9 research outputs found

### Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility

We derive generalizations of Dupire formula to the cases of general
stochastic drift and/or stochastic local volatility. First, we handle a case in
which the drift is given as difference of two stochastic short rates. Such a
setting is natural in foreign exchange context where the short rates correspond
to the short rates of the two currencies, equity single-currency context with
stochastic dividend yield, or commodity context with stochastic convenience
yield. We present the formula both in a call surface formulation as well as
total implied variance formulation where the latter avoids calendar spread
arbitrage by construction. We provide derivations for the case where both short
rates are given as single factor processes and present the limits for a single
stochastic rate or all deterministic short rates. The limits agree with
published results. Then we derive a formulation that allows a more general
stochastic drift and diffusion including one or more stochastic local
volatility terms. In the general setting, our derivation allows the computation
and calibration of the leverage function for stochastic local volatility
models

### Calibrating Local Volatility Models with Stochastic Drift and Diffusion

We propose Monte Carlo calibration algorithms for three models: local
volatility with stochastic interest rates, stochastic local volatility with
deterministic interest rates, and finally stochastic local volatility with
stochastic interest rates. For each model, we include detailed derivations of
the corresponding SDE systems, and list the required input data and steps for
calibration. We give conditions under which a local volatility can exist given
European option prices, stochastic interest rate model parameters, and
correlations. The models are posed in a foreign exchange setting. The drift
term for the exchange rate is given as a difference of two stochastic short
rates, domestic and foreign, each modeled by a G1++ process. For stochastic
volatility, we model the variance for the exchange rate by a CIR process. We
include tests to show the convergence and the accuracy of the proposed
algorithms

### De Sitter in Extended Supergravity

We show that known de Sitter solutions in extended gauged supergravity
theories are interrelated via a web of supersymmetry-breaking truncations. In
particular, all N=8 models reduce to a subset of the N=4 possibilities.
Furthermore, a different subset of the N=4 models can be truncated to stable de
Sitter vacua in N=2 theories. In addition to relations between the known cases,
we also find new (un)stable models.Comment: 16 page

### Metastable de Sitter vacua in N=2 to N=1 truncated supergravity

We study the possibility of achieving metastable de Sitter vacua in general
N=2 to N=1 truncated supergravities without vector multiplets, and compare with
the situations arising in N=2 theories with only hypermultiplets and N=1
theories with only chiral multiplets. In N=2 theories based on a quaternionic
manifold and a graviphoton gauging, de Sitter vacua are necessarily unstable,
as a result of the peculiar properties of the geometry. In N=1 theories based
on a Kahler manifold and a superpotential, de Sitter vacua can instead be
metastable provided the geometry satisfies some constraint and the
superpotential can be freely adjusted. In N=2 to N=1 truncations, the crucial
requirement is then that the tachyon of the mother theory be projected out from
the daughter theory, so that the original unstable vacuum is projected to a
metastable vacuum. We study the circumstances under which this may happen and
derive general constraints for metastability on the geometry and the gauging.
We then study in full detail the simplest case of quaternionic manifolds of
dimension four with at least one isometry, for which there exists a general
parametrization, and study two types of truncations defining Kahler
submanifolds of dimension two. As an application, we finally discuss the case
of the universal hypermultiplet of N=2 superstrings and its truncations to the
dilaton chiral multiplet of N=1 superstrings. We argue that de Sitter vacua in
such theories are necessarily unstable in weakly coupled situations, while they
can in principle be metastable in strongly coupled regimes.Comment: 40 pages, no figure

### The lightest scalar in theories with broken supersymmetry

We study the scalar mass matrix of general supersymmetric theories with local
gauge symmetries, and derive an absolute upper bound on the lightest scalar
mass. This bound can be saturated by suitably tuning the superpotential, and
its positivity therefore represents a necessary and sufficient condition for
the existence of metastable vacua. It is derived by looking at the subspace of
all those directions in field space for which an arbitrary supersymmetric mass
term is not allowed and scalar masses are controlled by supersymmetry-breaking
splitting effects. This subspace includes not only the direction of
supersymmetry breaking, but also the directions of gauge symmetry breaking and
the lightest scalar is in general a linear combination of fields spanning all
these directions. We present explicit results for the simplest case of theories
with a single local gauge symmetry. For renormalizable gauge theories, the
lightest scalar is a combination of the Goldstino partners and its square mass
is always positive. For more general non-linear sigma models, on the other
hand, the lightest scalar can involve also the Goldstone partner and its square
mass is not always positive.Comment: 30 pages, 3 figures; v2 introduction expanded, paragraph added in
section 5 and two references adde

### On Fayet-Iliopoulos terms and de Sitter vacua in supergravity: some easy pieces

We clarify a number of issues on Fayet-Iliopoulos (FI) terms in supergravity, keeping the formalism at a minumum and making use of explicit examples. We explain why, if the U(1) vector is massive everywhere in field space, FI terms are not genuine and can always be redefined away or introduced when they are not present. We formulate a simple anomaly-free model with a genuine FI term, a classically stable de Sitter (dS) vacuum and no global symmetries. We explore the relation between N=2 and N=1 FI terms by discussing N=1 truncations of N=2 models with classically stable dS vacua