We propose a new approach to estimating the loss-given-default distribution. More precisely, we obtain the default-time distribution of the leverage ratio (defined as the ratio of a firm's assets over its debt) by examining its last passage time to a certain level. In fact, the use of the last passage time is particularly relevant because it is not a stopping time: this corresponds to the fact that the timing and extent of severe firm-value deterioration, when default approaching, is neither observed nor easily estimated. We calibrate the model parameters to the credit market, so that we can illustrate the loss-given-default distribution implied in the quoted CDS spreads.

Managing a book of options on several underlying involves controlling positions of several thousands of financial assets. It is one of the most challenging financial problems involving both pricing and microstructural modeling. An options market maker has to manage both long- and short-dated options having very different dynamics. In particular, short-dated options inventories cannot be managed as a part of an aggregated inventory, which prevents the use of dimensionality reduction techniques such as a factorial approach or first-order Greeks approximation. In this paper, we show that a simple analytical approximation of the solution of the market maker's problem provides significantly higher flexibility than the existing algorithms designing options market making strategies.

We develop a duality theory for the problem of maximising expected lifetime utility from inter-temporal wealth over an infinite horizon, under the minimal no-arbitrage assumption of No Unbounded Profit with Bounded Risk (NUPBR). We use only deflators, with no arguments involving equivalent martingale measures, so do not require the stronger condition of No Free Lunch with Vanishing Risk (NFLVR). Our formalism also works without alteration for the finite horizon version of the problem. As well as extending work of Bouchard and Pham to any horizon and to a weaker no-arbitrage setting, we obtain a stronger duality statement, because we do not assume by definition that the dual domain is the polar set of the primal space. Instead, we adopt a method akin to that used for inter-temporal consumption problems, developing a supermartingale property of the deflated wealth and its path that yields an infinite horizon budget constraint and serves to define the correct dual variables. The structure of our dual space allows us to show that it is convex, without forcing this property by assumption. We proceed to enlarge the primal and dual domains to confer solidity to them, and use supermartingale convergence results which exploit Fatou convergence, to establish that the enlarged dual domain is the bipolar of the original dual space. The resulting duality theorem shows that all the classical tenets of convex duality hold. Moreover, at the optimum, the deflated wealth process is a potential converging to zero. We work out examples, including a case with a stock whose market price of risk is a three-dimensional Bessel process, so satisfying NUPBR but not NFLVR.

The latent order book of \cite{donier2015fully} is one of the most promising agent-based models for market impact. This work extends the minimal model by allowing agents to exhibit mean-reversion, a commonly observed pattern in real markets. This modification leads to new order book dynamics, which we explicitly study and analyze. Underlying our analysis is a mean-field assumption that views the order book through its \textit{average} density. We show how price impact develops in this new model, providing a flexible family of solutions that can potentially improve calibration to real data. While no closed-form solution is provided, we complement our theoretical investigation with extensive numerical results, including a simulation scheme for the entire order book.

Motivated by an equilibrium problem, we establish the existence of a solution for a family of Markovian backward stochastic differential equations with quadratic nonlinearity and discontinuity in $Z$. Using unique continuation and backward uniqueness, we show that the set of discontinuity has measure zero. In a continuous-time stochastic model of an endowment economy, we prove the existence of an incomplete Radner equilibrium with nondegenerate endogenous volatility.

Financial markets across all asset classes are known to exhibit trends. These trends have been exploited by traders for decades. Here, we empirically measure when trends revert, based on 30 years of daily futures prices for equity indices, interest rates, currencies and commodities. We find that trends tend to revert once they reach a critical level of statistical significance. Based on polynomial regression, we carefully measure this critical level. We find that it is universal across asset classes and has a universal scaling behavior, as the trend's time horizon runs from a few days to several years. The corresponding regression coefficients are small, but statistically highly significant, as confirmed by bootstrapping and out-of-sample testing. Our results signal to investors when to exit a trend. They also reveal how markets have become more efficient over the decades. Moreover, they point towards a potential deep analogy between financial markets and critical phenomena: our analysis supports the conjecture that financial markets can be modeled as statistical mechanical ensembles of Buy/Sell orders near critical points. In this analogy, the trend strength plays the role of an order parameter, whose dynamcis is described by a Langevin equation.

In quantitative finance, modeling the volatility structure of underlying assets is a key component in the pricing of options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887-904, 2016], seek to fit observed market data based on the observation that the log-realized variance behaves like a fractional Brownian motion with small Hurst parameter, $H < 1/2$, over reasonable timescales. Both time series data of asset prices and option derived price data indicate that $H$ often takes values close to $0.1$ or smaller, i.e. rougher than Brownian Motion. This change greatly improves the fit to time series data of underlying asset prices as well as to option prices while maintaining parsimoniousness. However, the non-Markovian nature of the driving fractional Brownian motion in the rough Bergomi model poses severe challenges for theoretical and numerical analyses as well as for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi model, the strong rate of convergence is only $H$. We prove rate $H + 1/2$ for the weak convergence of the Euler method and, surprisingly, in the case of quadratic payoff functions we obtain rate one. Indeed, the problem of weak convergence for rough Bergomi is very subtle; we provide examples demonstrating that the rate of convergence for payoff functions well approximated by second-order polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof relies on Taylor expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments.