In this paper we present a theoretical framework for determining dynamic ask
and bid prices of derivatives using the theory of dynamic coherent
acceptability indices in discrete time. We prove a version of the First
Fundamental Theorem of Asset Pricing using the dynamic coherent risk measures.
We introduce the dynamic ask and bid prices of a derivative contract in markets
with transaction costs. Based on these results, we derive a representation
theorem for the dynamic bid and ask prices in terms of dynamically consistent
sequence of sets of probability measures and risk-neutral measures.

We extend the lifecycle model (LCM) of consumption over a random horizon
(a.k.a. the Yaari model) to a world in which (i.) the force of mortality obeys
a diffusion process as opposed to being deterministic, and (ii.) a consumer can
adapt their consumption strategy to new information about their mortality rate
(a.k.a. health status) as it becomes available. In particular, we derive the
optimal consumption rate and focus on the impact of mortality rate uncertainty
vs.

We analyse time series of CDS spreads for a set of major US and European
institutions on a pe- riod overlapping the recent financial crisis. We extend
the existing methodology of {\epsilon}-drawdowns to the one of joint
{\epsilon}-drawups, in order to estimate the conditional probabilities of
abrupt co-movements among spreads. We correct for randomness and for finite
size effects and we find significant prob- ability of joint drawups for certain
pairs of CDS. We also find significant probability of trend rein- forcement,
i.e. drawups in a given CDS followed by drawups in the same CDS.

When estimating the risk of a P&L from historical data or Monte Carlo
simulation, the robustness of the estimate is important. We argue here that
Hampel's classical notion of qualitative robustness is not suitable for risk
measurement and we propose and analyze a refined notion of robustness that
applies to tail-dependent law-invariant convex risk measures on Orlicz space.
This concept of robustness captures the tradeoff between robustness and
sensitivity and can be quantified by an index of qualitative robustness.

This paper proposes a use of an ordinal classifier to evaluate the financial
solidity of non-life insurance companies as strong, moderate, weak, and
insolvency. This study constructed an efficient classification model that can
be used by regulators to evaluate the financial solidity and to determine the
priority of further examination as an early warning system. The proposed model
is beneficial to policy-makers to create guidelines for the solvency
regulations and roles of the government in protecting the public against
insolvency.

While defaults are rare events, losses can be substantial even for credit
portfolios with a large number of contracts. Therefore, not only a good
evaluation of the probability of default is crucial, but also the severity of
losses needs to be estimated. The recovery rate is often modeled independently
with regard to the default probability, whereas the Merton model yields a
functional dependence of both variables. We use Moody's Default and Recovery
Database in order to investigate the relationship of default probability and
recovery rate for senior secured bonds.

Overrides of credit ratings are important correctives of ratings that are
determined by statistical rating models. Financial institutions and banking
regulators agree on this because on the one hand errors with ratings of
corporates or banks can have fatal consequences for the lending institutions
and on the other hand errors by statistical methods can be minimised but not
completely avoided. Nonetheless, rating overrides can be misused in order to
conceal the real riskiness of borrowers or even entire portfolios.

Margin system for margin loans using cash and stock as collateral is
considered in this paper, which is the line of defence for brokers against risk
associated with margin trading. The conditional probability of negative return
is used as risk measure, and a recursive algorithm is proposed to realize this
measure under a Markov chain model. Optimal margin system is chosen from those
systems which satisfy the constraint of the risk measure.

The importance of adequately modeling credit risk has once again been
highlighted in the recent financial crisis. Defaults tend to cluster around
times of economic stress due to poor macro-economic conditions, {\em but also}
by directly triggering each other through contagion. Although credit default
swaps have radically altered the dynamics of contagion for more than a decade,
models quantifying their impact on systemic risk are still missing. Here, we
examine contagion through credit default swaps in a stylized economic network
of corporates and financial institutions.

We propose a dynamical model for the estimation of Operational Risk in
banking institutions. Operational Risk is the risk that a financial loss occurs
as the result of failed processes. Examples of operational losses are the ones
generated by internal frauds, human errors or failed transactions. In order to
encompass the most heterogeneous set of processes, in our approach the losses
of each process are generated by the interplay among random noise, interactions
with other processes and the efforts the bank makes to avoid losses.

The important application of semi-static hedging in financial markets
naturally leads to the notion of quasi self-dual processes. The focus of our
study is to give new characterizations of quasi self-duality for exponential
L\'evy processes such that the resulting market does not admit arbitrage
opportunities. We derive a set of equivalent conditions for the stochastic
logarithm of quasi self-dual martingale models and derive a further
characterization of these models not depending on the L\'evy-Khintchine
parametrization.

We provide a dual representation of quasiconvex conditional risk measures $%
\rho $ defined on $L^{0}$ modules of the $L^{p}$ type. This is a consequence of
more general result which extend the usual Penot-Volle representation for
quasiconvex real valued maps. We establish, in the conditional setting, a
complete duality between quasiconvex risk measures and the appropriate class of
dual functions.

The adverse effects of financial crises in terms of output losses or output
growth below its potential can be treated like losses from catastrophic events
which have a low likelihood but a large impact in the event that they occur.

Propagation of balance-sheet or cash-flow insolvency across financial
institutions may be modeled as a cascade process on a network representing
their mutual exposures. We derive rigorous asymptotic results for the magnitude
of contagion in a large financial network and give an analytical expression for
the asymptotic fraction of defaults, in terms of network characteristics. Our
results extend previous studies on contagion in random graphs to inhomogeneous
directed graphs with a given degree sequence and arbitrary distribution of
weights.

The global financial system has become highly connected and complex. Has been
proven in practice that existing models, measures and reports of financial risk
fail to capture some important systemic dimensions. Only lately, advisory
boards have been established in high level and regulations are directly
targeted to systemic risk. In the same direction, a growing number of
researchers employ network analysis to model systemic risk in financial
networks. Current approaches are concentrated on interbank payment network
flows in national and international level.

In this paper we look at the efficacy of different risk measures on energy
markets and across several different stock market indices. We use both the
Value at Risk and the Tail Conditional Expectation on each of these data sets.
We also consider several different durations and levels for historical risk
measures. Through our results we make some recommendations for a robust risk
management strategy that involves historical risk measures.

The banking systems that deal with risk management depend on underlying risk
measures. Following the Basel II accord, there are two separate methods by
which banks may determine their capital requirement. The Value at Risk measure
plays an important role in computing the capital for both approaches. In this
paper we analyze the errors produced by using this measure. We discuss other
measures, demonstrating their strengths and shortcomings. We give examples,
showing the need for the information from multiple risk measures in order to
determine a bank's loss distribution.

In this paper, we introduce a multivariate extension of the classical
univariate Value- at-Risk (VaR). This extension may be useful to understand how
solvency capital re- quirement computed for a given financial institution may
be affected by the presence of additional risks. We also generalize the
bivariate Conditional-Tail-Expectation (CTE), previously introduced by Di
Bernardino et al. (2011), in a multivariate set- ting and we study its
behavior. Several properties have been derived.

Threats on the stability of a financial system may severely affect the
functioning of the entire economy, and thus considerable emphasis is placed on
the analyzing the cause and effect of such threats. The financial crisis in the
current and past decade has shown that one important cause of instability in
global markets is the so-called financial contagion, namely the spreading of
instabilities or failures of individual components of the network to other,
perhaps healthier, components.

Starting from the requirement that risk measures of financial portfolios
should be based on their losses, not their gains, we define the notion of
loss-based risk measure and study the properties of this class of risk
measures. We characterize loss-based risk measures by a representation theorem
and give examples of such risk measures.

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.

This article makes use of the apparent indifference that the market has been
devoting to the developments made on the fundamentals of quantitative finance,
to introduce novel insight for better understanding market evolution.We show
how these drops and crises emerge as a natural result of local economical
principles ruling trades between economical agents and present evidence that
heavy-tails of the return distributions are bounded by constraints associated
with the topology of agent relations. Finally, we discuss how these constraints
may be helpful for properly evaluate model risk.

In this work, we consider the hedging error due to discrete trading in models
with jumps. Extending an approach developped by Fukasawa (2011) for continuous
processes, we propose a framework enabling to (asymptotically) optimize the
discretization times. More precisely, a discretization rule is said to be
optimal if for a given cost function, no strategy has (asymptotically, for
large cost) a lower mean square discretization error for a smaller cost. We
focus on discretization rules based on hitting times and give explicit
expressions for the optimal rules within this class.

A simple, yet reasonably accurate, analytical technique is proposed for
multi-factor structural credit portfolio models. The accuracy of the technique
is demonstrated by benchmarking against Monte Carlo simulations. The approach
presented here may be of high interest to practitioners looking for
transparent, intuitive, easy to implement and high performance credit portfolio
model.

In this paper, we detail the main simulation methods used in practice to
measure one-year reserve risk, and describe the bootstrap method providing an
empirical distribution of the Claims Development Result (CDR) whose variance is
identical to the closed-form expression of the prediction error proposed by
W\"uthrich et al. (2008). In particular, we integrate the stochastic modeling
of a tail factor in the bootstrap procedure. We demonstrate the equivalence
with existing analytical results and develop closed-form expressions for the
error of prediction including a tail factor.

This research presents an analysis of the demographic risk related to future
membership patterns in pension funds with restricted entrance, financed under a
pay-as-you-go scheme. The paper, therefore, proposes a stochastic model for
investigating the behaviour of the demographic variable "new entrants" and the
influence it exerts on the financial dynamics of such funds. Further
information on pension funds of Italian professional categories and an
application to the Cassa Nazionale di Previdenza e Assistenza dei Dottori
Commercialisti (CNPADC) are then provided.

This paper presents two cases of random banking data generators based on
migration matrices and scoring rules. The banking data generator is a new hope
in researches of finding the proving method of comparisons of various credit
scoring techniques. There is analyzed the influence of one cyclic
macro--economic variable on stability in the time account and client
characteristics. Data are very useful for various analyses to understand in the
better way the complexity of the banking processes and also for students and
their researches.

We analyze the counterparty risk embedded in CDS contracts, in presence of a
bilateral margin agreement. First, we investigate the pricing of collateralized
counterparty risk and we derive the bilateral Credit Valuation Adjustment
(CVA), unilateral Credit Valuation Adjustment (UCVA) and Debt Valuation
Adjustment (DVA). We propose a model for the collateral by incorporating all
related factors such as the thresholds, haircuts and margin period of risk. We
derive the dynamics of the bilateral CVA in a general form with related jump
martingales.

The problem behind this paper is the proper measurement of the degree of
quality/acceptability/distance to arbitrage of trades. We are narrowing the
class of coherent acceptability indices introduced by Cherny and Madan (2007)
by imposing an additional mathematical property. For this, we introduce the
notion of a concave distortion semigroup as a family $(\Psi_t)_{t\ge0}$ of
concave increasing functions $[0,1]\to[0,1]$ satisfying the semigroup property
$$ \Psi_s\circ\Psi_t=\Psi_{s+t},\quad s,t\ge0.

This paper discusses the financial risks faced by the UK Pension Protection
Fund (PPF) and what, if anything, it can do about them. It draws lessons from
the regulatory regimes under which other financial institutions, such as banks
and insurance companies, operate and asks why pension funds are treated
differently. It also reviews the experience with other government-sponsored
insurance schemes, such as the US Pension Benefit Guaranty Corporation, upon
which the PPF is modelled.

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.

We present a novel procedure for scaling relatively high frequency tail
probability and quantile estimates for the conditional distribution of returns.

Spectral risk measures (SRMs) are risk measures that take account of user
riskaversion, but to date there has been little guidance on the choice of
utility function underlying them. This paper addresses this issue by examining
alternative approaches based on exponential and power utility functions. A
number of problems are identified with both types of spectral risk measure.

This paper presents non-parametric estimates of spectral risk measures
applied to long and short positions in 5 prominent equity futures contracts. It
also compares these to estimates of two popular alternative measures, the
Value-at-Risk (VaR) and Expected Shortfall (ES). The spectral risk measures are
conditioned on the coefficient of absolute risk aversion, and the latter two
are conditioned on the confidence level. Our findings indicate that all risk
measures increase dramatically and their estimators deteriorate in precision
when their respective conditioning parameter increases.

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).

This paper applies the Extreme-Value (EV) Generalised Pareto distribution to
the extreme tails of the return distributions for the S&P500, FT100, DAX, Hang
Seng, and Nikkei225 futures contracts. It then uses tail estimators from these
contracts to estimate spectral risk measures, which are coherent risk measures
that reflect a user's risk-aversion function. It compares these to VaR and
Expected Shortfall (ES) risk measures, and compares the precision of their
estimators.

This paper empirically analyses risk in the Euro relative to other
currencies. Comparisons are made between a sub period encompassing the final
transitional stage to full monetary union with a sub period prior to this.
Stability in the face of speculative attack is examined using Extreme Value
Theory to obtain estimates of tail exchange rate changes. The findings are
encouraging. The Euro's common risk measures do not deviate substantially from
other currencies.

Both in practice and in the academic literature, models for setting margin
requirements in futures markets classically use daily closing price changes.
However, as well documented by research on high-frequency data, financial
markets have recently shown high intraday volatility, which could bring more
risk than expected. This paper tries to answer two questions relevant for
margin committees in practice: is it right to compute margin levels based on
closing prices and ignoring intraday dynamics? Is it justified to implement
intraday margin calls?

This paper applies an AR(1)-GARCH (1, 1) process to detail the conditional
distributions of the return distributions for the S&P500, FT100, DAX, Hang
Seng, and Nikkei225 futures contracts. It then uses the conditional
distribution for these contracts to estimate spectral risk measures, which are
coherent risk measures that reflect a user's risk-aversion function.

The social role of any company is to get the maximum profitability with the
less risk. Due to Basel III, banks should now raise their minimum capital
levels on an individual basis, with the aim of lowering the probability for a
large crash to occur. Such implementation assumes that with higher minimum
capital levels it becomes more probable that the value of the assets drop
bellow the minimum level and consequently expects the number of bank defaults
to drop also.

We study the problem of portfolio insurance from the point of view of a fund
manager, who guarantees to the investor that the portfolio value at maturity
will be above a fixed threshold. If, at maturity, the portfolio value is below
the guaranteed level, a third party will refund the investor up to the
guarantee. In exchange for this protection, the third party imposes a limit on
the risk exposure of the fund manager, in the form of a convex monetary risk
measure.

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.

This paper introduces a new model risk measure based on hedging strategies to
estimate model risk and provision calculation under uncertainty of volatility.
This measure allows comparing different products and models (pricing
hypothesis) under a homogeneous framework and conclude which one is the best.
The model risk measure is defined in terms of the expected value and standard
deviation of the loss given by the hedging strategy at a given time horizon. It
has been assumed that the market volatility surface is driven by Heston's
dynamics calibrated to market for a given time horizon.

The current research on credit risk is primarily focused on modeling default
probabilities. Recovery rates are often treated as an afterthought; they are
modeled independently, in many cases they are even assumed constant. This is
despite of their pronounced effect on the tail of the loss distribution. Here,
we take a step back, historically, and start again from the Merton model, where
defaults and recoveries are both determined by an underlying process. Hence,
they are intrinsically connected.

In order to protect brokers from customer defaults in a volatile market, an
active margin system is proposed for the transactions of margin lending in
China. The probability of negative return under the condition that collaterals
are liquidated in a falling market is used to measure the risk associated with
margin loans, and a recursive algorithm is proposed to calculate this
probability under a Markov chain model. The optimal maintenance margin ratio
can be given under the constraint of the proposed risk measurement for a
specified amount of initial margin.

The downside risk of a portfolio of (equity)assets is generally substantially
higher than the downside risk of its components. In particular in times of
crises when assets tend to have high correlation, the understanding of this
di?erence can be crucial in managing systemic risk of a portfolio. In this
paper we gen- eralize Merton's option formula in the presence jumps to the
multi-asset case. It is shown how common jumps across assets provide an
intuitive and powerful tool to describe systemic risk that is consistent with
data.

Set-valued risk measures on $L^p_d$ with $0 \leq p \leq \infty$ for conical
market models are defined, primal and dual representation results are given.
The collection of initial endowments which allow to super-hedge a multivariate
claim are shown to form the values of a set-valued sublinear (coherent) risk
measure. Scalar risk measures with multiple eligible assets also turn out to be
a special case within the set-valued framework.

This paper studies multidimensional dynamic risk measure induced by
conditional $g$-expectation. A notion of multidimensional $g$-expectation is
proposed to provide a multidimensional version of nonlinear expectations.

This paper investigates the impact of parameter uncertainty on capital
estimate in the well-known extended Loss Given Default (LGD) model with
systematic dependence between default and recovery. We demonstrate how the
uncertainty can be quantified using the full posterior distribution of model
parameters obtained from Bayesian inference via Markov chain Monte Carlo
(MCMC). Results show that the parameter uncertainty and its impact on capital
can be very significant.

We consider an insurance company in the case when the premium rate is a
bounded non-negative random function $c_\zs{t}$ and the capital of the
insurance company is invested in a risky asset whose price follows a geometric
Brownian motion with mean return $a$ and volatility $\sigma>0$. If
$\beta:=2a/\sigma^2-1>0$ we find exact the asymptotic upper and lower bounds
for the ruin probability $\Psi(u)$ as the initial endowment $u$ tends to
infinity, i.e. we show that $C_*u^{-\beta}\le\Psi(u)\le C^*u^{-\beta}$ for
sufficiently large $u$.

The problem of quantile hedging for multiple assets derivatives in the
Black-Scholes model with correlation is considered. Explicit formulas for the
probability maximizing function and the cost reduction function are derived.
Applicability of the results for the widely traded derivatives as digital,
quantos, outperformance and spread options is shown.

Under the Basel II standards, the Operational Risk (OpRisk) advanced
measurement approach allows a provision for reduction of capital as a result of
insurance mitigation of up to 20%. This paper studies the behaviour of
different insurance policies in the context of capital reduction for a range of
possible extreme loss models and insurance policy scenarios in a multi-period,
multiple risk settings.

If the probability of default parameters (PDs) fed as input into a credit
portfolio model are estimated as through-the-cycle (TTC) PDs stressed market
conditions have little impact on the results of the capital calculations
conducted with the model. At first glance, this is totally different if the PDs
are estimated as point-in-time (PIT) PDs. However, it can be argued that the
reflection of stressed market conditions in input PDs should correspond to the
use of reduced correlation parameters or even the removal of correlations in
the model.

Within the context of risk integration, we introduce in risk measurement
stochastic holding period (SHP) models. This is done in order to obtain a
`liquidity-adjusted risk measure' characterized by the absence of a fixed time
horizon. The underlying assumption is that - due to changes on market liquidity
conditions - one operates along an `operational time' to which the P&L process
of liquidating a market portfolio is referred. This framework leads to a
mixture of distributions for the portfolio returns, potentially allowing for
skewness, heavy tails and extreme scenarios.

Financial leverage can be regarded as an object of choice or a decision. We
show how this optics allows perceiving the recently introduced metrics of
see-through-leverage, which proved to be very useful in understanding the
phenomenology of the recent economic crisis.

The paper is motivated by a problem concerning the monotonicity of insurance
premiums with respect to their loading parameter: the larger the parameter, the
larger the insurance premium is expected to be. This property, usually called
loading monotonicity, is satisfied by premiums that appear in the literature.
The increased interest in constructing new insurance premiums has raised a
question as to what weight functions would produce loading-monotonic premiums.
In this paper we demonstrate a decisive role of log-supermodularity in
answering this question.

Analytical, free of time consuming Monte Carlo simulations, framework for
credit portfolio systematic risk metrics calculations is presented. Techniques
are described that allow calculation of portfolio-level systematic risk
measures (standard deviation, VaR and Expected Shortfall) as well as allocation
of risk down to individual transactions. The underlying model is the industry
standard multi-factor Merton-type model with arbitrary valuation function at
horizon (in contrast to the simplistic default-only case).

After the shocking series of bankruptcies started in 2008, the public does
not trust anymore the classical methods of assessing business risks. The global
economic severe downturn caused demand for both developed and emerging
economies' exports to drop and the crisis became truly global. However, this
current crisis offers opportunities for those companies able to play well their
cards. Entering new markets has always been a hazardous entrepreneurial
attempt, but also a rewarding one, in the case of success.

One possible way of risk management for an insurance company is to develop an
early and appropriate alarm system before the possible ruin. The ruin is
defined through the status of the aggregate risk process, which in turn is
determined by premium accumulation as well as claim settlement outgo for the
insurance company. The main purpose of this work is to design an effective
alarm system, i.e. to define alarm times and to recommend augmentation of
capital of suitable magnitude at those points to prevent or reduce the chance
of ruin.

We extend the Vasi\v{c}ek loan portfolio model to a setting where liabilities
fluctuate randomly and asset values may be subject to systemic jump risk. We
derive the probability distribution of the percentage loss of a uniform
portfolio and analyze its properties. We find that the impact of liability risk
is ambiguous and depends on the correlation between the continuous aggregate
factor and the asset-liability ratio as well as on the default intensity.

We examine three methods of constructing correlated Student-$t$ random
variables. Our motivation arises from simulations that utilise heavy-tailed
distributions for the purposes of stress testing and economic capital
calculations for financial institutions. We make several observations regarding
the suitability of the three methods for this purpose.

This paper considers nonlinear optimal stochastic control of insurance
company with proportional reinsurance policy under small bankrupt probability
constraints. The company controls the reinsurance rate and dividend payout
process to maximize the expected present value of the dividends until the time
of bankruptcy. However, if the optimal dividend barrier is too low to be
acceptable, it will make the company result in bankruptcy soon. In addition,
although risk and return should be highly correlated, over-risking is not a
good recipe for high return.

Based on a point of view that solvency and security are first, this paper
considers optimal control and financial valuation problems of a large insurance
company facing positive transaction cost asked by reinsurer. The company
controls proportional reinsurance and dividend pay-out policy to maximize the
expected present value of the dividend pay-outs until the time of bankruptcy.
The paper aims at finding explicitly value function and an optimal control
policy of the company by using stochastic analysis and PDE methods.

Discrete time hedging in a complete diffusion market is considered. The hedge
portfolio is rebalanced when the absolute difference between delta of the hedge
portfolio and the derivative contract reaches a threshold level. The rate of
convergence of the expected squared hedging error as the threshold level
approaches zero is analyzed. The results hinge to a great extent on a theorem
stating that the difference between the hedge ratios normalized by the
threshold level tends to a triangular distribution as the threshold level tends
to zero.

The model is aimed to discriminate the 'good' and the 'bad' companies in
Russian corporate sector based on their financial statements data (Russian
Accounting Standards). The data sample consists of 126 Russian public
companies- issuers of Ruble bonds which represent about 36% of total number of
corporate bonds issuers. 25 companies have defaulted on their debt in 2008-2009
which represent around 30% of default cases. 29% companies in the sample have
credit ratings assigned compared to 34% in the parent population. No SPV
companies were included in the sample.

We shall provide in this paper good deal pricing bounds for contingent claims
induced by the shortfall risk with some loss function. Assumptions we impose on
loss functions and contingent claims are very mild. We prove that the upper and
lower bounds of good deal pricing bounds are expressed by convex risk measures
on Orlicz hearts. In addition, we obtain its representation with the minimal
penalty function.

The intention with this paper is to provide all the estimation concepts and
techniques that are needed to implement a two-phases approach to the parametric
estimation of probability of default (PD) curves. In the first phase of this
approach, a raw PD curve is estimated based on parameters that reflect
discriminatory power. In the second phase of the approach, the raw PD curve is
calibrated to fit a target unconditional PD.

We analyze the errors arising from discrete readjustment of the hedging
portfolio when hedging options in exponential Levy models, and establish the
rate at which the expected squared error goes to zero when the readjustment
frequency increases.

This paper gives an overview of the theory of dynamic convex risk measures
for random variables in discrete time setting. We summarize robust
representation results of conditional convex risk measures, and we characterize
various time consistency properties of dynamic risk measures in terms of
acceptance sets, penalty functions, and by supermartingale properties of risk
processes and penalty functions.

We develop a generalization of the Black-Cox structural model of default
risk. The extended model captures uncertainty related to firms ability to avoid
default even if companys liabilities momentarily exceeding its assets.
Diffusion in a linear potential with the radiation boundary condition is used
to mimic a companys default process. The exact solution of the corresponding
Fokker-Planck equation allows for derivation of analytical expressions for the
cumulative probability of default and the relevant hazard rate.

Inspired by the bankruptcy of Lehman Brothers and its consequences on the
global financial system, we develop a simple model in which the Lehman default
event is quantified as having an almost immediate effect in worsening the
credit worthiness of all financial institutions in the economic network. In our
stylized description, all properties of a given firm are captured by its
effective credit rating, which follows a simple dynamics of co-evolution with
the credit ratings of the other firms in our economic network.

We find the minimum probability of lifetime ruin of an investor who can
invest in a market with a risky and a riskless asset and who can purchase a
reversible life annuity. The surrender charge of a life annuity is a proportion
of its value. Ruin occurs when the total of the value of the risky and riskless
assets and the surrender value of the life annuity reaches zero.

The economic equities maximization criterion (MFPE) leads to the choice of
financial portfolio, which maximizes the ratio of the expected value of the
insurance company on the capital. This criterion is presented in the framework
of a non-life insurance company and is applied within the framework of the
French legislation and in a lawful context inspired of the works in progress
about the European project Solvency 2. In the French regulation case, the
required solvency margin does not depend of the asset allocation.

We introduce a general framework for models of cascade and contagion
processes on networks, to identify their commonalities and differences. In
particular, models of social and financial cascades, as well as the fiber
bundle model, the voter model, and models of epidemic spreading are recovered
as special cases. To unify their description, we define the net fragility of a
node, which is the difference between its fragility and the threshold that
determines its failure.

In this paper we introduce a novel approach to risk estimation based on
nonlinear factor models - the "StressVaR" (SVaR). Developed to evaluate the
risk of hedge funds, the SVaR appears to be applicable to a wide range of
investments. Its principle is to use the fairly short and sparse history of the
hedge fund returns to identify relevant risk factors among a very broad set of
possible risk sources. This risk profile is obtained by calibrating a
collection of nonlinear single-factor models as opposed to a single
multi-factor model.

We apply a Coupled Markov Chain approach to model rating transitions and
thereby default probabilities of companies. We estimate parameters by applying
a maximum likelihood estimation using a large set of historical ratings. Given
the parameters the model can be used to simulate scenarios for joint rating
changes of a set of companies, enabling the use of contemporary risk management
techniques. We obtain scenarios for the payment streams generated by CDX
contracts and portfolios of such contracts.

Analytical, free of time consuming Monte Carlo simulations, framework for
credit portfolio systematic risk metrics calculations is presented. Techniques
are described that allow calculation of portfolio-level systematic risk
measures (standard deviation, VaR and Expected Shortfall) as well as allocation
of risk down to individual transactions. The underlying model is the industry
standard multi-factor Merton-type model with arbitrary valuation function at
horizon (in contrast to the simplistic default-only case).

The aim of this research is to give a simple framework to evaluate/quantize
the "transparency" of a firm. We assume that the process of the firm value is
only observable once in a while but is strongly correlated with the stock price
which is observable and tradable. This hybrid type structure make the
transparency "observable". The implication of the present study is that the
depth of the shock to the market caused by the precise accounting information
does reflect the degree of transparency. Furthermore, it can be quantized
resorting to the calibration method.

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.

High precision analytical approximation is proposed for variance-covariance
based risk allocation in a portfolio of risky assets. A general case of a
single-period multi-factor Merton-type model with stochastic recovery is
considered. The accuracy of the approximation as well as its speed are compared
to and shown to be superior to those of Monte Carlo simulation.