We propose a generalization of the classical notion of the $V@R_{\lambda}$
that takes into account not only the probability of the losses, but the balance
between such probability and the amount of the loss. This is obtained by
defining a new class of law invariant risk measures based on an appropriate
family of acceptance sets. The $V@R_{\lambda}$ and other known law invariant
risk measures turn out to be special cases of our proposal.

The paper concerns primal and dual representations as well as time
consistency of set-valued dynamic risk measures. Set-valued risk measures
appear naturally when markets with transaction costs are considered and capital
requirements can be made in a basket of currencies or assets. Time consistency
of scalar risk measures can be generalized to set-valued risk measures in
different ways.

There is empirical evidence that recovery rates tend to go down just when the
number of defaults goes up in economic downturns. This has to be taken into
account in estimation of the capital against credit risk required by Basel II
to cover losses during the adverse economic downturns; the so-called "downturn
LGD" requirement. This paper presents estimation of the LGD credit risk model
with default and recovery dependent via the latent systematic risk factor using
Bayesian inference approach and Markov chain Monte Carlo method.

We present a computational method for measuring financial risk by estimating
the Value at Risk and Expected Shortfall from financial series. We have made
two assumptions: First, that the predictive distributions of the values of an
asset are conditioned by information on the way in which the variable evolves
from similar conditions, and secondly, that the underlying random processes can
be described using piecewise Gaussian processes.

Unlike other industries in which intellectual property is patentable, the
financial industry relies on trade secrecy to protect its business processes
and methods, which can obscure critical financial risk exposures from
regulators and the public. We develop methods for sharing and aggregating such
risk exposures that protect the privacy of all parties involved and without the
need for a trusted third party.

Regulation and risk management in banks depend on underlying risk measures.
In general this is the only purpose that is seen for risk measures. In this
paper we suggest that the reporting of risk measures can be used to determine
the loss distribution function for a financial entity. We demonstrate that a
lack of sufficient information can lead to ambiguous risk situations. We give
examples, showing the need for the reporting of multiple risk measures in order
to determine a bank's loss distribution.

The inverse first passage time problem asks whether, for a Brownian motion
$B$ and a nonnegative random variable $\zeta$, there exists a time-varying
barrier $b$ such that $\mathbb{P}\{B_s > b(s), \, 0 \le s \le t\} =
\mathbb{P}\{\zeta > t\}$. We study a "smoothed" version of this problem and ask
whether there is a "barrier" $b$ such that $\mathbb{E}[\exp(-\lambda \int_0^t
\psi(B_s - b(s)) \, ds)] = \mathbb{P}\{\zeta > t\}$, where $\lambda$ is a
killing rate parameter and $\psi: \mathbb{R} \to [0,1]$ is a non-increasing
function.

The benefits of diversifying risks are difficult to estimate quantitatively
because of the uncertainties in the dependence structure between the risks.
Also, the modelling of multidimensional dependencies is a non-trivial task.
This paper focuses on one such technique for portfolio aggregation, namely the
aggregation of risks within trees, where dependencies are set at each step of
the aggregation with the help of some copulas.

Karl Menger's 1934 paper on the St. Petersburg paradox contains mathematical
errors that invalidate his conclusion that unbounded utility functions,
specifically Bernoulli's logarithmic utility, fail to resolve modified versions
of the St. Petersburg paradox.

The problem of stock hedging is reconsidered in this paper, where a put
option is chosen from a set of available put options to hedge the market risk
of a stock. A formula is proposed to determine the probability that the
potential loss exceeds a predetermined level of Value-at-Risk, which is used to
find the optimal strike price and optimal hedge ratio. The assumptions that the
chosen put option finishes in-the-money and the constraint of hedging budget is
binding are relaxed in this paper.

In this paper we estimate the propagation of liquidity shocks through
interbank markets when the information about the underlying credit network is
incomplete. We show that techniques such as Maximum Entropy currently used to
reconstruct credit networks severely underestimate the risk of contagion by
assuming a trivial (fully connected) topology, a type of network structure
which can be very different from the one empirically observed.

We prove a law of large numbers for the loss from default and use it for
approximating the distribution of the loss from default in large, potentially
heterogenous portfolios. The density of the limiting measure is shown to solve
a non-linear SPDE, and the moments of the limiting measure are shown to satisfy
an infinite system of SDEs. The solution to this system leads to %the solution
to the SPDE through an inverse moment problem, and to the distribution of the
limiting portfolio loss, which we propose as an approximation to the loss
distribution for a large portfolio.

Financial volatility risk is addressed through a multiple round evolutionary
quantum game equilibrium leading to Multifractal Self-Organized Criticality
(MSOC) in the financial returns and in the risk dynamics. The model is
simulated and the results are compared with financial volatility data.

A risk of small defined-benefit pension schemes is that there are too few
members to eliminate idiosyncratic mortality risk, that is there are too few
members to effectively pool mortality risk. This means that when there are few
members in the scheme, there is an increased risk of the liability value
deviating significantly from the expected liability value, as compared to a
large scheme.

In this paper, we investigate two different frameworks for assessing the risk
in a multi-period decision process: a dynamically inconsistent formulation
(whereby a single, static risk measure is applied to the entire sequence of
future costs), and a dynamically consistent one, obtained by suitably composing
one-step risk mappings. We characterize the class of dynamically consistent
measures that provide a tight approximation for a given inconsistent measure,
and discuss how the approximation factors can be computed.

Recent academic work has developed a method to determine, in real time, if a
given stock is exhibiting a price bubble. Currently there is speculation in the
financial press concerning the existence of a price bubble in the aftermath of
the recent IPO of LinkedIn. We analyze stock price tick data from the short
lifetime of this stock through May 24, 2011, and we find that LinkedIn has a
price bubble.

The classical reduced-form and filtration expansion framework in credit risk
is extended to the case of multiple, non-ordered defaults, assuming that
conditional densities of the default times exist. Intensities and pricing
formulas are derived, revealing how information driven default contagion arises
in these models. We then analyze the impact of ordering the default times
before expanding the filtration.

We develop a dynamic point process model of correlated default timing in a
portfolio of firms, and analyze typical and atypical default profiles in the
limit as the size of the pool grows. In our model, a name defaults at a
stochastic intensity that is influenced by an idiosyncratic risk process, a
systematic risk process common to all names, and past defaults. We prove a law
of large numbers for the default rate in the pool, which describes the
"typical" behavior of defaults.

We study an asymptotic behaviour of the difference between value-at-risks
VaR(L) and VaR(L+S) for heavy-tailed random variables L and S as an application
to sensitivity analysis of quantitative operational risk management in the
framework of an advanced measurement approach (AMA) of Basel II. We have
different types of results according to the magnitude relationship of thickness
of tails of L and S. Especially if the tail of S is enough thinner than the one
of L, then VaR(L + S) - VaR(L) is asymptotically equivalent to an expected loss
of S when L and S are independent.

A key issue in the estimation of energy hedges is the hedgers' attitude
towards risk which is encapsulated in the form of the hedgers' utility
function. However, the literature typically uses only one form of utility
function such as the quadratic when estimating hedges. This paper addresses
this issue by estimating and applying energy market based risk aversion to
commonly applied utility functions including log, exponential and quadratic,
and we incorporate these in our hedging frameworks.

This paper examines the volatility and covariance dynamics of cash and
futures contracts that underlie the Optimal Hedge Ratio (OHR) across different
hedging time horizons. We examine whether hedge ratios calculated over a short
term hedging horizon can be scaled and successfully applied to longer term
horizons. We also test the equivalence of scaled hedge ratios with those
calculated directly from lower frequency data and compare them in terms of
hedging effectiveness.

Risk is an inherent feature of agricultural production and marketing and
accurate measurement of it helps inform more efficient use of resources. This
paper examines three tail quantile-based risk measures applied to the
estimation of extreme agricultural financial risk for corn and soybean
production in the US: Value at Risk (VaR), Expected Shortfall (ES) and Spectral
Risk Measures (SRMs). We use Extreme Value Theory (EVT) to model the tail
returns and present results for these three different risk measures using
agricultural futures market data.

Spectral risk measures are attractive risk measures as they allow the user to
obtain risk measures that reflect their risk-aversion functions. To date there
has been very little guidance on the choice of risk-aversion functions
underlying spectral risk measures. This paper addresses this issue by examining
two popular risk aversion functions, based on exponential and power utility
functions respectively. We find that the former yields spectral risk measures
with nice intuitive properties, but the latter yields spectral risk measures
that can have perverse properties.

This paper examines the precision of estimators of Quantile-Based Risk
Measures (Value at Risk, Expected Shortfall, Spectral Risk Measures). It first
addresses the question of how to estimate the precision of these estimators,
and proposes a Monte Carlo method that is free of some of the limitations of
existing approaches. It then investigates the distribution of risk estimators,
and presents simulation results suggesting that the common practice of relying
on asymptotic normality results might be unreliable with the sample sizes
commonly available to them.

Value at risk (VaR) is a risk measure that has been widely implemented by
financial institutions. This paper measures the correlation among asset price
changes implied from VaR calculation. Empirical results using US and UK equity
indexes show that implied correlation is not constant but tends to be higher
for events in the left tails (crashes) than in the right tails (booms).

Accurate forecasting of risk is the key to successful risk management
techniques. Using the largest stock index futures from twelve European bourses,
this paper presents VaR measures based on their unconditional and conditional
distributions for single and multi-period settings. These measures underpinned
by extreme value theory are statistically robust explicitly allowing for
fat-tailed densities. Conditional tail estimates are obtained by adjusting the
unconditional extreme value procedure with GARCH filtered returns.

Key to the imposition of appropriate minimum capital requirements on a daily
basis requires accurate volatility estimation. Here, measures are presented
based on discrete estimation of aggregated high frequency UK futures
realisations underpinned by a continuous time framework. Squared and absolute
returns are incorporated into the measurement process so as to rely on the
quadratic variation of a diffusion process and be robust in the presence of fat
tails.

Spectral risk measures are attractive risk measures as they allow the user to
obtain risk measures that reflect their subjective risk-aversion. This paper
examines spectral risk measures based on an exponential utility function, and
finds that these risk measures have nice intuitive properties.

This paper investigates the hedging effectiveness of a dynamic moving window
OLS hedging model, formed using wavelet decomposed time-series. The wavelet
transform is applied to calculate the appropriate dynamic minimum-variance
hedge ratio for various hedging horizons for a number of assets.

In recent years research on credit risk modelling has mainly focused on
default probabilities. Recovery rates are usually modelled independently, quite
often they are even assumed constant. Then, however, the structural connection
between recovery rates and default probabilities is lost and the tails of the
loss distribution can be underestimated considerably. The problem of
underestimating tail losses becomes even more severe, when calibration issues
are taken into account. To demonstrate this we choose a Merton-type structural
model as our reference system.

We consider a structural model for the estimation of credit risk based on
Merton's original model. By using Random-Matrix theory we demonstrate
analytically that the presence of correlations severely limits the effect of
diversification in a credit portfolio if the correlation are not identical
zero. The existence of correlations alters the tails of the loss distribution
tremendously, even if their average is zero. Under the assumption of randomly
fluctuating correlations, a lower bound for the estimation of the loss
distribution is provided.

As part of Basel II's incremental risk charge (IRC) methodology, this paper
summarizes our extensive investigations of constructing transition probability
matrices (TPMs) for unsecuritized credit products in the trading book. The
objective is to create monthly or quarterly TPMs with predefined sectors and
ratings that are consistent with the bank's Basel PDs. Constructing a TPM is
not a unique process.

Under the Basel II standards, the Operational Risk (OpRisk) advanced
measurement approach is not prescriptive regarding the class of statistical
model utilised to undertake capital estimation. It has however become well
accepted to utlise a Loss Distributional Approach (LDA) paradigm to model the
individual OpRisk loss process corresponding to the Basel II Business
line/event type. In this paper we derive a novel class of doubly stochastic
alpha-stable family LDA models.

In the present work we address the problem of evaluating the historical
performance of a trading strategy or a certain portfolio of assets. Common
indicators such as the Sharpe ratio and the risk adjusted return have
significant drawbacks. In particular, they are global indices, that is they do
not preserve any 'local' information about the performance dynamics either in
time or for a particular investment horizon. This information could be
fundamental for practitioners as the past performance can be affected by the
non-stationarity of financial market.

This paper generalizes the framework for arbitrage-free valuation of
bilateral counterparty risk to the case where collateral is included, with
possible re-hypotecation. We analyze how the payout of claims is modified when
collateral margining is included in agreement with current ISDA documentation.
We then specialize our analysis to interest-rate swaps as underlying portfolio,
and allow for mutual dependences between the default times of the investor and
the counterparty and the underlying portfolio risk factors.

This paper develops a structural credit risk model to characterize the
difference between the economic and recorded default times for a firm. Recorded
default occurs when default is recorded in the legal system. The economic
default time is the last time when the firm is able to pay off its debt prior
to the legal default time. It has been empirically documented that these two
times are distinct (see Guo, Jarrow, and Lin (2008)).

It is well known that any sufficiently regular one-dimensional payoff
function has an explicit static hedge by bonds, forward contracts and lots of
vanilla options. We show that the natural extension of the corresponding
representation leads to a static hedge based on the same instruments along with
traffic light options, which have recently been introduced in the market. One
big advantage of these replication strategies is the easy structure of the
hedge. Hence, traffic light options are particularly powerful building blocks
for more complicated bivariate options.

We consider dynamic sublinear expectations (i.e., time-consistent coherent
risk measures) whose scenario sets consist of singular measures corresponding
to a general form of volatility uncertainty. We derive a c\`adl\`ag nonlinear
martingale which is also the value process of a superhedging problem. The
superhedging strategy is obtained from a representation similar to the optional
decomposition. Furthermore, we prove an optional sampling theorem for the
nonlinear martingale and characterize it as the solution of a second order
backward SDE.

We analyze the size dependence and temporal stability of firm bankruptcy risk
in the US economy by applying Zipf scaling techniques. We focus on a single
risk factor-the debt-to-asset ratio R-in order to study the stability of the
Zipf distribution of R over time. We find that the Zipf exponent increases
during market crashes, implying that firms go bankrupt with larger values of R.
Based on the Zipf analysis, we employ Bayes's theorem and relate the
conditional probability that a bankrupt firm has a ratio R with the conditional
probability of bankruptcy for a firm with a given R value.

After September 2008, the advanced economies severe decline caused demand for
emerging economies' exports to drop and the crisis became truly global, much
deeper and broader than expected. In these times of global depression, most
countries and companies are affected, some more than others. The financial
crisis has turned out to be much deeper and broader than expected. Entering new
markets has always been a hazardous entrepreneurial attempt, but also a
rewarding one, in the case of success.

A new notion of stochastic ordering is introduced to compare multivariate
stochastic risk models with respect to extreme portfolio losses. In the
framework of multivariate regular variation comparison criteria are derived in
terms of ordering conditions on the spectral measures, which allows for
analytical or numerical verification in practical applications. Additional
comparison criteria in terms of further stochastic orderings are derived.

In this paper we present a theoretical framework for studying coherent
acceptability indices in a dynamic setup. We study dynamic coherent
acceptability indices and dynamic coherent risk measures, and we establish a
duality between them. We derive a representation theorem for dynamic coherent
risk measures in terms of so called dynamically consistent sequence of sets of
probability measures. Based on these results, we give a specific construction
of dynamic coherent acceptability indices.

The claim experience of the past is a very important information to calculate
the fair price of an insurance contract. In a lot of European countries for
instance the prices for motor car insurance depend on the number of claims the
driver has reported to the insurance company during the last years. Classically
these prices are calculated on the basis of a mixed Poisson model with a gamma
mixing distribution. The mixing distribution models the car drivers' qualities
across the insured portfolio.

In practice daily volatility of portfolio returns is transformed to longer
holding periods by multiplying by the square-root of time which assumes that
returns are not serially correlated. Under this assumption this procedure of
scaling can also be applied to contributions to volatility of the assets in the
portfolio. Trading at exchanges located in different time zones can lead to
significant serial cross-correlations of the returns of these assets when using
close prices as is usually done in practice. These serial correlations cause
the square-root-of-time rule to fail.

We consider an optimal control problem of a property insurance company with
proportional reinsurance strategy. The insurance business brings in catastrophe
risk, such as earthquake and flood. The catastrophe risk could be partly
reduced by reinsurance. The management of the company controls the reinsurance
rate and dividend payments process to maximize the expected present value of
the dividends before bankruptcy. This is the first time to consider the
catastrophe risk in property insurance model, which is more realistic.

In modern portfolio theory, the balancing of expected returns on investments
against uncertainties in those returns is aided by the use of utility
functions. The Kelly criterion offers another approach, rooted in information
theory, that always implies logarithmic utility. The two approaches seem
incompatible, too loosely or too tightly constraining investors' risk
preferences, from their respective perspectives.

This paper considers an optimal control of a large company with debt
liability under bankrupt probability constraints. The company, which faces
constant liability payments and has choices to choose various
production/business policies from an available set of control policies with
different expected profits and risks, controls the business policy and dividend
payout process to maximize the expected present value of the dividends until
the time of bankruptcy. However, if the dividend payout barrier is too low to
be acceptable, it may result in the company's bankruptcy soon.

A novel dynamical model for the study of operational risk in banks is
proposed. The equation of motion takes into account the interactions among
different bank's processes, the spontaneous generation of losses via a noise
term and the efforts made by the banks to avoid their occurrence. A scheme for
the estimation of some parameters of the model is illustrated, so that it can
be tailored on the internal organizational structure of a specific bank.

We consider the effect of recovery rates on a pool of credit assets. We allow
the recovery rate to depend on the defaults in a general way. Using the theory
of large deviations, we study the structure of losses in a pool consisting of a
continuum of types. We derive the corresponding rate function and show that it
has a natural interpretation as the favored way to rearrange recoveries and
losses among the different types. Numerical examples are also provided.

The purpose of this paper is to give a selective survey on recent progress in
random metric theory and its applications to conditional risk measures.

For purposes of Value-at-Risk estimation, we consider three multivariate
families of heavy-tailed distributions, which can be seen as multidimensional
versions of Paretian stable and Student's t distributions allowing different
marginals to have different tail thickness. After a discussion of relevant
estimation and simulation issues, we conduct a backtesting study on a set of
portfolios containing derivative instruments, using historical US stock price
data.

This paper considers optimal control problem of a large insurance company
under higher standard of solvency. The company controls proportional
reinsurance rate, dividend pay-outs and investing process to maximize the
expected present value of the dividend pay-outs until the time of bankruptcy.
This paper aims at describing the optimal return function as well as the
optimal policy.

The framework of this paper is that of uncertainty, that is when no reference
probability measure is given. To every convex regular risk measure $\rho$ on
${\cal C}_b(\Omega)$, we associate a canonical $c_{\rho}$-class of probability
measures. Furthermore the convex risk measure admits a dual representation in
terms of a weakly relatively compact set of probability measures absolutely
continuous with respect to some probability measure belonging to the canonical
$c_{\rho}$-class.

We show that any objective risk measurement algorithm mandated by central
banks for regulated financial entities will result in more risk being taken on
by those financial entities than would otherwise be the case. Furthermore, the
risks taken on by the regulated financial entities are far more systemically
concentrated than they would have been otherwise, making the entire financial
system more fragile.

Sustaining efficiency and stability by properly controlling the equity to
asset ratio is one of the most important and difficult challenges in bank
management. Due to unexpected and abrupt decline of asset values, a bank must
closely monitor its net worth as well as market conditions, and one of its
important concerns is when to raise more capital so as not to violate capital
adequacy requirements. In this paper, we model the tradeoff between avoiding
costs of delay and premature capital raising, and solve the corresponding
optimal stopping problem.

The quantitative aspirations of economists and financial analysts have for
many years been based on the belief that it should be possible to build models
of economic systems - and financial markets in particular - that are as
predictive as those in physics. While this perspective has led to a number of
important breakthroughs in economics, "physics envy" has also created a false
sense of mathematical precision in some cases. We speculate on the origins of
physics envy, and then describe an alternate perspective of economic behavior
based on a new taxonomy of uncertainty.

We review different approaches for measuring the impact of liquidity on CDS
prices. We start with reduced form models incorporating liquidity as an
additional discount rate. We review Chen, Fabozzi and Sverdlove (2008) and
Buhler and Trapp (2006, 2008), adopting different assumptions on how liquidity
rates enter the CDS premium rate formula, about the dynamics of liquidity rate
processes and about the credit-liquidity correlation.

The presence of non linear instruments is responsible for the emergence of
non Gaussian features in the price changes distribution of realistic
portfolios, even for Normally distributed risk factors. This is especially true
for the benchmark Delta Gamma Normal model, which in general exhibits
exponentially damped power law tails. We show how the knowledge of the model
characteristic function leads to Fourier representations for two standard risk
measures, the Value at Risk and the Expected Shortfall, and for their
sensitivities with respect to the model parameters.

We study the risk assessment of uncertain cash flows in terms of dynamic
convex risk measures for processes as introduced in Cheridito, Delbaen, and
Kupper (2006). These risk measures take into account not only the amounts but
also the timing of a cash flow. We discuss their robust representation in terms
of suitably penalized probability measures on the optional sigma-field. This
yields an explicit analysis both of model and discounting ambiguity. We focus
on supermartingale criteria for different notions of time consistency.

The impact of default events on the loss distribution of a credit portfolio
can be assessed by determining the loss distribution conditional on these
events. While it is conceptually easy to estimate loss distributions
conditional on default events by means of Monte Carlo simulation, it becomes
impractical for two or more simultaneous defaults as the conditioning event is
extremely rare. We provide an analytical approach to the calculation of the
conditional loss distribution for the CreditRisk+ portfolio model with
independent random loss given default distributions.

The present paper provides a multi-period contagion model in the credit risk
field. Our model is an extension of Davis and Lo's infectious default model. We
consider an economy of n firms which may default directly or may be infected by
other defaulting firms (a domino effect being also possible). The spontaneous
default without external influence and the infections are described by not
necessarily independent Bernoulli-type random variables. Moreover, several
contaminations could be required to infect another firm.

We provide a dual representation of quasiconvex maps between two lattices of
random variables in terms of conditional expectations. This generalizes the
dual representation of quasiconvex real valued functions and the dual
representation of conditional convex maps.

In the framework of Embedded Value new standards, namely the MCEV norms, the
latest principles published in June 2008 address the issue of market and
underwriting risks measurement by using stochastic models of projection and
valorization. Knowing that stochastic models particularly data-consuming, the
question which can arise is the treatment of insurance portfolios only
available in aggregate data or portfolios in situation of incomplete
information.

Decision making in the presence of randomness is an important problem,
particularly in finance. Often, decision makers base their choices on the
values of `risk measures' or `nonlinear expectations'; it is important to
understand how these decisions evolve through time. In this paper, we consider
how these decisions are affected by the use of a moving horizon, and the
possible inconsistencies that this creates. By giving a formal treatment of
time consistency without Bellman's equations, we show that there is a new sense
in which these decisions can be seen as consistent.

We introduce the general arbitrage-free valuation framework for counterparty
risk adjustments in presence of bilateral default risk, including default of
the investor. We illustrate the symmetry in the valuation and show that the
adjustment involves a long position in a put option plus a short position in a
call option, both with zero strike and written on the residual net value of the
contract at the relevant default times. We allow for correlation between the
default times of the investor, counterparty and underlying portfolio risk
factors.

In this paper, we present the principal components of an economic scenario
generator (ESG), both for the theoretical design and for practical
implementation. The choice of these components should be linked to the ultimate
vocation of the economic scenario generator, which can be either a tool for
pricing financial products or a tool for projection and risk management. We
then develop a study on some performance measure indicators of the ESG as an
input for the decision-making process, namely the indicators of stability and
bias absence.

In this paper we will discuss the optimal risk transfer problems when risk
measures are generated by G-expectations, and we present the relationship
between inf-convolution of G-expectations and the inf-convolution of drivers G.

The quantification of diversification benefits due to risk aggregation plays
a prominent role in the (regulatory) capital management of large firms within
the financial industry. However, the complexity of today's risk landscape makes
a quantifiable reduction of risk concentration a challenging task. In the
present paper we discuss some of the issues that may arise. The theory of
second-order regular variation and second-order subexponentiality provides the
ideal methodological framework to derive second-order approximations for the
risk concentration and the diversification benefit.

We present a simple model of firm rating evolution. We consider two sources
of defaults: individual dynamics of economic development and Potts-like
interactions between firms. We show that such a defined model leads to phase
transition, which results in collective defaults. The existence of the
collective phase depends on the mean interaction strength. For small
interaction strength parameters, there are many independent bankruptcies of
individual companies. For large parameters, there are giant collective defaults
of firm clusters.

We introduce the formalism of generalized Fourier transforms in the context
of risk management. We develop a general framework to efficiently compute the
most popular risk measures, Value-at-Risk and Expected Shortfall (also known as
Conditional Value-at-Risk). The only ingredient required by our approach is the
knowledge of the characteristic function describing the financial data in use.
This allows to extend risk analysis to those non-Gaussian models defined in the
Fourier space, such as Levy noise driven processes and stochastic volatility
models.