Dynamic Conditioning and Credit Correlation Baskets

C. Albanese and A. Vidler

Dynamic conditioning is a technique that allows one to formulate correlation models for large baskets without incurring in the curse of dimensionality. The individual price processes for each reference name can be described by a lattice model specified semi-parametrically or even nonparametrically
and which can realistically have about 1000 sites. The time discretization step is chosen so small to satisfy the Courant stability condition and is typically of about a few hours. This constraint ensures needed smoothness for the single name probability kernels which can thus be directly manipulated. A flexible multi-factor correlation model can be obtained by means of conditioning trees corresponding to binomial processes with jumps. There is one conditioning tree associated to each reference names, one associated to each industry sector and a global one to the basket itself. Since the conditioning trees are correlated, the underlying processes are also mutually correlated.
In this paper, we discuss a modeling framework for CDOs based on dynamic conditioning in greater detail than previously done in our other papers. We also show that the model calibrates well to index tranches throughout in the period from 2005 to the end of 2007 and yields instructive insights.

Stochastic Integrals and Abelian Processes

C. Albanese

We study triangulation schemes for the joint kernel of a diffusion process with uniformly continuous coecients 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 coecients. 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. 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 reobtained.

Kernel Convergence Estimates for Diffusions with Continuous Coefficients

C. Albanese

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 tends to 0. 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 Holder
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 Holder differentiability
except that the convergence rate is slower. Holder continuity
itself is not strictly necessary and can be relaxed by an hypothesis
of uniform continuity.

Operator Methods, Abelian Processes and Dynamic Conditioning

C. Albanese

A mathematical framework for Continuous Time Finance based on
operator algebraic methods offers a new direct and entirely
constructive perspective on the field. It also leads to new
numerical analysis techniques which can take advantage of the
emerging massively parallel GPU architectures which are uniquely
suited to execute large matrix manipulations.

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 mathematical framework self-consistent. 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.

Callable Swaps, Snowballs and Videogames

C. Albanese

Although economically more meaningful than the alternatives, short
rate models have been dismissed for financial engineering
applications in favor of market models as the latter are more
flexible and best suited to cluster computing implementations. In
this paper, we argue that the paradigm shift toward GPU
architectures currently taking place in the high performance
computing world can potentially change the situation and tilt the
balance back in favor of a new generation of short rate models. We
find that operator methods provide a natural mathematical framework
for the implementation of realistic short rate models that match
features of the historical process such as stochastic monetary
policy, calibrate well to liquid derivatives and provide new
insights on complex structures. In this paper, we show that callable
swaps, callable range accruals, target redemption notes (TARNs) and
various flavors of snowballs and snowblades can be priced with
methods numerically as precise, fast and stable as the ones based on
analytic closed form solutions by means of BLAS level-3 methods on
massively parallel GPU architectures.

Moment Methods for Exotic Volatility Derivatives

C. Albanese and A. Osseiran

The latest generation of volatility derivatives goes beyond variance
and volatility swaps and probe 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.

A Structural Model for Credit-Equity Derivatives and Bespoke CDOs

C. Albanese, A. Vidler

We present a new structural model for single name equity and credit derivatives which we also correlate across reference names to produce a model for bespoke synthetic CDOs. The model captures volatility and outlook risk along with correlation risk for small and for large moves separately. We show that the model calibrates well to both equity structured products and credit derivatives. As examples, we discuss a number of single name derivatives on IBM spanning the credit-equity spectrum and ranging from volatility swaps, to cliquets, CDS options and CDSs on leveraged loans with pre-payment risk. We also use the model to price tranches on the investment grade DJ.CDX.IG index along with tranches on the high yield index DJ.CDX.HY. We show that the model gives consistent and high precision pricing across all these derivative asset classes. We show that this can be achieved consistently, with the very
same parameter choices across these diverse derivative assets and making use of only minor explicit time dependencies.

Monetary Policy Risk and CMS Spreads

C. Albanese, M. Trovato

Central banks’ monetary policies are regarded by financial institutions as a key driver for the definition of their interest rate hedging strategies. However this valuable information is not directly incorporated in most derivative pricing models commonly used in financial institutions. We present a novel approach to interest rate modelling, which incorporates a direct specification of stochastic monetary policy within an arbitrage-free context. In particular, we take the 3-month spot LIBOR rate as modelling primitive and propose a three factor interest rate term structure model solved on a continuous-time lattice. The model is constructed with local volatility, stochastic volatility regimes and stochastic drift regimes and can be formally written as dL_t = μ_at (L_t)dt + s_bt (L_t) dW_t + jumps. The drift and the volatility terms are stochastic and driven by the processes at and bt, which can be made correlated to the rates themselves. We show that, with a nearly time-homogeneous parameterisation, the model can achieve a persistent smile structure across maturities, in agreement with the EUR market, and that the model implied correlation structure is consistent with historical estimates. We apply the model to callable swaps and callable CMS spread range accruals and analyze the impact of monetary policy to the valuation of these derivative contracts.

Convergence Estimates for Diffusions on Continuous Time Lattices

C. Albanese, A. Mijatovic

In this paper we introduce a discretization scheme based on a continuous-time Markov chain for the Black-Scholes diffusion process. Our principal aim is to find the optimal convergence rate for the probability density function of the discretized process as the distance h between the nodes of the state-space of the Markov chain goes to zero. The main theorem of
the paper (theorem 4.1) states that the probability kernel $P^h_t(x, y)$ of the discretized process converges at the rate $O(h^2)$ to the probability density function $p_t(x, y)$ of the diffusion process. We also show that this convergence is uniform in the state variables x and y and that
the proposed discretization scheme converges at a rate which is no faster than $O(h^2)$.

A stochastic monetary policy interest rate model

C. Albanese, M. Trovato

We present a three factor interest rate term structure model solved on a continuous-time lattice and constructed with local volatility, asymmetric jumps, stochastic volatility regimes and stochastic monetary policy. We take the 3-month spot LIBOR rate as modelling primitive. An analysis of the historical data suggests that the process is subject to drift regimes. As the 3-month spot LIBOR rate is not an asset price process, the drift is not constrained by the no-arbitrage condition. We incorporate a direct specification of the drift process in our model: this is a novel approach in interest rate modelling and it represents one of the main contributions of this work. The model can be formally written as dL_t = μ_at (L_t)dt + s_bt (L_t) dW_t + jumps where the drift and the volatility terms are stochastic and driven by the processes at, bt, correlated to the rates themselves. We show that the model can achieve a persistent smile structure across maturities with a nearly time homogeneous parameterisation, in good qualitative agreement with the EUR market. We also show that the model is able to explain most yield curve shapes of interest, and that these are crucially affected by the drift regimes, as one would intuititively expect. Furthermore we explain that drift modelling provides a powerful tool in order to control the long term behaviour of the process without affecting considerably the dynamics at earlier times. In fact, whilst jumps are predominant at short maturities, stochastic volatility has the greatest impact at medium maturities, wheareas drift is the main driver in the long end. In addition, we argue that direct drift modelling is economically meaningful and allows one to impose economic views on central banks monetary policy into the model.

A Numerical Method for Pricing Electricity Derivatives for
Jump-Diffusion Processes Based on Continuous Time Lattices

C. Albanese, H. Lo, S. Tompaidis

We present a numerical method for pricing derivatives on electricity prices. The method is based on approximating the generator of the underlying process and can be applied for stochastic processes that are combinations of diffusions and jump processes. The method is accurate even in the case of processes with fast mean-reversion and jumps of large magnitude. We illustrate the speed and accuracy of the method by pricing European and Bermudan options and calculating the hedge ratios of European options for the Geman-Roncoroni model for electricity prices.

A stochastic volatility model for callable CMS swaps and translation invariant path dependent derivatives

C. Albanese, M. Trovato

We present a stochastic volatility term structure model based on a continuous time lattice which allows for a numerically stable and quite efficient methodology to price fixed income exotics. We present numerical applications to bermudan swaptions and callable CMS swaps. We then extend the model to translation invariant path dependent payo¤s by means of an efficient dimensional reduction technique and show its application to callable range accruals and target redemption notes.

Spectral methods for volatility derivatives

C. Albanese, H. Lo, A. Mijatovic

IIn 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 opened 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 define a stochastic volatility model with jumps and local volatility, which is almost stationary, and calibrate it to the European options on the S&P 500 for a broad range of strikes and maturities. We then extend the model, by lifting the corresponding Markov generator, to keep track of relevant path information, namely the realized variance. The lifted generator is too large a matrix to be diagonalized numerically. We overcome this difficulty by developing 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 the European options, general accrued variance payoffs as well as forward-starts and VIX options.

Dynamic Credit Correlation Modelling

C. Albanese, O. Chen, A. Dalessandro, A. Vidler

As the market for credit baskets and single tranche bespoke CDOs keeps growing very rapidly, we witness a lively debate about the merits and shortcomings of the base correlation model, which is currently the recognized market standard. Difficulties such as the lack of arbitrage-freedom and the witnessed impossibility to calibrate in some market situations are motivations to research a different standard for mark-to-market and risk management. To contribute to this ongoing debate, we describe here a new modeling framework based on a structural, bottom-up approach. Points of interest for this model are that it can be made consistent with many data sources such as rating transition probabilities, historical default probabilities, single name credit spread curves and equilibrium recovery swap rates. Remarkably enough, we find that the model can be calibrated simultaneously to synchronous datasets for the iTraxx and CDX term structures for tranche spreads across the entire capital structure, including the index basis, and for maturities up to 10 years. The model makes use of an innovative mathematical framework based on spectral analysis and provides numerically noiseless spreads and hedge ratios. As far as execution times are concerned, the model is at least as fast if not faster than the most simplified analytic versions of the base correlation model.

A stochastic volatility model for risk-reversals in foreign exchange

C. Albanese, A. Mijatovic

It is a widely recognised fact that risk-reversals play a central role in pricing of derivatives in foreign exchange markets. It is also known that the values of risk-reversals vary stochastically with time. In this paper we introduce a stochastic volatility model with jumps and local volatility, defined on a continuous time lattice, which provides a way of modeling this kind of risk using a numerically efficient algorithm.

A Stochastic Volatility Model for Bermuda Swaptions and Callable Constant Maturity Swaps

C. Albanese, M. Trovato

It is widely recognized that fixed income exotics should be priced by means of a stochastic volatility model. Callable constant maturity swaps (CMS) are a particularly interesting case due to the sensitivity of swap rates to implied swaption volatilities for very deep out of the money strikes. In this paper, we introduce a stochastic volatility term structure model based on a continuous time lattice which allows for a numerically stable and quite efficient methodology to price fixed income exotics in this class.

Transformations of Markov Processes and Classification Scheme for Solvable Driftless Diffusions

C. Albanese, A. Kuznetsov

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.

Pricing Equity Default Swaps

C. Albanese, O.Chen

Appeared in Risk Magazine

Pricing credit-equity hybrids is a challenging task as the established pricing methodologies for equity options and credit derivatives are quite different. Equity default swaps provide an illuminating example of the clash of methodologies: from the equity derivatives viewpoint they are digital American puts with payments in installments and thus would naturally be priced by means of a local volatility model, but from the credit viewpoint they share features with credit default swaps and thus should be priced with a model allowing for jumps and possibly jump to default. The question arises of whether the two model classes can be consistent. In this paper we answer this question in the negative and find that market participants appear to be pricing equity default swaps by means of local volatility models not including jumps. We arrive at this conclusion by comparing a CEV model with an absorbing default barrier and a credit barrier model together with a credit-to-equity mapping that is calibrated to achieve consistency between equity option data, credit default swap spreads and historical credit transition probabilities and default frequencies.

Discrete Credit Barrier Models

C. Albanese, O. Chen

Quantitative Finance, to appear

The model introduced in this article is designed to pr ovide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that takes on values on a discrete lattice and represents credit quality. Rating transitions are associated to barrier crossings and default events are associated with an absorbing state. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift is estimated to consistently match the average spread curves corresponding to all the various ratings .

Laplace Transforms for Integrals of Stochastic Processes

C. Albanese, S. Lawi

To appear in Markov Processes and Related Fields

Laplace transforms for integrals of stochastic processes have been known in analytically closed form in just a handful of cases: 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 modeling 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 generating functions for stochastic integrals and transition 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-Wilson classification tree.

Affine Lattice Models

C. Albanese, A. Kuznetsov

International Journal of Theoretical and Applied Finance, to appear
We introduce a n ew class of lattice models based on a Poisson approximation scheme for affine processes, whereby the approximant process itself is affine. A key property of this class of lattice models is that the location of the time nodes can be chosen in a payoff dependent way and one has the flexibility of setting them only at the relevant dates. Time stepping invariance relies on the ability of computing node-to-node discounted transition probabilities in analytical closed form. The method is quite general and far reaching and it is introduced in this article in the framework of the broadly used single-factor, affine sho rt rate models. To illustrate the use of affine lattice models in these cases, we analyze in detail the example of Bermuda swaptions.

Discretization Schemes for Subordinated Processes

C. Albanese, A. Kuznetsov

We introduce a Poisson approximation scheme for jump processes and use it to construct numerical discretizations for the corresponding partial integro-differential equations. Transition probabilities are computed analytically as expansions in orthogonal polynomials to ensure that results don’t depend on the size of time steps and to maintain local no-arbitrage conditions.

Unifying the Three Volatility Models

C. Albanese, A. Kuznetsov

Risk Magazine, Risk Magazine (2004), Volume 17(3), p.94-98

This article describes a method for building analytically tractable option pricing models which combine state dependent vo latility, stochastic volatility and jumps. Starting from a Laguerre representation of the pricing kernel, we show how to account for jumps and stochastic volatility by altering the time dependent coefficients of a series expansion. This operation is easy to implement analytically and gives rise to numerically efficient formulas for the pricing kernel. Stemmed from a line of research on barrier models for credit derivatives, the method is presented here in the broader context of option pricing theory.

Implied Migration Rates from Credit Barrier Models

C. Albanese, O. Chen

Journal of Banking and Finance, to appear

The risk neutral credit migration process captures quantitative information which is relevant to the pricing theory and risk management of credit derivatives. In this article, we derive implied migration rates by means of a recently introduced credit barrier model which is calibrated on the basis of aggregate information such as credit migration rates and credit spread curves. The model is characterized by an underlying stochastic process that represents credit quality and default events are associated to barrier crossings. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift and implied recovery rates are estimated to consistently match the average spread curves corresponding to all the various ratings. The implied migration rates obtained with our credit barrier model are then compared with those obtained via the Jarrow-Lando-Turnbull model by the Kijima-Komoribayashi model in a detailed example.

Credit Barrier Models

C. Albanese, J. Campolieti, O. Chen, A. Zavidonov

Risk Magazine, June 2003

The model introduced in this article is designed to provide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that represents credit quality and default events are associated to barrier crossings. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift and implied recovery rates are estimated to consistently match the average spread curves corresponding to all the various ratings.

Spectral Risk Measures for Credit Portfolios

C. Albanese, S. Lawi

"Risk Measures for Banking and Regulation", ed. G.Szego

In this article, we experiment with several different risk measures such as standard deviation, value-at-risk, expected shortfall and power-law spectral measures. We consider several families of test portfolios, one with a typical market risk profit-and-loss profile, and the others containing defaultable bonds of various credit ratings and various degree of diversification. We find that the risk measures are roughly equivalent on the market risk portfolios but differ significantly on the credit ones. In fact, value-at-risk as well as some coherent risk measures including expected shortfall have counter-intuitive fea tures as a function of the deg ree of diversification for return distributions deviating substantially from the normal. Certain spectral risk measures possess the most intuitive properties regarding diversification.

Black-Scholes Goes Hypergeometric

C. Albanese, J. Campolieti, P. Carr, A. Lipton

Risk Magazine, December 2001

We introduce a general pricing formula that extends Black-Scholes’ and contains as particular cases most analytically solvable models in the literature, including the quadratic and the constant-elasticity of-variance (CEV) models for European and barrier options. In addition, large families of new solutions can be found, containing as many as seven free para meters.

Jumping in Line

C. Albanese, S. Jaimungal, D. Rubisov

Risk Magazine, February 2001

We introduce an efficient numerical method to price derivative claims assuming the underlying follows a jump process of the variance Gamma (VG) type. The algorithm is based on the method of lines and i nvolves the solution of ordinary differential equations and a Richardson extrapolation method. The method is applied to equity, Bermudan and barrier options with VG jumps.

A Two State Jump Model

C. Albanese, S. Jaimungal, D. Rubisov

Quantitative Finance, April 2003

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. The model is capable of justifying the observed implied volatility skews for options at all maturities. Furthermore, the term structure of implied varianc e-rate appears to be an increasing function of the time to maturity, in agreement with empirical evidence. As in GARCH type models, jump sizes are positively correlated to volatility. Explicit extensions of the VG pricing formulas for European ptions are developed, in which complex combinatorial expressions arise whose valuation is hardly feasible. The main result of the article is a resummation algorithm based on the method of lines, which greatly reduces the algorithmic complexity of the pricing formulas. 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.

A New Fourier Transform Algorithm for Value-at-risk

C. Albanese, K. Jackson, P. Wiberg

Quantitative Finance, to appear

Methods to computing value-at-risk gradients with respect to portfolio positions have many applications. They include calculation of capital/reward efficient frontiers, hedging of der i vative portfolios and optimal replication. We present a new algorithm for computing value-at-risk and its gradients. If the return can be decomposed as a sum of independent portfolio marginals, the pay-off distribution can be computed with multiple convolutions. This principal-component-type decomposition is also useful for calibrating fat-tailed distributions. We conclude with two applications of hedging with value-at-risk.

Integrability by Quadratures of Pricing Equations

C. Albanese, G. Campolieti

We introduce a canonical transformation method for finding solutions to pricing problems by quadratures. The method is systematic and allows one to derive in a unified framework the exact solutions in the pricing literature. As an application, we construct a new families of pricing models based on the squared Bessel process which extends the constant-variance-of-volatility (CEV) model and is integrable by quadratures.

Dimension Reduction for Value-at-risk

C. Albanese, K. Jackson, P. Wiberg

Journal of Risk Finance, June 2002

In this article we present two new portfolio-dependent methods for dimension reduction in models for market risk, the risk of a decrease in the value of a portfolio due to adverse market movements.

Small Transaction Cost Asymptotics and Dynamic Hedging

C. Albanese, S. Tompaidis

European Journal of Operational Research, to appear

Transaction costs are one of the major impediments to the implementation of dynamic hedging strategies. We consider an alternative to utility maximization, similar to the “good-deal” pricing framework in incomplete markets. We perform a dynamic riskreward analysis for a family of non-self-financing strategies of practical importance: deterministic time hedging; i.e., hedging at predetermined, fixed, times. In the limit of small relative transaction costs, we carry out the asymptotic analysis and find that transaction costs affect the hedge ratios and that the time between trades is related in a simple way to the local sensitivities of the replication target.