Quantitative Finance

Bank behaviour is important for pricing XVA because it links different counterparties and thus breaks the usual XVA pricing assumption of counterparty independence. Consider a typical case of a bank hedging a client trade via a CCP. On client default the hedge (effects) will be removed (rebalanced). On the other hand, if the hedge counterparty defaults the hedge will be replaced. Thus if the hedge required initial margin then the default probability driving MVA is from the client not from the hedge counterparty. This is the opposite of usual assumptions where counterparty XVAs are computed independent of each other. Replacement of the hedge counterparty means multiple CVA costs on the hedge side need inclusion. Since hedge trades are generally at riskless mid (or worse) these costs are paid on the client side, and must be calculated before the replacement hedge counterparties are known. We call these counterparties anonymous counterparties. The effects on CVA and MVA will generally be exclusive because MVA largely removes CVA, and CVA is hardly relevant for CCPs. Effects on KVA and FVA will resemble those on MVA. We provide a theoretical framework, including anonymous counterparties, and numerical examples. Pricing XVA by considering counterparties in isolation is inadequate and behaviour must be taken into account.

We address the modelling of commodities that are supposed to have positive price but, on account of a possible failure in the physical delivery mechanism, may turn out not to. This is done by explicitly incorporating a `delivery liability' option into the contract. As such it is a simple generalisation of the established Black model.

Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see \cite{Tank} for an overview), and more recently rough volatility models \cite{AlosLeon, Fukasawa}. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models.

We derive sharp bounds for the prices of VIX futures using the full information of S&P 500 smiles. To that end, we formulate the model-free sub/superreplication of the VIX by trading in the S&P 500 and its vanilla options as well as the forward-starting log-contracts. A dual problem of minimizing/maximizing certain risk-neutral expectations is introduced and shown to yield the same value. The classical bounds for VIX futures given the smiles only use a calendar spread of log-contracts on the S&P 500. We analyze for which smiles the classical bounds are sharp and how they can be improved when they are not. In particular, we introduce a family of functionally generated portfolios which often improves the classical bounds while still being tractable; more precisely, determined by a single concave/convex function on the line. Numerical experiments on market data and SABR smiles show that the classical lower bound can be improved dramatically, whereas the upper bound is often close to optimal.

We construct a data-driven statistical indicator for quantifying the tail risk perceived by the EURGBP option market surrounding Brexit-related events. We show that under lognormal SABR dynamics this tail risk is closely related to the so-called martingale defect and provide a closed-form expression for this defect which can be computed by solving an inverse calibration problem. In order to cope with the the uncertainty which is inherent to this inverse problem, we adopt a Bayesian statistical parameter estimation perspective. We probe the resulting posterior densities with a combination of optimization and adaptive Markov chain Monte Carlo methods, thus providing a careful uncertainty estimation for all of the underlying parameters and the martingale defect indicator. Finally, to support the feasibility of the proposed method, we provide a Brexit "fever curve" for the year 2019.

The introduction of CCPs in most derivative transactions will dramatically change the landscape of derivatives pricing, hedging and risk management, and, according to the TABB group, will lead to an overall liquidity impact about 2 USD trillions. In this article we develop for the first time a comprehensive approach for pricing under CCP clearing, including variation and initial margins, gap credit risk and collateralization, showing concrete examples for interest rate swaps. Mathematically, the inclusion of asymmetric borrowing and lending rates in the hedge of a claim lead to nonlinearities showing up in claim dependent pricing measures, aggregation dependent prices, nonlinear PDEs and BSDEs. This still holds in presence of CCPs and CSA. We introduce a modeling approach that allows us to enforce rigorous separation of the interconnected nonlinear risks into different valuation adjustments where the key pricing nonlinearities are confined to a funding costs component that is analyzed through numerical schemes for BSDEs. We present a numerical case study for Interest Rate Swaps that highlights the relative size of the different valuation adjustments and the quantitative role of initial and variation margins, of liquidity bases, of credit risk, of the margin period of risk and of wrong way risk correlations.

We depart from the usual methods for pricing contracts with the counterparty credit risk found in most of the existing literature. In effect, typically, these models do not account for either systemic effects or at-first-default contagion and postulate that the contract value at default equals either the risk-free value or the pre-default value. We propose instead a fairly general framework, which allows us to perform effective Credit Value Adjustment (CVA) computations for a contract with bilateral counterparty risk in the presence of systemic and wrong or right way risks. Our general methodology focuses on the role of alternative settlement clauses, but it is also aimed to cover various features of margin agreements. A comparative analysis of numerical results reported in the final section supports our initial conjecture that alternative specifications of settlement values have a non-negligible impact on the CVA computation for contracts with bilateral counterparty risk. This emphasizes the practical importance of more sophisticated models that are capable of fully reflecting the actual features of financial contracts, as well as the influence of the market environment.

In this paper we propose a tractable quadratic programming formulation for calculating the equilibrium term structure of electricity prices. We rely on a theoretical model described in [21], but extend it so that it reflects actually traded electricity contracts, transaction costs and liquidity considerations. Our numerical simulations examine the properties of the term structure and its dependence on various parameters of the model. The proposed quadratic programming formulation is applied to calculate the equilibrium term structure of electricity prices in the UK power grid consisting of a few hundred power plants. The impact of ramp up and ramp down constraints are also studied.

The accuracy of least squares calibration using option premiums and particle filtering of price data to find model parameters is determined. Derivative models using exponential Lévy processes are calibrated using regularized weighted least squares with respect to the minimal entropy martingale measure. Sequential importance resampling is used for the Bayesian inference problem of time series parameter estimation with proposal distribution determined using extended Kalman filter. The algorithms converge to their respective global optima using a highly parallelizable statistical optimization approach using a grid of initial positions. Each of these methods should produce the same parameters. We investigate this assertion.

We determine an explicit formula for the Laplace transform of the price of an option on a maximal interest rate when the instantaneous rate satisfies Cox-Ingersoll-Ross's model. This generalizes considerably one result of Leblanc-Scaillet.