304 research outputs found
Constant Maturity Credit Default Swap Pricing with Market Models
In this work we derive an approximated no-arbitrage market valuation formula
for Constant Maturity Credit Default Swaps (CMCDS). We move from the CDS
options market model in Brigo (2004), and derive a formula for CMCDS that is
the analogous of the formula for constant maturity swaps in the default free
swap market under the LIBOR market model. A "convexity adjustment"-like
correction is present in the related formula. Without such correction, or with
zero correlations, the formula returns an obvious deterministic-credit-spread
expression for the CMCDS price. To obtain the result we derive a joint dynamics
of forward CDS rates under a single pricing measure, as in Brigo (2004).
Numerical examples of the "convexity adjustment" impact complete the paper
Probability-free models in option pricing: statistically indistinguishable dynamics and historical vs implied volatility
We investigate whether it is possible to formulate option pricing and hedging
models without using probability. We present a model that is consistent with
two notions of volatility: a historical volatility consistent with statistical
analysis, and an implied volatility consistent with options priced with the
model. The latter will be also the quadratic variation of the model, a pathwise
property. This first result, originally presented in Brigo and Mercurio (1998,
2000), is then connected with the recent work of Armstrong et al (2018), where
using rough paths theory it is shown that implied volatility is associated with
a purely pathwise lift of the stock dynamics involving no probability and no
semimartingale theory in particular, leading to option models without
probability. Finally, an intermediate result by Bender et al. (2008) is
recalled. Using semimartingale theory, Bender et al. showed that one could
obtain option prices based only on the semimartingale quadratic variation of
the model, a pathwise property, and highlighted the difference between
historical and implied volatility. All three works confirm the idea that while
historical volatility is a statistical quantity, implied volatility is a
pathwise one. This leads to a 20 years mini-anniversary of pathwise pricing
through 1998, 2008 and 2018, which is rather fitting for a talk presented at
the conference for the 45 years of the Black, Scholes and Merton option pricing
paradigm.Comment: Paper presented at the conference "Options: 45 Years after the
publication of the Black-Scholes-Merton Model", Jerusalem, 4-5 December 201
The direct L2 geometric structure on a manifold of probability densities with applications to Filtering
In this paper we introduce a projection method for the space of probability
distributions based on the differential geometric approach to statistics. This
method is based on a direct L2 metric as opposed to the usual Hellinger
distance and the related Fisher Information metric. We explain how this
apparatus can be used for the nonlinear filtering problem, in relationship also
to earlier projection methods based on the Fisher metric. Past projection
filters focused on the Fisher metric and the exponential families that made the
filter correction step exact. In this work we introduce the mixture projection
filter, namely the projection filter based on the direct L2 metric and based on
a manifold given by a mixture of pre-assigned densities. The resulting
prediction step in the filtering problem is described by a linear differential
equation, while the correction step can be made exact. We analyze the
relationship of a specific class of L2 filters with the Galerkin based
nonlinear filters, and highlight the differences with our approach, concerning
particularly the continuous--time observations filtering problems
Dangers of Bilateral Counterparty Risk: the fundamental impact of closeout conventions
We analyze the practical consequences of the bilateral counterparty risk
adjustment. We point out that past literature assumes that, at the moment of
the first default, a risk-free closeout amount will be used. We argue that the
legal (ISDA) documentation suggests in many points that a substitution closeout
should be used. This would take into account the risk of default of the
survived party. We show how the bilateral counterparty risk adjustment changes
strongly when a substitution closeout amount is considered. We model the two
extreme cases of default independence and co-monotonicity, which highlight pros
and cons of both risk free and substitution closeout formulations, and allow us
to interpret the outcomes as dramatic consequences on default contagion.
Finally, we analyze the situation when collateral is present
Consistent single- and multi-step sampling of multivariate arrival times: A characterization of self-chaining copulas
This paper deals with dependence across marginally exponentially distributed
arrival times, such as default times in financial modeling or inter-failure
times in reliability theory. We explore the relationship between dependence and
the possibility to sample final multivariate survival in a long time-interval
as a sequence of iterations of local multivariate survivals along a partition
of the total time interval. We find that this is possible under a form of
multivariate lack of memory that is linked to a property of the survival times
copula. This property defines a "self-chaining-copula", and we show that this
coincides with the extreme value copulas characterization. The self-chaining
condition is satisfied by the Gumbel-Hougaard copula, a full characterization
of self chaining copulas in the Archimedean family, and by the Marshall-Olkin
copula. The result has important practical implications for consistent
single-step and multi-step simulation of multivariate arrival times in a way
that does not destroy dependency through iterations, as happens when
inconsistently iterating a Gaussian copula
Interest-Rate Modelling in Collateralized Markets: Multiple curves, credit-liquidity effects, CCPs
The market practice of extrapolating different term structures from different
instruments lacks a rigorous justification in terms of cash flows structure and
market observables. In this paper, we integrate our previous consistent theory
for pricing under credit, collateral and funding risks into term structure
modelling, integrating the origination of different term structures with such
effects. Under a number of assumptions on collateralization, wrong-way risk,
gap risk, credit valuation adjustments and funding effects, including the
treasury operational model, and via an immersion hypothesis, we are able to
derive a synthetic master equation for the multiple term structure dynamics
that integrates multiple curves with credit/funding adjustments
Projection based dimensionality reduction for measure valued evolution equations in statistical manifolds
We propose a dimensionality reduction method for infinite-dimensional
measure-valued evolution equations such as the Fokker-Planck partial
differential equation or the Kushner-Stratonovich resp. Duncan-Mortensen-Zakai
stochastic partial differential equations of nonlinear filtering, with
potential applications to signal processing, quantitative finance, heat flows
and quantum theory among many other areas. Our method is based on the
projection coming from a duality argument built in the exponential statistical
manifold structure developed by G. Pistone and co-authors. The choice of the
finite dimensional manifold on which one should project the infinite
dimensional equation is crucial, and we propose finite dimensional exponential
and mixture families. This same problem had been studied, especially in the
context of nonlinear filtering, by D. Brigo and co-authors but the
structure on the space of square roots of densities or of densities themselves
was used, without taking an infinite dimensional manifold environment space for
the equation to be projected. Here we re-examine such works from the
exponential statistical manifold point of view, which allows for a deeper
geometric understanding of the manifold structures at play. We also show that
the projection in the exponential manifold structure is consistent with the
Fisher Rao metric and, in case of finite dimensional exponential families, with
the assumed density approximation. Further, we show that if the sufficient
statistics of the finite dimensional exponential family are chosen among the
eigenfunctions of the backward diffusion operator then the statistical-manifold
or Fisher-Rao projection provides the maximum likelihood estimator for the
Fokker Planck equation solution. We finally try to clarify how the finite
dimensional and infinite dimensional terminology for exponential and mixture
spaces are related.Comment: Added maximum likelihood theorem and projection approximation
analysis. Updated version to appear in: Nielsen, F., Critchley, F., & Dodson,
K. (Eds), Computational Information Geometry for Image and Signal Processing,
Springer Verlag, 201
Arbitrage-free Pricing of Credit Index Options: The no-armageddon pricing measure and the role of correlation after the subprime crisis
In this work we consider three problems of the standard market approach to
pricing of credit index options: the definition of the index spread is not
valid in general, the usually considered payoff leads to a pricing which is not
always defined, and the candidate numeraire one would use to define a pricing
measure is not strictly positive, which would lead to a non-equivalent pricing
measure.
We give a general mathematical solution to the three problems, based on a
novel way of modeling the flow of information through the definition of a new
subfiltration. Using this subfiltration, we take into account consistently the
possibility of default of all names in the portfolio, that is neglected in the
standard market approach. We show that, while the related mispricing can be
negligible for standard options in normal market conditions, it can become
highly relevant for different options or in stressed market conditions.
In particular, we show on 2007 market data that after the subprime credit
crisis the mispricing of the market formula compared to the no arbitrage
formula we propose has become financially relevant even for the liquid
Crossover Index Options.Comment: Updated version accepted for publication in Mathematical Financ
Credit Default Swap Calibration and Equity Swap Valuation under Counterparty Risk with a Tractable Structural Model
In this paper we develop a tractable structural model with analytical default
probabilities depending on some dynamics parameters, and we show how to
calibrate the model using a chosen number of Credit Default Swap (CDS) market
quotes. We essentially show how to use structural models with a calibration
capability that is typical of the much more tractable credit-spread based
intensity models. We apply the structural model to a concrete calibration case
and observe what happens to the calibrated dynamics when the CDS-implied credit
quality deteriorates as the firm approaches default. Finally we provide a
typical example of a case where the calibrated structural model can be used for
credit pricing in a much more convenient way than a calibrated reduced form
model: The pricing of counterparty risk in an equity swap.Comment: Reduced version in Proceedings of the FEA 2004 Conference at MIT,
Cambridge, Massachusetts, November 8-10, and in: Pykhtin, M. (Editor),
Counterparty Credit Risk Modeling: Risk Management, Pricing and Regulation.
Risk Books, 2005, Londo
Coordinate-free Stochastic Differential Equations as Jets
We explain how It\^o Stochastic Differential Equations (SDEs) on manifolds
may be defined using 2-jets of smooth functions. We show how this relationship
can be interpreted in terms of a convergent numerical scheme. We show how jets
can be used to derive graphical representations of It\^o SDEs. We show how jets
can be used to derive the differential operators associated with SDEs in a
coordinate free manner. We relate jets to vector flows, giving a geometric
interpretation of the It\^o--Stratonovich transformation. We show how
percentiles can be used to give an alternative coordinate free interpretation
of the coefficients of one dimensional SDEs. We relate this to the jet
approach. This allows us to interpret the coefficients of SDEs in terms of "fan
diagrams". In particular the median of a SDE solution is associated to the
drift of the SDE in Stratonovich form for small times
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