Quantitative Finance

Various valuation adjustments, or XVAs, can be written in terms of non-linear PIDEs equivalent to FBSDEs. In this paper we develop a Fourier-based method for solving FBSDEs in order to efficiently and accurately price Bermudan derivatives, including options and swaptions, with XVA under the flexible dynamics of a local Lévy model: this framework includes a local volatility function and a local jump measure. Due to the unavailability of the characteristic function for such processes, we use an asymptotic approximation based on the adjoint formulation of the problem.

It is well known that the minimal superhedging price of a contingent claim is too high for practical use. In a continuous-time model uncertainty framework, we consider a relaxed hedging criterion based on acceptable shortfall risks. Combining existing aggregation and convex dual representation theorems, we derive duality results for the minimal price on the set of upper semicontinuous discounted claims.

Taleb (2018) claimed a novel approach to evaluating the quality of probabilistic election forecasts via no-arbitrage pricing techniques and argued that popular forecasts of the 2016 U.S. Presidential election had violated arbitrage boundaries. We show that under mild assumptions all such political forecasts are arbitrage-free and that the heuristic that Taleb's argument was based on is false.

This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black--Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature. It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black--Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk. The proposed theory has asymptotic behaviors that are associated with the weak and strong arbitrage bubble limits. For the case where the arbitrage bubble's volatility is zero (deterministic bubble), the weak limit corresponds to the usual Black-Scholes model. The strong limit case also give a Black--Scholes model, but the underlying asset's mean value replaces the interest rate. When the bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise to option price dynamics that are similar to the Black--Scholes model. Explicit formulas are derived for Gaussian and lognormal stochastic bubbles. Consequently, the Black--Scholes model can be considered to be a "low energy" limit of a more general stochastic model.

In this work we present an equilibrium formulation for price impacts. This is motivated by the Buhlmann equilibrium in which assets are sold into a system of market participants and can be viewed as a generalization of the Esscher premium. Existence and uniqueness of clearing prices for the liquidation of a portfolio are studied. We also investigate other desired portfolio properties including monotonicity and concavity. Price per portfolio unit sold is also calculated. In special cases, we study price impacts generated by market participants who follow the exponential utility and power utility.

The objective of this paper is to develop a duality between a novel Martingale Entropy Optimal Transport problem (D) and an associated optimization problem (P). In (D) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by (Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018) but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as in Liero et al. The Problem (D) differs from the corresponding problem in Liero et al. not only by the martingale constraint, but also because we admit less restrictive relaxation terms D, which may not have a divergence formulation. In Problem (P) the objective functional, associated via Fenchel coniugacy to the terms D, is not any more linear, as in OT or in MOT. This leads to a novel optimization problem which also has a clear financial interpretation as a non linear subhedging value. Our results allow us to establish a novel nonlinear robust pricing-hedging duality in financial mathematics, which covers a wide range of known robust results in its generality.

This paper solves the consumption-investment problem under Epstein-Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize optimal consumption and investment strategies through backward stochastic differential equations (BSDEs). Compared with classical results on a fixed horizon, our characterization involves an additional stochastic process to account for the uncertainty of the horizon. As demonstrated in a Markovian setting, this added uncertainty drastically alters optimal strategies from the fixed-horizon case. The main results are obtained through the development of new techniques for BSDEs with superlinear growth on unbounded random horizons.

We consider the problem of the optimal trading strategy in the presence of a price predictor, linear trading costs and a quadratic risk control. The solution is known to be a band system, a policy that induces a no-trading zone in the positions space. Using a path-integral method introduced in a previous work, we give equations for the upper and lower edges of this band, and solve them explicitly in the case of an Ornstein-Uhlenbeck predictor. We then explore the shape of this solution and derive its asymptotic behavior for large values of the predictor, without requiring trading costs to be small.

This paper presents a continuous-time model of intraday trading, pricing, and liquidity with dynamic TWAP and VWAP benchmarks. The model is solved in closed-form for the competitive equilibrium and also for non-price-taking equilibria. The intraday trajectories of TWAP trading targets cause predictable intraday patterns of price pressure, and randomness in VWAP target trajectories induces additional randomness in intraday price-pressure patterns. TWAP and VWAP trading both reduce market liquidity and increase price volatility relative to just terminal trading targets alone. The model is computationally tractable, which lets us provide a number of numerical illustrations.

In this article, we consider the problem of equilibrium price formation in an incomplete securities market consisting of one major financial firm and a large number of minor firms. They carry out continuous trading via the securities exchange to minimize their cost while facing idiosyncratic and common noises as well as stochastic order flows from their individual clients. The equilibrium price process that balances demand and supply of the securities, including the functional form of the price impact for the major firm, is derived endogenously both in the market of finite population size and in the corresponding mean field limit.