Claudio Albanese

A two-state jump model

Abstract We introduce a pricing model for equity options in which sample paths follow a variance-gamma (VG) jump model whose parameters evolve according to a two-state Markov chain process. As in GARCH type models, jump sizes are positively correlated to volatility. The model is capable of justifying the observed implied volatility skews for options at all maturities. Furthermore, the term structure of implied VG kurtosis is an increasing function of the time to maturity, in agreement with empirical evidence. Explicit pricing formulae, extending the known VG formulae, for European options are derived. In addition, a resummation algorithm, based on the method of lines, which greatly reduces the algorithmic complexity of the pricing formulae, is introduced. This algorithm is also the basis of approximate numerical schemes for American and Bermudan options, for which a state dependent exercise boundary can be computed.

Stochastic Integrals and Abelian Processes

We study triangulation schemes for the joint kernel of a diusion process with uniformly continuous coecients and an adapted, non-resonant Abelian process. The proto- typical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coecients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summa- tions we discuss. We discretize the space coordinate in uniform steps and assume that time is either con- tinuous or finely discretized as in a fully explicit Euler method and the Courant condition is satisfied. We show that the Fourier transform of the joint kernel of a diusion and a stochastic integral converges in a uniform graph norm associated to the Markov generator. Convergence also implies smoothness properties for the Fourier transform of the joint kernel. Stochastic integrals are straightforward to define for finite triangulations and the convergence result gives a new and entirely constructive way of defining stochastic integrals in the continuum. The method relies on a reinterpretation and extension of the classic theorems by Feynman-Kac, Girsanov, Ito and Cameron-Martin, which are also reobtained. We make use of a path-wise analysis without relying on a probabilistic interpretation. The Fourier representation is needed to regularize the hypo-elliptic character of the joint process of a diusion and an adapted stochastic integral. The argument extends as long as the Fourier analysis framework can be generalized. This condition leads to the notion of non-resonant Abelian process.

Time quantization and q deformations

We extend to quantum mechanics the technique of stochastic subordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. As examples, we considered two versions of the q-deformed harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules.

Capital Valuation Adjustment and Funding Valuation Adjustment

In the aftermath of the 2007 global financial crisis, banks started reflecting into derivative pricing the cost of capital and collateral funding through XVA metrics. Here XVA is a catch-all acronym whereby X is replaced by a letter such as C for credit, D for debt, F for funding, K for capital and so on, and VA stands for valuation adjustment. This behaviour is at odds with economies where markets for contingent claims are complete, whereby trades clear at fair valuations and the costs for capital and collateral are both irrelevant to investment decisions. In this paper, we set forth a mathematical formalism for derivative portfolio management in incomplete markets for banks. A particular emphasis is given to the problem of finding optimal strategies for retained earnings which ensure a sustainable dividend policy.

VaR Optimisation and Regression Sensitivities

Infinitesimal sensitivities, computed as derivatives of pricing functions, are often used to compute hedge ratios for high frequency rebalancing strategies executed by front desk trading operations. However, for the purpose of risk calculations requiring a 2-week VaR metric with shocks of size typical of stress periods, infinitesimal sensitivities are arguably unfit for purpose. In this article, we propose to quantify a metric for Risk-Not-In-VAR (RniVAR) by computing upper bounds on the errors in the sensitivities expansion. We find that regression based sensitivities can be substantially compressed and made more robust by means of Krylov regularisation. They also have a better quality P\&L explain than infinitesimal sensitivities computed by taking infinitesimal derivatives. There is a trade-off between compression ratios for regularised sensitivities and the size of RniVAR. The latter is added as a conservative adjustment to the IM formula to make it consistent with back-testing benchmarks, while regularisation de-noises dynamic rebalancing by avoiding over-fitting and makes hedging more robust.

Credit Limits, Stress Testing and Model Risk for Capital Metrics

The topics of Economic Capital modelling, reverse stress testing and credit limits are inextricably intertwined as they all focus on exceptional loss events. In this paper, we use the KVA framework in to frame these three topics within a single unified approach. We propose setting credit limits based on an incremental KVA metric interpreted as a measure of capital consumption for each individual client. Compared to Potential Future Exposure (PFE), incremental KVA is more risk sensitive as i) it is portfolio sensitive and detects cross-selling opportunities, ii) captures wrong-way-risk, iii) accounts for idiosyncratic features such as granularity and credit risk concentration, iv) measures also other risks besides the default of the counterparty such as CVA and FVA mark-to-market losses and is suitable to generalisations driven by regulatory changes such as MVA.A reverse stress testing exercise based on KVA metrics reveals stress scenarios of two kinds: the ones where losses are due to an idiosyncratic vulnerability of the portfolio and the ones tied to systemic risk and the credit cycle. The latter are completely missed by the PFE, although crucially important during periods of market distress.Since different pricing models tend to diverge the most for stressed scenarios, model risk has a material impact on capital projections and, consequently, on the KVA metric. We discuss model risk comparing Gaussian interest rate models admitting arbitrarily negative rates with alternative models where rates are bounded from below.

Regression Sensitivities for Initial Margin Calculations

Implementations of the Standard Initial Margin Model (SIMM) and the Sensitivity Based Approach (SBA) in the Fundamental Review of the Trading Book (FRTB), both call for the calculation of sensitivities with respect to a standardised set of risk factors. Since standard factors are generally collinear and pricing functions are possibly rough, finding sensitivities qualifies as a mathematically ill-posed problem for which analytical derivatives do not provide a robust solution. Numerical instabilities are particularly problematic since they hamper reconciliation and make collateral optimisation strategies inefficient.In this article, we introduce a method for calculating sensitivities based on ridge regressions to keep sensitivities small and stable. We find that a drift term and FX cross-gammas significantly improves the accuracy of the P&L explain achieved in the SIMM methodology. The method implies rigorous upper bounds on errors in P&L explain, on which basis we adjust Initial Margin conservatively in order to pass back-testing benchmarks.

Poisson Kernels as Expansions in

This paper concerns stochastic processes on chains of arbitrary length whose Poisson kernel can be expressed in terms of the q‐Racah polynomials, the most general q‐deformed orthogonal polynomials in the discrete series of the Askey scheme. We give a new interpretation of this kernel as the probability transition density for a subordinated Markov process with only nearest neighbor hops. As an application, we give an elementary proof and extend a positivity result for a class of Poisson kernels which Gasper and Rahman established with direct methods.