subgroup, indicating a critical need for appropriate services. With funding from the California HIV/AIDS Research Programs, this academic-community partnership developed, implemented, and evaluated a PrEP project that co-located HIV services with gender affirming care in a Federally Qualified Heath Center (FQHC). Trans and Latinx staff led intervention adaptation and activities. Methods This paper engages qualitative methods to describe how a PrEP demonstration project- Triunfo- successfully engaged Spanish-speaking transgender Latinas in services. We conducted 13 in-depth interviews with project participants and five interviews with providers and clinic staff. Interviews were conducted in Spanish or English. We conducted six months of ethnographic observation of intervention activities and recorded field notes. We conducted thematic analysis. Results Beneficial elements of the intervention centered around three intertwined themes: creating trusted space, providing comprehensive patient navigation, and offering social support “entre nosotras” (“between us women/girls”). The combination of these factors contributed to the intervention’s success supporting participants to initiate and persist on PrEP, many of whom had previously never received healthcare. Participants shared past experiences with transphobia and concerns around discrimination in a healthcare setting. Developing trust proved foundational to making participants feel welcome and “en casa/ at home” in the healthcare setting, which began from the moment participants entered the clinic and continued throughout their interactions with staff and providers. A gender affirming, bilingual clinician and peer health educators (PHE) played a critical part in intervention development, participant recruitment, and patient navigation. Conclusions Our research adds nuance to the existing literature on peer support services and navigation by profiling the multifaced roles that PHE served for participants. PHE proved instrumental to empowering participants to overcome structural and other barriers to healthcare, successfully engaging a group who previously avoided healthcare in clinical settings.
Catastrophe (CAT) bond markets are incomplete and hence carry uncertainty in instrument pricing. As such various pricing approaches have been proposed, but none treat the uncertainty in catastrophe occurrences and interest rates in a sufficiently flexible and statistically reliable way within a unifying asset pricing framework. Consequently, little is known empirically about the expected risk-premia of CAT bonds. The primary contribution of this paper is to present a unified Bayesian CAT bond pricing framework based on uncertainty quantification of catastrophes and interest rates. Our framework allows for complex beliefs about catastrophe risks to capture the distinct and common patterns in catastrophe occurrences, and when combined with stochastic interest rates, yields a unified asset pricing approach with informative expected risk premia. Specifically, using a modified collective risk model -- Dirichlet Prior-Hierarchical Bayesian Collective Risk Model (DP-HBCRM) framework -- we model catastrophe risk via a model-based clustering approach. Interest rate risk is modeled as a CIR process under the Bayesian approach. As a consequence of casting CAT pricing models into our framework, we evaluate the price and expected risk premia of various CAT bond contracts corresponding to clustering of catastrophe risk profiles. Numerical experiments show how these clusters reveal how CAT bond prices and expected risk premia relate to claim frequency and loss severity.
We design a system for risk-analyzing and pricing portfolios of non-performing consumer credit loans. The rapid development of credit lending business for consumers heightens the need for trading portfolios formed by overdue loans as a manner of risk transferring. However, the problem is nontrivial technically and related research is absent. We tackle the challenge by building a bottom-up architecture, in which we model the distribution of every single loan's repayment rate, followed by modeling the distribution of the portfolio's overall repayment rate. To address the technical issues encountered, we adopt the approaches of simultaneous quantile regression, R-copula, and Gaussian one-factor copula model. To our best knowledge, this is the first study that successfully adopts a bottom-up system for analyzing credit portfolio risks of consumer loans. We conduct experiments on a vast amount of data and prove that our methodology can be applied successfully in real business tasks.
This paper prices and replicates the financial derivative whose payoff at $T$ is the wealth that would have accrued to a $\$1$ deposit into the best continuously-rebalanced portfolio (or fixed-fraction betting scheme) determined in hindsight. For the single-stock Black-Scholes market, Ordentlich and Cover (1998) only priced this derivative at time-0, giving $C_0=1+σ\sqrt{T/(2π)}$. Of course, the general time-$t$ price is not equal to $1+σ\sqrt{(T-t)/(2π)}$. I complete the Ordentlich-Cover (1998) analysis by deriving the price at any time $t$. By contrast, I also study the more natural case of the best levered rebalancing rule in hindsight. This yields $C(S,t)=\sqrt{T/t}\cdot\,\exp\{rt+σ^2b(S,t)^2\cdot t/2\}$, where $b(S,t)$ is the best rebalancing rule in hindsight over the observed history $[0,t]$. I show that the replicating strategy amounts to betting the fraction $b(S,t)$ of wealth on the stock over the interval $[t,t+dt].$ This fact holds for the general market with $n$ correlated stocks in geometric Brownian motion: we get $C(S,t)=(T/t)^{n/2}\exp(rt+b'Σb\cdot t/2)$, where $Σ$ is the covariance of instantaneous returns per unit time. This result matches the $\mathcal{O}(T^{n/2})$ "cost of universality" derived by Cover in his "universal portfolio theory" (1986, 1991, 1996, 1998), which super-replicates the same derivative in discrete-time. The replicating strategy compounds its money at the same asymptotic rate as the best levered rebalancing rule in hindsight, thereby beating the market asymptotically. Naturally enough, we find that the American-style version of Cover's Derivative is never exercised early in equilibrium.
We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.
Vanna-Volga is a popular method for the interpolation/extrapolation of volatility smiles. The technique is widely used in the FX markets context, due to its ability to consistently construct the entire Lognormal smile using only three Lognormal market quotes. However, the derivation of the Vanna-Volga method itself is free of distributional assumptions. With this is mind, it is surprising there have been no attempts to apply the method to Normal volatilities (the current standard for interest rate markets). We show how the method can be modified to build Normal volatility smiles. As it turns out, only minor modifications are required compared to the Lognormal case. Moreover, as the inversion of Normal volatilities from option prices is easier in the Normal case, the smile construction can occur at a machine-precision level using analytical formulae, making the approximations via Taylor-series unnecessary. Apart from being based on practical and intuitive hedging arguments, the Vanna-Volga has further important advantages. In comparison to the Normal SABR model, the Vanna-Volga can easily fit both classical convex and atypical concave smiles (frowns). Concave smile patterns are sometimes observed around ATM strikes in the interest rate markets, particularly in the situations of anticipated jumps (with an unclear outcome) in interest rates. Besides, concavity is often observed towards the lower/left end of the Normal volatility smiles of interest rates. At least in these situations, the Vanna-Volga can be expected to interpolate/extrapolate better than SABR.
We consider arbitrage free valuation of European options in Black-Scholes and Merton markets, where the general structure of the market is known, however the specific parameters are not known. In order to reflect this subjective uncertainty of a market participant, we follow a Bayesian approach to option pricing. Here we use historic discrete or continuous observations of the market to set up posterior distributions for the future market. Given a subjective physical measure for the market dynamics, we derive the existence of arbitrage free pricing rules by constructing subjective option pricing measures. The non-uniqueness of such measures can be proven using the freedom of choice of prior distributions. The subjective market measure thus turns out to model an incomplete market. In addition, for the Black-Scholes market we prove that in the high frequency limit (or the long time limit) of observations, Bayesian option prices converge to the standard BS-Option price with the true volatility. In contrast to this, in the Merton market with normally distributed jumps Bayesian prices do not converge to standard Merton prices with the true parameters, as only a finite number of jump events can be observed in finite time. However, we prove that this convergence holds true in the limit of long observation times.
An option market maker incurs funding costs when carrying and hedging inventory. To hedge a net long delta inventory, for example, she pays a fee to borrow stock from the securities lending market. Because of haircuts, she posts additional cash margin to the lender which needs to be financed at her unsecured debt rate. This paper incorporates funding asymmetry (borrowed cash and invested cash earning different interest rates) and realistic stock financing cost into the classic option pricing theory. It is shown that an option position can be dynamically replicated and self financed in the presence of these funding costs. Noting that the funding amounts and costs are different for long and short positions, we extend Black-Scholes partial differential equations (PDE) per position side. The PDE's nonlinear funding cost terms create a free funding boundary and would result in the bid price for a long position on an option lower than the ask price for a short position. An iterative Crank-Nicholson finite difference method is developed to compute European and American vanilla option prices. Numerical results show that reasonable funding cost parameters can easily produce same magnitude of bid/ask spread of less liquid, longer term options as observed in the market place. Portfolio level pricing examples show the netting effect of hedges, which could moderate impact of funding costs.