It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment s+ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: σBS(k, T ) 2 T ∼ Ψ(s+ − 1) × k (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Drȃgulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443-453], and then show the validity of a refined expansion of the type σBS(k, T ) 2 T = (β1k 1/2 + β2 + . . . ) 2 , where all constants are explicitly known as functions of s+, the Heston model parameters, spot vol and maturity T . In the case of the "zero-correlation" Heston model such an expansion was derived by Gulisashvili and Stein [Appl. Math. Optim. 61, 3 (2010), 287-315]. Our methods and results may prove useful beyond the Heston model: the entire quantitative analysis is based on affine principles: at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of log ST (equivalently: Mellin transform of ST ); what matters is that these transforms satisfy ordinary differential equations of Riccati type. Secondly, our analysis reveals a new parameter ("critical slope"), defined in a model free manner, which drives the second and higher order terms in tail-and implied volatility expansions.

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.

The driving of tropical precipitation by the variability of the underlying sea surface temperature (SST) plays a critical role in the atmospheric general circulation. To assess the precipitation sensitivity to SST variability, it is necessary to observe and understand the relationship between precipitation and SST. However, the precipitation–SST relationships from any coupled atmosphere–ocean system can be difficult to interpret given the challenge of disentangling the SST-forced atmospheric response and the atmospheric intrinsic variability. This study demonstrates that the two components can be isolated using uncoupled atmosphere-only simulations, which extract the former when driven by time-varying SSTs and the latter when driven by climatological SSTs. With a simple framework that linearly combines the two types of uncoupled simulations, the coupled precipitation–SST relationships are successfully reproduced. Such a framework can be a useful tool for quantitatively diagnosing tropical air–sea interactions. The precipitation sensitivity to SST variability is investigated with the use of uncoupled simulations with prescribed SST anomalies, where the influence of atmospheric intrinsic variability on SST is deactivated. Through a focus on local precipitation–SST relationships, the precipitation sensitivity to local SST variability is determined to be predominantly controlled by the local background SST. In addition, the strength of the precipitation response increases monotonically with the local background SST, with a very sharp growth at high SSTs. These findings are supported by basic principles of moist static stability, from which a simple formula for precipitation sensitivity to local SST variability is derived.

We quantify the sensitivity of the Eisenberg-Noe clearing vector to estimation errors in the bilateral liabilities of a financial system. The interbank liabilities matrix is a crucial input to the computation of the clearing vector. However, in practice central bankers and regulators must often estimate this matrix because complete information on bilateral liabilities is rarely available. As a result, the clearing vector may suffer from estimation errors in the liabilities matrix. We quantify the clearing vector's sensitivity to such estimation errors and show that its directional derivatives are, like the clearing vector itself, solutions of fixed point equations. We describe estimation errors utilizing a basis for the space of matrices representing permissible perturbations and derive analytical solutions to the maximal deviations of the Eisenberg-Noe clearing vector. This allows us to compute upper bounds for the worst case perturbations of the clearing vector. Moreover, we quantify the probability of observing clearing vector deviations of a certain magnitude, for uniformly or normally distributed errors in the relative liability matrix.Applying our methodology to a dataset of European banks, we find that perturbations to the relative liabilities can result in economically sizeable differences that could lead to an underestimation of the risk of contagion. Our results are a first step towards allowing regulators to quantify errors in their simulations.

The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.