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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 (BSDEs) 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 t…
We show that an estimate by de la Peña, Ibragimov and Jordan for $\mathbb{E}(X-c)^+$, with $c$ a constant and $X$ a random variable of which the mean, the variance, and $\mathbb{P}(X \leq c)$ are known, implies an estimate by Scarf on the infimum of $\mathbb{E}(X \wedge c)$ over the set of positive random variables $X$ with fixed mean and variance. This also shows, as a consequence, that the former estimate implies a…
We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E[\max\{U_1(Y), π(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$. Terms of this form arise in financial risk estimation and option pricing. When $U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under gener…
We consider the numerical approximation of $\mathbb{P}[G\in Ω]$ where the $d$-dimensional random variable $G$ cannot be sampled directly, but there is a hierarchy of increasingly accurate approximations $\{G_\ell\}_{\ell\in\mathbb{N}}$ which can be sampled. The cost of standard Monte Carlo estimation scales poorly with accuracy in this setup since it compounds the approximation and sampling cost. A direct application…
We consider a family of fractional Brownian fields $\{B^{H}\}_{H\in (0,1)}$ on $\mathbb{R}^{d}$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $ψ$ such that the covariance of $$ X^{H}(x) = Γ(H)^{\frac{1}{2}} \left( B^{H}(x) - \int_{\mathbb{R}^{d}} B^{H}(u) ψ(u, x)du\right), $$ converges to the covariance of a log-correlated Gaussian field when $H \downarrow 0$. We then u…
Implementing a set of microeconomic criteria, we develop price dynamics equations using a function of demand/supply with key symmetry properties. The function of demand/supply can be linear or nonlinear. The type of function determines the nature of the tail of the distribution based on the randomness in the supply and demand. For example, if supply and demand are normally distributed, and the function is assumed to …
We investigate the problem of computing a nested expectation of the form $\mathbb{P}[\mathbb{E}[X|Y] \!\geq\!0]\!=\!\mathbb{E}[\textrm{H}(\mathbb{E}[X|Y])]$ where $\textrm{H}$ is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested ex…
Risk contagion concerns any entity dealing with large scale risks. Suppose (X,Y) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of Y on X. This can be measured in a variety of ways. In this paper, we study two such measures: the quantity E[max(X-t,0)|Y > t] called Marginal Mean Excess (MME) as well…
Let $Ω$ be a Polish space with Borel $σ$-field $\mathcal{F}$ and countably generated sub $σ$-field $\mathcal{G}\subset\mathcal{F}$. Denote by $\mathcal{L}(\mathcal{F})$ the set of all bounded $\mathcal{F}$-upper semianalytic functions from $Ω$ to the reals and by $\mathcal{L}(\mathcal{G})$ the subset of $\mathcal{G}$-upper semianalytic functions. Let $\mathcal{E}(\cdot|\mathcal{G})\colon\mathcal{L}(\mathcal{F})\to\ma…
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimension $d$ is arbitrary.
The conservative wealth-exchange process derived from trade interactions is modeled as a multiplicative stochastic transference of value, where each interaction multiplies the wealth of the poorest of the two intervening agents by a random gain eta=(1+kappa), with kappa a random return. Analyzing the kinetic equation for the wealth distribution P(w,t), general properties are derived for arbitrary return distributions…
In this manuscript, we analytically and numerically study statistical properties of an heteroskedastic process based on the celebrated ARCH generator of random variables whose variance is defined by a memory of $q_{m}$-exponencial, form ($e_{q_{m}=1}^{x}=e^{x}$). Specifically, we inspect the self-correlation function of squared random variables as well as the kurtosis. In addition, by numerical procedures, we infer t…
Invoice or payment dilution is the gap between the approved invoice amount and the actual collection is a significant source of non credit risk and margin loss in supply chain finance. Traditionally, this risk is managed through the buyer's irrevocable payment undertaking (IPU), which commits to full payment without deductions. However, IPUs can hinder supply chain finance adoption, particularly among sub-invested gr…