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"Interest Rates and The Credit Crunch: New Formulas and Market Models"
We start by describing the major changes that occurred in
the quotes of market rates after the 2007 subprime mortgage crisis. We
comment on their lost analogies and consistencies, and hint on a
possible, simple way to formally reconcile them. We then show how to
price interest rate swaps under the new market practice of using
different curves for generating future LIBOR rates and for discounting
cash flows. Straightforward modifications of the market formulas for
caps and swaptions will also be derived.
Finally, we will introduce a new LIBOR market model, which will be
based on modeling the joint evolution of FRA rates and forward rates
belonging to the discount curve. We will start by analyzing the basic
lognormal case and then add stochastic volatility. The dynamics of FRA
rates under different measures will be obtained and closed form
formulas for caplets and swaptions derived in the lognormal and Heston
(1993) cases |
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“Smile-consistent
CMS adjustments in closed form: introducing the
Vanna-Volga approach"
In this
article, we
introduce the Vanna-Volga approach for an alternative valuation of CMS
convexity adjustments. Our pricing procedure leads to closed-form
formulas that are extremely simple to implement and that retrieve,
within bid-ask spreads, market data of CMS swap spreads. |
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"No-arbitrage conditions for
cash-settled swaptions"
In this note, we derive no-arbitrage conditions that must be satisfied
by the pricing function of cash-settled swaptions. The specific
examples of a flat implied volatility and of a smile generated by the
SABR functional form will be analyzed |
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"Consistent Pricing of FX
Options"
We deal with the problem of inferring implied volatilities for non
quoted strikes and analyze a possible solution in a foreign exchange
(FX) option market. In such a market, in fact, there are only
three active quotes for each market maturity, thus presenting us with
the problem of a consistent determination of the other implied
volatilities. FX brokers and market makers typically address this issue
by using an empirical procedure to construct the whole smile for a
given maturity. Volatility quotes are then provided in terms of the
option’s delta, for ranges from the 5 delta put to the 5
delta call. In
this article, we will review this market procedure for a given currency
and prove related robustness and consistency results. |
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"Swaption skews and convexity
adjustments"
We test both the SABR model and the shifted-lognormal mixture model as
far as the joint calibration to swaption smiles and CMS swap spreads is
concerned. Such a joint calibration is essential to consistently
recover implied volatilities for non-quoted strikes and CMS adjustments
for any expiry-tenor pair. |
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"Mixing Gaussian Models to Price CMS Derivatives"
In
this article, we propose a simple interest rate model, which can well
accommodate swaption smiles, while recovering market prices of CMS swap
spreads. The model is based on a (possibly multi-factor) Gaussian short
rate model coupled with parameter uncertainty. Examples of calibration
to real market data will be presented as well as the pricing of some
typical CMS-based derivatives. |
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"Pricing Inflation-Indexed
Options with
Stochastic Volatility"
In order to recover smile-consistent prices for inflation-indexed caps
and floors, we propose a stochastic volatility model for forward
consumer price indices, with volatility dynamics as in Heston (1993).
Closed-form formulas for inflation-indexed caplets and floorlets based
on the Carr and Madan (1998) Fourier transform approach are then
derived. An example of calibration to market data and numerical details
concerning our pricing procedure are finally given. |
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"Consistent Pricing and
Hedging of an FX
Options Book"
In the foreign exchange (FX) options market away-from-the-money options
are quite actively traded, and quotes for the same type of instruments
are available everyday with very narrow spreads (at least for the main
currencies). This makes it possible to devise a procedure for
extrapolating the implied volatilities of non-quoted options, providing
us with reliable data to which to calibrate our favorite model.
In this article, we test the goodness of the Brigo, Mercurio and
Rapisarda (2004) model as far as some fundamental practical
implications are concerned. This model, which is based on a geometric
Brownian motion with time-dependent coefficients that are not known
initially and whose value is randomly drawn at an infinitesimal future
time, can accommodate very general volatility surfaces and, in case of
the FX options market, can lead to a perfect fit to the main volatility
quotes.
We first show the fitting capability of the model with an example from
real market data. We then support the goodness of our calibration by
providing a diagnostic on the forward volatilities implied by the
model. We also compare the model prices of some exotic options with the
corresponding ones given by a market practice. Finally, we show how to
derive bucketed sensitivities to volatility and how to hedge
accordingly a typical options book. |
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"Pricing Inflation-Indexed
Derivatives"
We start by briefly reviewing the approach proposed by Jarrow and
Yildirim (2003) for modelling inflation and nominal rates in a
consistent way. Their methodology is applied to the pricing of general
inflation-indexed swaps and options. We then introduce two different
market model approaches to price inflation swaps, caps and floors.
Analytical formulas are explicitly derived. Finally, an example of
calibration to swap market data is considered.
|
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"A Shifted Lognormal LIBOR
Model with
Uncertain Parameters"
The lognormal
LIBOR market
model leads to flat implied volatility surfaces. To accommodate the
typical smile effect in the caps and swaptions markets, one then
usually resort to local- or stochastic-volatility models, possibly with
jumps. This models are generally difficult to implement and rather time
consuming when used to price and hedge exotic derivatives. However, it
is possible to introduce stochasticity in the volatility in a very
simple and intuitive manner, so as to calibrate the market implied
volatility surfaces while preserving a great deal of analytical
tractability. This is achieved by assuming that the forward rate
dynamics are given by displaced geometric Brownian motions where the
model parameters are not known at the initial time, but are discrete
random variables whose values are drawn at an infinitesimal time. We
refer to our model as to a Shifted Lognormal LIBOR Model with Uncertain
Parameters (SLLMUP).
In this article, we analyze
the SLLMUP
analytical tractability by deriving caps and swaptions prices in closed
form. We then illustrate
how the model can accommodate market caps data and how the
instantaneous correlation parameters can be used for a calibration to swaptions
prices. We
finally analyze some important model's implications: i) we infer the
swaptions smile implied by our joint calibration, and ii) we plot the
evolution of some forward volatilities implied by the model.
|
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"Smile at the Uncertainty"
The success of the
Black-Scholes
formula is mainly due to the possibility of synthesizing option prices
through a unique parameter, the implied volatility. The Black-Scholes
model,
however, can not be used to price simultaneously all options in a given
market because of the smile/skew effect commonly observed in practice.
Moreover, historical analysis shows that volatilities are indeed
stochastic.
Stochastic volatility models, therefore, seem to be a more realistic
choice when modelling asset price dynamics for valuing derivative
securities.
However, only few examples retain enough analytical tractability.
The purpose of this paper is to propose a stochastic-volatility model
that is analytically tractable as much as Black and Scholes’s
and for
which
Vega hedging can be defined in a natural way. The model is based on an
uncertain volatility whose random value is drawn, on an infinitesimal
future
time, from a finite distribution.
Our uncertain volatility model is equivalent to assuming a number of
different possible scenarios for the asset forward volatility, which
can
therefore be hedged accordingly. An application to the FX market is
explicitly
considered.
|
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"A
Multi-Stage Uncertain Volatility Model".
We consider a simple
uncertain-volatility
model for the asset price underlying a given option market. The asset
price
volatility is assumed to follow a discrete (actually finite) Markov
chain,
which changes value on some fixed future times. The volatility chain is
independent of the Brownian motion governing the future evolution of
the
asset.
Modeling the volatility evolution in this way is equivalent to assuming
different possible scenarios for the asset forward volatility. Closed
form
formulas for the claims that are explicitly priced under the
Black-Scholes
paradigm, are derived accordingly.
|
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"Parameterizing Correlations:
A
Geometric Interpretation".
Finance as a whole is
linked
to correlations among the prices of assets (stocks, bonds, . . . ) or
more
generally of basic financial quantities (including interest rates,
credit
spreads, . . . ). Correlations are usually expressed in terms of
matrices,
which must satisfy four basic properties: A1) all their entries must
lie
in the interval [-1;1]; A2) diagonal entries must be equal to one; A3)
the matrix must be symmetric; A4) the matrix must be positive
semidefinite.
In many applications, it is fundamental to be able to safely build
and manipulate correlation structures, with which to model the joint
behaviour
of financial variables, both for pricing purposes and for risk
management.
This implies the need for a parameterization of a generic correlation
matrix
– a parameterization guaranteeing that the matrix will
fulfill all four
properties above. In this paper we will revise the “standard
angles
parameterization”
(SAP) procedure described for example in Pinheiro and Bates [5],
recently
applied to financial modeling by Rebonato and Jaeckel [8], and discuss
a geometric interpretation to it.
Moreover, thanks to the new geometrical view, we will recast the SAP
in a more parsimonious way, leading to the “triangular angles
parameterization”
(TAP), which is actually the form of SAP described in [5]. Finally, we
consider an application to a demanding problem, that of sensibly
parameterizing
instantaneous correlations in the LMM.
|
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"Pricing the Smile in a
Forward LIBOR
Market Model".
We introduce two general
classes
of analytically-tractable diffusions for modeling forward LIBOR rates
under
their canonical measure.
The first class is based on the assumption of forward-rate densities
given by the mixture of known basic densities. We consider two
fundamental
examples: i) a mixture of lognormal densities, and ii) a mixture of
densities
associated to “hyperbolic-sine” processes. We
derive explicit dynamics,
prove existence and uniqueness results for the solution to the related
SDEs and obtain closed-form formulas for caps prices.
The second class is based on assuming a smooth functional dependence,
at expiry, between a forward rate and an associated Brownian motion.
This
class is highly tractable: it implies explicit dynamics, known
marginal
and transition densities and explicit caplet prices at any time. As an
example, we analyze the dynamics given by a linear combination of
geometric
Brownian motions (GBM) with perfectly correlated
(decorrelated)
returns.
We finally construct a specific model in the second class that
reproduces
exactly the market caplet volatilities given in input.
Examples of the implied-volatility curves produced by the considered
models are also shown. |
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"Lognormal-Mixture Dynamics
under
Different Means".
We prove existence and
uniqueness
of the strong solution to the SDE whose drift rate is a given constant
and whose diffusion coefficient is defined so as to imply a marginal
density
that is given by a mixture of lognormal densities. Such densities,
which
can have different means, must fulfill the no-arbitrage conditions that
typically arise in mathematical finance when using the given SDE for
modeling
the price dynamics of some financial asset. |
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"Alternative Asset-Price
Dynamics and
Volatility Smile".
We propose a general class
of
analytically tractable models for the dynamics of an asset price
leading
to smiles or skews in the implied volatility structure. The considered
asset can be an exchange rate, a stock index, or even a forward Libor
rate.
The class is based on an explicit SDE under a given forward measure.
The
models we propose feature i) explicit asset-price dynamics, ii)
virtually
unlimited number of parameters, iii) analytical formulas for European
options.
We then consider the fundamental case where the asset price density
is given, at every time, by a mixture of lognormal densities. We also
derive
an explicit approximation of the implied volatility function in terms
of
the option moneyness. We finally introduce two other examples: the
first
is still based on lognormal densities, but it allows for different
means
in the distributions, the second is instead based on processes of
hyperbolic-sine
type. |
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"Approximated Moment-Matching
Dynamics
for Basket-Options Simulation".
The aim is this paper is
to
present two moment matching procedures for basket-options pricing and
to
test their distributional approximations via distances on the space of
probability densities, the Kullback-Leibler information (KLI) and the
Hellinger
distance (HD). We are interested in
measuring
the
KLI and the HD between the real simulated basket terminal distribution
and the distributions used for the approximation, both in the lognormal
and shifted-lognormal families. We isolate influences of instantaneous
volatilities and instantaneous correlations, in order to assess which
configurations
of these variables have a major impact on the KLI and HD and therefore
on the quality of the approximation. A number of numerical
investigations
is carried out.
|
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"Lognormal-Mixture
Dynamics and Calibration to Market Volatility Smiles".
We introduce a general
class
of analytically tractable models for the dynamics of an asset price
based
on the assumption that the asset-price density is given by the mixture
of known basic densities. We consider the lognormal-mixture model as a
fundamental example, deriving explicit dynamics, closed form formulas
for
option prices and analytical approximations for the implied volatility
function. We then introduce the asset-price model that is obtained by
shifting
the previous lognormal-mixture dynamics and investigate its analytical
tractability. We finally consider a specific example of calibration to
real market option data. |
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"On
Deterministic-Shift
extensions of Short-Rate Models".
In the present paper we
show
how to extend any time-homogeneous short-rate model (such as Vasicek
(1977),
Cox-Ingersoll-Ross (1985), Dothan (1978)) to a model that can reproduce
any observed yield curve, through a procedure that preserves
the
possible analytical tractability of the original model. In
the
case
of the Vasicek (1977) model, our extension is equivalent to that of
Hull
and White (1990), whereas in the case of the Cox-Ingersoll-Ross (1985)
(CIR) model, our extension is more analytically tractable and
avoids
problems concerning the use of numerical solutions. Our approach can
also
be applied to the Dothan (1978) or Rendleman and Bartter
(1980)
model,
thus yielding a "quasi" lognormal short-rate model which fits any given
yield curve and for which there exist analytical formulae for
prices
of zero coupon bonds. We also consider the extension of
time-homogeneous
models without analytical formulae but whose
tree-construction
procedures
are particularly appealing, such as the Exponential Vasicek's. We
explain
why the CIR++ extended CIR model is the more interesting model obtained
through our procedure. We also give explicit analytical formulae for
bond
options, hence swaptions, caps and floors, and we explain how
the
model can be used for Monte Carlo evaluation of European path-dependent
interest-rate derivatives. We finally hint at the same extension for
multifactor
models and explain its strong points for concrete applications. |
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"Discrete
Time vs Continuous Time Stock-price Dynamics and implications for
Option
Pricing".
In the present paper we
construct
stock price processes with the same marginal log-normal law as that of
a geometric Brownian motion and also with the same transition density
(and
returns' distributions) between any two instants in a given
discrete-time
grid. We then illustrate how option prices based on such processes
differ
from Black and Scholes', in that option prices can be either
arbitrarily
close to the option intrinsic value or arbitrarily close to the
underlying
stock price. We also explain that this is due to the particular way one
models the stock-price process in between the grid time instants which
are relevant for trading. The theoretical result concerning scalar
stochastic
differential equations with prescribed diffusion coefficient whose
densities
evolve in a prescribed exponential family, on which part of the paper
is
based, is presented in detail. |