arXiv subjects

Claudio Albanese

Publications and source records attributed to Claudio Albanese.

13 recordsLinked to original sources

arXiv2020

XVA Analysis From the Balance Sheet

XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007--09 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key in identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly. It is then plugged in the dynamic and trade incremental context of a real derivative banking portfolio. The corresponding cost-of-capital XVA strategy ensures to bank shareholders a submartingale equity process corresponding to a target hurdle rate on their capital at risk, consistently between and throughout deals. Set on a forward/backward SDE formulation, this strategy can be solved efficiently using GPU computing combined with deep learning regression methods in a whole bank balance sheet context. A numerical case study emphasizes the workability and added value of the ensuing pathwise XVA computations.

q-fin.RM↗

arXiv2016

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.

q-fin.CP↗

arXiv2012

Restructuring Counterparty Credit Risk

We introduce an innovative theoretical framework to model derivative transactions between defaultable entities based on the principle of arbitrage freedom. Our framework extends the traditional formulations based on Credit and Debit Valuation Adjustments (CVA and DVA). Depending on how the default contingency is accounted for, we list a total of ten different structuring styles. These include bipartite structures between a bank and a counterparty, tri-partite structures with one margin lender in addition, quadri-partite structures with two margin lenders and, most importantly, configurations where all derivative transactions are cleared through a Central Counterparty (CCP). We compare the various structuring styles under a number of criteria including consistency from an accounting standpoint, counterparty risk hedgeability, numerical complexity, transaction portability upon default, induced behaviour and macro-economic impact of the implied wealth allocation.

q-fin.RM↗

arXiv2009

Spectral methods for volatility derivatives

In the first quarter of 2006 Chicago Board Options Exchange (CBOE) introduced, as one of the listed products, options on its implied volatility index (VIX). This created the challenge of developing a pricing framework that can simultaneously handle European options, forward-starts, options on the realized variance and options on the VIX. In this paper we propose a new approach to this problem using spectral methods. We use a regime switching model with jumps and local volatility defined in \cite{FXrev} and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. The main idea of this paper is to "lift" (i.e. extend) the generator of the underlying process to keep track of the relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by applying a new semi-analytic algorithm for block-diagonalization. This method enables us to evaluate numerically the joint distribution between the underlying stock price and the realized variance, which in turn gives us a way of pricing consistently European options, general accrued variance payoffs and forward-starting and VIX options.

q-fin.PR↗

arXiv2007

Stochastic Mechanics as a Gauge Theory

We show that non-relativistic Quantum Mechanics can be faithfully represented in terms of a classical diffusion process endowed with a gauge symmetry of group Z_4. The representation is based on a quantization condition for the realized action along paths. A lattice regularization is introduced to make rigorous sense of the construction and then removed. Quantum mechanics is recovered in the continuum limit and the full U(1) gauge group symmetry of electro-magnetism appears. Anti-particle representations emerge naturally, albeit the context is non-relativistic. Quantum density matrices are obtained by averaging classical probability distributions over phase-action variables. We find that quantum conditioning can be described in classical terms but not through the standard notion of sub sigma-algebras. Delicate restrictions arise by the constraint that we are only interested in the algebra of gauge invariant random variables. We conclude that Quantum Mechanics is equivalent to a theory of gauge invariant classical stochastic processes we call Stochastic Mechanics.

math.PR↗

arXiv2007

Stochastic Integrals and Abelian Processes

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coefficients and an adapted, non-resonant Abelian process. The prototypical example of Abelian process to which our methods apply is given by stochastic integrals with uniformly continuous coeffcients. The range of applicability includes also a broader class of processes of practical relevance, such as the sup process and certain discrete time summations we discuss. We discretize the space coordinate in uniform steps and assume that time is either continuous 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 diffusion 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 re-obtained. 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 diffusion 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.

math.PR↗

arXiv2007

Operator Methods, Abelian Processes and Dynamic Conditioning

A mathematical framework for Continuous Time Finance based on operator algebraic methods offers a new direct and entirely constructive perspective on the field and leads to new numerical analysis techniques. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can effectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions.

math.PR↗

arXiv2007

Kernel Convergence Estimates for Diffusions with Continuous Coefficients

We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step small enough for the method to be stable. We find sharp uniform bounds for the convergence rate as a function of the degree of smoothness which we conjecture. The bounds also apply to the time derivative of the kernel and its first two space derivatives. Our proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. Convergence rates depend on the degree of smoothness and H\"older differentiability of the coefficients. We find that the fastest convergence rate is of order $O(h^2)$ and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of H\"older differentiability except that the convergence rate is slower. H\"older continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

math.NA↗

arXiv2007

Moment Methods for Exotic Volatility Derivatives

The latest generation of volatility derivatives goes beyond variance and volatility swaps and probes our ability to price realized variance and sojourn times along bridges for the underlying stock price process. In this paper, we give an operator algebraic treatment of this problem based on Dyson expansions and moment methods and discuss applications to exotic volatility derivatives. The methods are quite flexible and allow for a specification of the underlying process which is semi-parametric or even non-parametric, including state-dependent local volatility, jumps, stochastic volatility and regime switching. We find that volatility derivatives are particularly well suited to be treated with moment methods, whereby one extrapolates the distribution of the relevant path functionals on the basis of a few moments. We consider a number of exotics such as variance knockouts, conditional corridor variance swaps, gamma swaps and variance swaptions and give valuation formulas in detail.

q-fin.ST↗

arXiv2007

Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions

We propose a new classification scheme for diffusion processes for which the backward Kolmogorov equation is solvable in analytically closed form by reduction to hypergeometric equations of the Gaussian or confluent type. The construction makes use of transformations of diffusion processes to eliminate the drift which combine a measure change given by Doob's h-transform and a diffeomorphism. Such transformations have the important property of preserving analytic solvability of the process: the transition probability density for the driftless process can be expressed through the transition probability density of original process. We also make use of tools from the theory of ordinary differential equations such as Liouville transformations, canonical forms and Bose invariants. Beside recognizing all analytically solvable diffusion process known in the previous literature fall into this scheme and we also discover rich new families of analytically solvable processes.

math.PR↗

arXiv2007

Laplace Transforms for Integrals of Markov Processes

Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitional probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree.

math.PR↗

arXiv2004

Mirror Inversion of Quantum States in Linear Registers

Transfer of data in linear quantum registers can be significantly simplified with pre-engineered but not dynamically controlled inter-qubit couplings. We show how to implement a mirror inversion of the state of the register in each excitation subspace with respect to the centre of the register. Our construction is especially appealing as it requires no dynamical control over individual inter-qubit interactions. If, however, individual control of the interactions is available then the mirror inversion operation can be performed on any substring of qubits in the register. In this case a sequence of mirror inversions can generate any permutation of a quantum state of the involved qubits.

quant-ph↗

arXiv2003

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.

hep-th↗