Explicit robust hedging strategies for convex or concave payoffs under a
continuous semimartingale model with uncertainty and small transaction costs
are constructed. In an asymptotic sense, the upper and lower bounds of the
cumulative volatility enable us to super-hedge convex and concave payoffs
respectively. The idea is a combination of Mykland's conservative delta hedging
and Leland's enlarging volatility. We use a specific sequence of stopping times
as rebalancing dates, which can be superior to equidistant one even when there
is no model uncertainty.

The credit crisis of 2007 and 2008 has thrown much focus on the models used
to price mortgage backed securities. Many institutions have relied heavily on
the credit ratings provided by credit agency. The relationships between
management of credit agencies and debt issuers may have resulted in conflict of
interest when pricing these securities which has lead to incorrect risk
assumptions and value expectations from institutional buyers. Despite the
existence of sophisticated models, institutional buyers have relied on these
ratings when considering the risks involved with these products.

The SABR model is a stochastic volatility model not admitting a closed form
solution. Hagan, Kumar, Leniewski and Woodward have given an approximate
solution by means of perturbative techniques. A more precise approximation was
obtained by Henry-Labord\`ere using the heat kernel expansion method. The
latter relies on deep and hard theorems from Riemannian geometry which are
almost totally unknown to people working in finance, who however are those
primarily interested in these results.

This work focuses on the indifference pricing of American call option
underlying a non-traded stock, which may be partially hedgeable by another
traded stock. Under the exponential forward measure, the indifference price is
formulated as a stochastic singular control problem. The value function is
characterized as the unique solution of a partial differential equation in a
Sobolev space. Together with some regularities and estimates of the value
function, the existence of the optimal strategy is also obtained.

The purpose of this paper is two-fold. First is to extend the notions of an
n-dimensional semimartingale and its stochastic integral to a piecewise
semimartingale of stochastic dimension. The properties of the former carry over
largely intact to the latter, avoiding some of the pitfalls of
infinite-dimensional stochastic integration.

In this paper we describe how to include funding and margining costs into a
risk-neutral pricing framework for counterparty credit risk. We consider
realistic settings and we include in our models the common market practices
suggested by the ISDA documentation without assuming restrictive constraints on
margining procedures and close-out netting rules. In particular, we allow for
asymmetric collateral and funding rates, and exogenous liquidity policies and
hedging strategies. Re-hypothecation liquidity risk and close-out amount
evaluation issues are also covered.

In the forthcoming ISDA Standard Credit Support Annex (SCSA), the trades
denominated in non-G5 currencies as well as those include multiple currencies
are expected to be allocated to the USD silo, where the contracts are
collateralized by USD cash, or a different currency with an appropriate
interest rate overlay to achieve the same economic effects. In this paper, we
have presented a simple generic valuation framework for the clean price under
the USD silo with the the detailed procedures for the initial term structure
construction.

This paper considers exponential utility indifference pricing for a
multidimensional non-traded assets model and provides two approximations for
the utility indifference price: a linear approximation by Picard iteration and
a semigroup approximation by splitting techniques. The key tool is the
probabilistic representation for the utility indifference price by the solution
of fully coupled linear forward-backward stochastic differential equations. We
apply our methodology to study the counterparty risk of derivatives in
incomplete markets.

In the present paper we show that the Binomial-tree approach for pricing,
hedging, and risk assessment of Convertible bonds in the framework of the
Tsiveriotis-Fernandes model has serious drawbacks. Key words: Convertible
bonds, Binomial tree, Tsiveriotis-Fernandes model, Convertible bond pricing,
Convertible bond Greeks, Convertible Arbitrage, Delta-hedging of Convertible
bonds, Risk Assessment of Convertible bonds.

We present a dialogue on Counterparty Credit Risk touching on Credit Value at
Risk (Credit VaR), Potential Future Exposure (PFE), Expected Exposure (EE),
Expected Positive Exposure (EPE), Credit Valuation Adjustment (CVA), Debit
Valuation Adjustment (DVA), DVA Hedging, Closeout conventions, Netting clauses,
Collateral modeling, Gap Risk, Re-hypothecation, Wrong Way Risk, Basel III,
inclusion of Funding costs, First to Default risk, Contingent Credit Default
Swaps (CCDS) and CVA restructuring possibilities through margin lending.

We extend the now classic structural credit modeling approach of Black and
Cox to a class of "two-factor" models that unify equity securities such as
options written on the stock price, and credit products like bonds and credit
default swaps. In our approach, the two sides of the stylized balance sheet of
a firm, namely the asset value and debt value, are assumed to follow a two
dimensional Markov process.

We develop a single-period model for a large economic agent who trades with
market makers at their utility indifference prices. A key role is played by a
pair of conjugate saddle functions associated with the description of Pareto
optimal allocations in terms of the utility function of a representative market
maker.

We provide sufficient conditions for the existence and uniqueness of
solutions to a stochastic differential equation which arises in a price impact
model. These conditions are stated as smoothness and boundedness requirements
on utility functions or Malliavin differentiability of payoffs and endowments.

This paper studies the optimal timing to liquidate defaultable securities in
a general intensity-based credit risk model under stochastic interest rate. We
incorporate the potential price discrepancy between the market and investors,
which is characterized by risk-neutral valuation under different default risk
premia specifications. To quantify the value of optimally timing to sell, we
introduce the {delayed liquidation premium} which is closely related to the
stochastic bracket between the market price and a pricing kernel.

In a Markovian stochastic volatility model, we consider financial agents
whose investment criteria are modelled by forward exponential performance
processes. The problem of contingent claim indifference valuation is first
addressed and a number of properties are proved and discussed. Special
attention is taken on the comparison between the forward and backward
indifference valuation. In addition, we initiate the problem of optimal risk
sharing on this forward setting and we solve it when the agents' forward
performance criteria are exponential.

We consider evaluation methods for payoffs with an inherent financial risk as
encountered for instance for portfolios held by pension funds and insurance
companies. Pricing such payoffs in a way consistent to market prices typically
involves combining actuarial techniques with methods from mathematical finance.
We propose to extend standard actuarial principles by a new market-consistent
evaluation procedure which we call `two step market evaluation.' This procedure
preserves the structure of standard evaluation techniques and has many other
appealing properties.

Consider an American option that pays G(X^*_t) when exercised at time t,
where G is a positive increasing function, X^*_t := \sup_{s\le t}X_s, and X_s
is the price of the underlying security at time s. Assuming zero interest
rates, we show that the seller of this option can hedge his position by trading
in the underlying security if he begins with initial capital
X_0\int_{X_0}^{\infty}G(x)x^{-2}dx (and this is the smallest initial capital
that allows him to hedge his position).

The pricing and hedging of a general class of options (including American,
Bermudan and European options) on multiple assets are studied in the context of
currency markets where trading in all assets is subject to proportional
transaction costs, and where the existence of a riskfree numeraire is not
assumed. Probabilistic dual representations are obtained for the bid and ask
prices of such options, together with constructions of hedging strategies,
optimal stopping times and approximate martingale representations for both long
and short option positions.

This paper studies pricing of stock options for the case when the evolution
of the risk-free assets or bond is stochastic. We show that, in the typical
scenario, the martingale measure is not unique, that there are non-replicable
claims, and that the martingale prices can vary significantly; for instance,
for a European put option, any positive real number is a martingale price for
some martingale measure. In addition, the second moment of the hedging error
for a strategy calculated via a given martingale measure can take any arbitrary
positive value under some equivalent measure.

This note studies an issue relating to essential smoothness that can arise
when the theory of large deviations is applied to a certain option pricing
formula in the Heston model. The note identifies a gap, based on this issue, in
the proof of Corollary 2.4 in \cite{FordeJacquier10} and describes how to
circumvent it. This completes the proof of Corollary 2.4 in
\cite{FordeJacquier10} and hence of the main result in \cite{FordeJacquier10},
which describes the limiting behaviour of the implied volatility smile in the
Heston model far from maturity.

In financial markets valuable information is rarely circulated homogeneously,
because of time required for information to spread. However, advances in
communication technology means that the 'lifetime' of an important piece of
information is typically short. Hence, viewed as a tradable asset, information
shares the characteristics of a nondurable commodity: while it can be stored
and transmitted freely, its worth diminishes rapidly in time.

In this work, we have presented a simple analytical approximation scheme for
generic non-linear FBSDEs. By treating the interested systems as the linear
decoupled FBSDE perturbed with a non-linear generator, we have shown that it is
possible to carry out recursive approximation to an arbitrarily higher order of
expansion. We have also provided two concrete examples to demonstrate how it
works and shown its accuracy relative to the results directly obtained from
numerical techniques, such as PDE and Monte Carlo simulation.

We derive a small-time expansion for out-of-the-money call options under an
exponential Levy model, using the small-time expansion for the distribution
function given in Figueroa-Lopez & Houdre (2009), combined with a change of
numeraire via the Esscher transform. In particular, we find that the effect of
a non-zero volatility $\sigma$ of the Gaussian component of the driving
L\'{e}vy process is to increase the call price by $1/2\sigma^2 t^2
e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the L\'evy density.

This paper studies the valuation of game-type credit default swaps (CDSs)
that allow the protection buyer and seller to raise or reduce the respective
position once prior to default. This leads to the analytical and numerical
studies of a stochastic game with optimal stopping subject to early termination
resulting from a default. Under a structural credit risk model based on
spectrally negative Levy processes, we analyze the existence of the Nash
equilibrium and derive the associated saddle point.

This paper is devoted to pricing American options using Monte Carlo and the
Malliavin calculus. Unlike the majority of articles related to this topic, in
this work we will not use localization fonctions to reduce the variance. Our
method is based on expressing the conditional expectation E[f(St)/Ss] using the
Malliavin calculus without localization. Then the variance of the estimator of
E[f(St)/Ss] is reduced using closed formulas, techniques based on a
conditioning and a judicious choice of the number of simulated paths.

Recent work of Dupire (2005) and Carr & Lee (2010) has highlighted the
importance of understanding the Skorokhod embedding originally proposed by Root
(1969) for the model-independent hedging of variance options. Root's work shows
that there exists a barrier from which one may define a stopping time which
solves the Skorokhod embedding problem. This construction has the remarkable
property, proved by Rost (1976), that it minimises the variance of the stopping
time among all solutions.

In this paper we analyse financial implications of exchangeability and
similar properties of finite dimensional random vectors. We show how these
properties are reflected in prices of some basket options in view of the
well-known put-call symmetry property and the duality principle in option
pricing. A particular attention is devoted to the case of asset prices driven
by Levy processes.

Modelling stock prices via jump processes is common in financial markets. In
practice, to hedge a contingent claim one typically uses the so-called
delta-hedging strategy. This strategy stems from the Black--Merton--Scholes
model where it perfectly replicates contingent claims. From the theoretical
viewpoint, there is no reason for this to hold in models with jumps. However in
practice the delta-hedging strategy is widely used and its potential
shortcoming in models with jumps is disregarded since such models are typically
incomplete and hence most contingent claims are non-attainable.

We introduce a new approach for the numerical pricing of American options.
The main idea is to choose a finite number of suitable excessive functions
(randomly) and to find the smallest majorant of the gain function in the span
of these functions. The resulting problem is a linear semi-infinite programming
problem, that can be solved using standard algorithms. This leads to good upper
bounds for the original problem. For our algorithms no discretization of space
and time and no simulation is necessary. Furthermore it is applicable even for
high-dimensional problems.

We present an explicit hedging strategy, which enables to prove arbitrageness
of market incorporating at least two assets depending on the same random
factor. The implied Black-Scholes volatility, computed taking into account the
form of the graph of the option price, related to our strategy, demonstrates
the "skewness" inherent to the observational data.

The importance of collateralization through the change of funding cost is now
well recognized among practitioners. In this article, we have extended the
previous studies of collateralized derivative pricing to more generic
situation, that is asymmetric and imperfect collateralization as well as the
associated CVA. We have presented approximate expressions for various cases
using Gateaux derivative which allow straightforward numerical analysis.
Numerical examples for CCS (cross currency swap) and IRS(interest rate swap)
with asymmetric collateralization were also provided.

The coupled nonlinear volatility and option pricing model presented recently
by Ivancevic is investigated, which generates a leverage effect, i.e., stock
volatility is (negatively) correlated to stock returns, and can be regarded as
a coupled nonlinear wave alternative of the Black-Scholes option pricing model.
In this short report, we analytically propose the two-component financial rogue
waves of the coupled nonlinear volatility and option pricing model without an
embedded w-learning. Moreover, we exhibit their dynamical behaviors for chosen
different parameters.

This paper investigates market-consistent valuation of insurance liabilities
in the context of, for instance, Solvency II and to some extent IFRS 4. We
propose an explicit and consistent framework for the valuation of insurance
liabilities which incorporates the Solvency II approach as a special case.

This paper studies the valuation of a class of credit default swaps (CDSs)
with the embedded option to switch to a different premium and notional
principal anytime prior to a credit event. These are early exercisable
contracts that give the protection buyer or seller the right to step-up,
step-down, or cancel the CDS position. The pricing problem is formulated under
a structural credit risk model based on Levy processes. This leads to the
analytic and numerical studies of an optimal stopping problem subject to early
termination due to default.

We obtain the maximum entropy distribution for an asset from call and digital
option prices. A rigorous mathematical proof of its existence and exponential
form is given, which can also be applied to legitimise a formal derivation by
Buchen and Kelly. We give a simple and robust algorithm for our method and
compare our results to theirs. We present numerical results which show that our
approach implies very realistic volatility surfaces even when calibrating only
to at-the-money options. Finally, we apply our approach to options on the S&P
500 index.

We investigate financial markets under model risk caused by uncertain
volatilities. For this purpose we consider a financial market that features
volatility uncertainty. To have a mathematical consistent framework we use the
notion of G-expectation and its corresponding G-Brownian motion recently
introduced by Peng (2007). Our financial market consists of a riskless asset
and a risky stock with price process modeled by a geometric G-Brownian motion.
We adapt the notion of arbitrage to this more complex situation and consider
stock price dynamics which exclude arbitrage opportunities.

This paper is devoted to the pricing of Barrier options by optimal quadratic
quantization method. From a known useful representation of the premium of
barrier options one deduces an algorithm similar to one used to estimate
nonlinear filter using quadratic optimal functional quantization. Some
numerical tests are fulfilled in the Black-Scholes model and in a local
volatility model and a comparison to the so called Brownian Bridge method is
also done.

We propose a general class of models for the simultaneous treatment of
equity, corporate bonds, government bonds and derivatives. The noise is
generated by a general affine Markov process. The framework allows for
stochastic volatility, jumps, the possibility of default and correlations
between different assets. We extend the notion of a discounted moment
generation function of the log stock price to the case where the underlying can
default and show how to calculate it in terms of a coupled system of
generalized Riccati equations.

In this paper, we present a new method for calculating the limit of early
exercise boundary at expiry. We price American style of general derivative
using a formula expressed as a sum of the value of European style of derivative
and so called American premium. We use the latter expression to calculate an
analytic formula for limit of early exercise boundary at expiry. Method applied
on American style plain vanilla, Asian and lookback options yields identical
results with already known values.

We present a novel approach to the pricing of financial instruments in
emission markets, for example, the EU ETS. The proposed hybrid model is
positioned between existing complex full equilibrium models and pure
risk-neutral models. Using an exogenously specified demand for a polluting good
it gives a causal explanation for the accumulation of CO2 emissions and takes
into account the feedback effect from the cost of carbon to the rate at which
the market emits CO2.

We analyze the practical consequences of the bilateral counterparty risk
adjustment. We point out that past literature assumes that, at the moment of
the first default, a risk-free closeout amount will be used. We argue that the
legal (ISDA) documentation suggests in many points that a substitution closeout
should be used. This would take into account the risk of default of the
survived party. We show how the bilateral counterparty risk adjustment changes
strongly when a substitution closeout amount is considered.

For a long time interest-rate models were built on a single yield curve used
both for discounting and forwarding. However, the crisis that has affected
financial markets in the last years led market players to revise this
assumption and accommodate basis-swap spreads, whose remarkable widening can no
longer be neglected. In recent literature we find many proposals of multi-curve
interest-rate models, whose calibration would typically require market quotes
for all yield curves. At present this is not possible since most of the quotes
are missing or extremely illiquid.

We introduce a class of randomly time-changed fast mean-reverting stochastic
volatility models and, using spectral theory and singular perturbation
techniques, we derive an approximation for the prices of European options in
this setting. Three examples of random time-changes are provided and the
implied volatility surfaces induced by these time-changes are examined as a
function of the model parameters.

In this paper we propose a copula contagion mixture model for correlated
default times. The model includes the well known factor, copula, and contagion
models as its special cases. The key advantage of such a model is that we can
study the interaction of different models and their pricing impact.
Specifically, we model the marginal default times to follow some contagion
intensity processes coupled with copula dependence structure.

We provide a general method to compute a Taylor expansion in time of implied
volatility for stochastic volatility models, using a heat kernel expansion.
Beyond the order 0 implied volatility which is already known, we compute the
first order correction exactly at all strikes from the scalar coefficient of
the heat kernel expansion. Furthermore, the first correction in the heat kernel
expansion gives the second order correction for implied volatility, which we
also give exactly at all strikes. As an application, we compute this asymptotic
expansion at order 2 for the SABR model.

We consider stochastic volatility models using piecewise constant parameters.
We suggest a hybrid optimization algorithm for fitting the models to a
volatility surface and provide some numerical results. Finally, we provide an
outlook on how to further improve the calibration procedure.

Options that allow the holder to extend the maturity by paying an additional
fixed amount found many applications in finance. Closed-form solution for these
options first appeared in Longstaff (1990) for the case when underlying asset
follows a geometric Brownian motion with the constant interest rate and
volatility. Unfortunately there are several typographical errors in the
published formula for the holder-extendible put. These are subsequently
repeated in textbooks, other papers and software. This short paper presents a
correct formula.

The aim of this study was to develop methods for evaluating the
American-style option prices when the volatility of the underlying asset is
described by a stochastic process. As part of this problem were developed
techniques for modeling the early exercise surface of the American option.
These methods of present work are compared to the complexity of modeling and
computation speed. The paper presents the semi-analytic expression for the
price of American options with stochastic volatility. The results of numerical
computations and their calibration are also presented.

In this paper new analytical and numerical approaches to valuating
path-dependent options of European type have been developed. The model of
stochastic volatility as a basic model has been chosen. For European options we
could improve the path integral method, proposed B. Baaquie, and generalized it
to the case of path-dependent options, where the payoff function depends on the
history of changes in the underlying asset. The dependence of the implied
volatility on the parameters of the stochastic volatility model has been
studied.

In the information-based approach to asset pricing the market filtration is
modelled explicitly as a superposition of signals concerning relevant market
factors and independent noise. The rate at which the signal is revealed to the
market then determines the overall magnitude of asset volatility. By letting
this information flow rate random, we obtain an elementary stochastic
volatility model within the information-based approach. Such an extension is
economically justified on account of the fact that in real markets information
flow rates are rarely measurable.

Bilateral CVA as currently implement has the counterintuitive effect of
profiting from one's own widening CDS spreads, i.e. increased risk of default,
in practice. The unified picture of CVA and liquidity introduced by Morini &
Prampolini 2010 has contributed to understanding this. However, there are two
significant omissions for practical implementation that come from the same
source, i.e. positions not booked in usual position-keeping systems. The first
omission is firm-level positions that change value upon firm default.

We study specific nonlinear transformations of the Black-Scholes implied
volatility to show remarkable properties of the volatility surface. Model-free
bounds on the implied volatility skew are given. Pricing formulas for the
European options which are written in terms of the implied volatility are
given. In particular, we prove elegant formulas for the fair strikes of the
variance swap and the gamma swap.

In this paper we consider a new class of dynamic pricing principles and
recursive utilities. We start with the interpretation of the generator of a
backward stochastic differential equation as an infinitesimal pricing rule or
an instantaneous utility. With this interpretation the generator has an
economic meaning and describes the subjective views of the investor concerning
the expected change in the price or the utility. We give a motivation for
considering non-Markovian generators of BSDEs which leads us to the study of
so-called time-delayed backward stochastic differential equations.

We study the general model of self-financing trading strategies in illiquid
markets introduced by Schoenbucher and Wilmott, 2000. A hedging strategy in the
framework of this model satisfies a nonlinear partial differential equation
(PDE) which contains some function g(alpha). This function is deep connected to
an utility function. We describe the Lie symmetry algebra of this PDE and
provide a complete set of reductions of the PDE to ordinary differential
equations (ODEs). In addition we are able to describe all types of functions
g(alpha) for which the PDE admits an extended Lie group.

The asymptotic behavior of the implied volatility associated with a general
call pricing function has been extensively studied in the last decade. The main
topics discussed in this paper are Lee's moment formulas for the implied
volatility, and Piterbarg's conjecture, describing how the implied volatility
behaves in the case where all the moments of the stock price are finite. We
find various conditions guaranteeing the existence of the limit in Lee's moment
formulas.

Using spectral decomposition techniques and singular perturbation theory, we
develop a systematic method to approximate the prices of a variety of options
in a fast mean-reverting stochastic volatility setting. Four examples are
provided in order to demonstrate the versatility of our method. These include:
European options, up-and-out options, double-barrier knock-out options, and
options which pay a rebate upon hitting a boundary. For European options, our
method is shown to produce option price approximations which are equivalent to
those developed in [5].

We study a financial model with a non-trivial price impact effect. In this
model we consider the interaction of a large investor trading in an illiquid
security, and a market maker who is quoting prices for this security. We assume
that the market maker quotes the prices such that by taking the other side of
the investor's demand, the market maker will arrive at maturity with the
maximal expected utility of the terminal wealth.

This paper considers the modelling of collateralized debt obligations (CDOs).
We propose a top-down model via forward rates generalizing Filipovi\'c,
Overbeck and Schmidt (2009) to the case where the forward rates are driven by a
finite dimensional L\'evy process. The contribution of this work is twofold: we
provide conditions for absence of arbitrage in this generalized framework.
Furthermore, we study the relation to market models by embedding them in the
forward rate framework.

We show that in a large class of stochastic volatility models with additional
skew-functions (local-stochastic volatility models) the tails of the cumulative
distribution of the log-returns behave as exp(-c|y|), where c is a positive
constant depending on time and on model parameters. We obtain this estimate
proving a stronger result: using some estimates for the probability that Ito
processes remain around a deterministic curve from Bally et al.

A version of indifference pricing methodology is proposed that shows how to
include statistical regularities of nonstochastic randomness in pricing
relations. The problem of pricing of a European option is considered as a
decision making problem, with price being a decision. Classical relations
(forward contract value and Black-Scholes formula, in particular) are obtained
as particular cases. We show that in the general case of nonstochastic
randomness the minimal expected profit of uncovered European option position is
always negative.

We consider option pricing in a regime-switching market. As the market is
incomplete, there is no unique price for a derivative. We apply the good-deal
bounds idea to obtain ranges for the price of a derivative. As an illustration,
we calculate the good-deal pricing bounds for a European call option. We
examine the stability of the good-deal pricing bounds for the European call
option when we change the market model's parameters. We find that the pricing
bounds depend strongly on the market parameters.

In this paper incomplete-information models are developed for the pricing of
securities in a stochastic interest rate setting. In particular we consider
credit-risky assets that may include random recovery upon default. The market
filtration is generated by a collection of information processes associated
with economic factors, on which interest rates depend, and information
processes associated with market factors used to model the cash flows of the
securities. We use information-sensitive pricing kernels to give rise to
stochastic interest rates.

In this paper we investigate novel applications of a new class of equations
which we call time-delayed backward stochastic differential equations. We show
that many pricing and hedging problems concerning structured products,
participating products or variable annuities can be handled by this equations.
Time-delayed BSDEs may appear when we want to find a strategy and a portfolio
which should replicate the liability whose pay-off depends on the applied
investment strategy or the values of the portfolio.

This paper works out fair values of stock loan model with automatic
termination clause. This stock loan is treated as a generalized perpetual
American option with an automatic termination clause and possibly negative
interest rate. Since it helps a bank to control the risk, banks should charge
less service fees compared to stock loans without automatic termination
clauses. The automatic termination clause is in fact a stop order set by the
bank.

We generalize Merton's asset valuation approach to systems of multiple
financial firms where cross-ownership of equities and liabilities is present.
The liabilities, which may include debts and derivatives, can be of differing
seniority. We derive equations for the prices of equities and recovery claims
under no-arbitrage. An existence result and a uniqueness result are proven.
Examples and an algorithm for the simultaneous calculation of all no-arbitrage
prices are provided.

We consider fractional Black-Scholes market with proportional transaction
costs. When transaction costs are present, one trades periodically i.e. we have
the discrete trading with equidistance $n^{-1}$ between trading times. We
derive a non trivial hedging error for a class of European options with convex
payoff in the case when the transaction costs coefficients decrease as
$n^{-(1-H)}$. We study the expected hedging error and asymptotic behavior of
the hedge as $H \to 1/2$

This paper describes a flexible and tractable bottom-up dynamic correlation
modelling framework with a consistent stochastic recovery specification. The
stochastic recovery specification only models the first two moments of the spot
recovery rate as its higher moments have almost no contribution to the loss
distribution and CDO tranche pricing.

We consider the problem of numerical approximation for forward-backward
stochastic differential equations with drivers of quadratic growth (qgFBSDE).
To illustrate the significance of qgFBSDE, we discuss a problem of cross
hedging of an insurance related financial derivative using correlated assets.
For the convergence of numerical approximation schemes for such systems of
stochastic equations, path regularity of the solution processes is
instrumental. We present a method based on the truncation of the driver, and
explicitly exhibit error estimates as functions of the truncation height.

We performed a comprehensive analysis on the price bounds of CDO tranche
options, and illustrated that the CDO tranche option prices can be effectively
bounded by the joint distribution of default time (JDDT) from a default time
copula. Systemic and idiosyncratic factors beyond the JDDT only contribute a
limited amount of pricing uncertainty. The price bounds of tranche option
derived from a default time copula are often very narrow, especially for the
senior part of the capital structure where there is the most market interests
for tranche options.

In this paper, a finite-state mean-reverting model for the short-rate, based
on the continuous time Ehrenfest process, will be examined. Two explicit
pricing formulae for zero-coupon bonds will be derived in the general and the
special symmetric cases. Its limiting relationship to the Vasicek model will be
examined with some numerical results.

We consider the pricing of derivatives written on the discrete realized
variance of an underlying security. In the literature, the realized variance is
usually approximated by its continuous-time limit, the quadratic variation of
the underlying log-price. Here, we characterize the short-time limits of call
options on both objects. We find that the difference strongly depends on
whether or not the stock price process has jumps.

The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term
structure models, long-term yields and forward rates can never fall. We present
a refined version of the DIR theorem, where we identify the reciprocal of the
maturity date as the maximal order that long-term rates at earlier dates can
dominate long-term rates at later dates. The viability assumption imposed on
the market model is weaker than those appearing previously in the literature.

Gaussian copulas are widely used in the industry to correlate two random
variables when there is no prior knowledge about the co-dependence between
them. The perturbed Gaussian copula approach allows introducing the skew
information of both random variables into the co-dependence structure. The
analytical expression of this copula is derived through an asymptotic expansion
under the assumption of a common fast mean reverting stochastic volatility
factor.

We study the pricing of credit derivatives with asymmetric information. The
managers have complete information on the value process of the firm and on the
default threshold, while the investors on the market have only partial
observations, especially about the default threshold. Different information
structures are distinguished using the framework of enlargement of filtrations.
We specify risk neutral probabilities and we evaluate default sensitive
contingent claims in these cases.

It turns out that in the bivariate Black-Scholes economy Margrabe type
options exhibit symmetry properties leading to semi-static hedges of rather
general barrier options. Some of the results are extended to variants obtained
by means of Brownian subordination. In order to increase the liquidity of the
hedging instruments for certain currency options, the duality principle can be
applied to set up hedges in a foreign market by using only European vanilla
options sometimes along with a risk-less bond.

Constant Proportion Portfolio Insurance (CPPI) is an investment strategy
designed to give participation in the performance of a risky asset while
protecting the invested capital. This protection is however not perfect and the
gap risk must be quantified. CPPI strategies are path-dependent and may have
American exercise which makes their valuation complex. A naive description of
the state of the portfolio would involve three or even four variables.

We apply a quadratic hedging scheme developed by Foellmer, Schweizer, and
Sondermann to European contingent products whose underlying asset is modeled
using a GARCH process and show that local risk-minimizing strategies with
respect to the physical measure do exist, even though an associated minimal
martingale measure is only available in the presence of bounded innovations.
More importantly, since those local risk-minimizing strategies are in general
convoluted and difficult to evaluate, we introduce Girsanov-like risk-neutral
measures for the log-prices that yield more tractable and usefu

We study the effect of parameters uncertainties on a stochastic diffusion
model, in particular the impact on the pricing of contingent claims, thanks to
Dirichlet Forms methods. We apply recent techniques, developed by Bouleau, to
hedging procedures in order to compute the sensitivities of SDE trajectories
with respect to parameter perturbations. We show that this model can reproduce
a bid-ask spread. We also prove that, if the stochastic differential equation
admits a closed form representation, also the sensitivities have closed form
representations.

The generalized 5D Black-Scholes differential equation with stochastic
volatility is derived. The projections of the stochastic evolutions associated
with the random variables from an enlarged space or superspace onto an ordinary
space can be achieved via higher-dimensional operators. The stochastic nature
of the securities and volatility associated with the 3D Merton-Garman equation
can then be interpreted as the effects of the extra dimensions. We showed that
the Merton-Garman equation is the first excited state, i.e.

We present a multivariate stochastic volatility model with leverage, which is
flexible enough to recapture the individual dynamics as well as the
interdependencies between several assets while still being highly analytically
tractable.

Consider a frictionless market trading a finite number of co-maturing
European call and put options written on a risky asset plus an instrument with
path-dependent payoff known as a weighted variance swap, e.g. a vanilla
variance swap or a corridor variance swap. The question we ask is: Do the
traded prices admit an arbitrage opportunity? We determine necessary and
sufficient model-free conditions for the price of a continuously monitored
weighted variance swap to be consistent with absence of arbitrage.

In this note, we would like to find the laws of electrodynamics in simple
economic systems. In this direction, we identify the chief economic variables
and parameters, scalar and vector, which are amenable to be put directly into
the crouch of the laws of electrodynamics, namely Maxwell's equations.
Moreover, we obtain Phillp's curve, recession and Black-Scholes formula, as
sample applications.

We propose a probabilistic framework for pricing derivatives, which
acknowledges that information and beliefs are subjective. Market prices can be
translated into implied probabilities. In particular, futures imply returns for
these implied probability distributions. We argue that volatility is not risk,
but uncertainty. Non-normal distributions combine the risk in the left tail
with the opportunities in the right tail -- unifying the "risk premium" with
the possible loss. Risk and reward must be part of the same picture and
expected returns must include possible losses due to risks.

We derive an arbitrage free relationship between recovery swap rates, digital
default swap spreads and conventional CDS spreads, and argue that the fair
forward recovery rate used in recovery swaps must contain a convexity premium
over the expected recovery value.

In this paper we develop an algorithm to calculate the prices and Greeks of
barrier options in a hyper-exponential additive model with piecewise constant
parameters. We obtain an explicit semi-analytical expression for the
first-passage probability. The solution rests on a randomization and an
explicit matrix Wiener-Hopf factorization. Employing this result we derive
explicit expressions for the Laplace-Fourier transforms of the prices and
Greeks of barrier options.

In this three-part series of papers, we argue that the conventional spread
measures are not well defined for credit-risky bonds and introduce a set of
credit term structures which correct for the biases associated with the
strippable cash flow valuation assumption. We demonstrate that the resulting
estimates are significantly more robust and remain meaningful even when applied
to deeply distressed bonds.

In the third part of this series we introduce consistent relative value
measures for CDS-Bond basis trades using the bond-implied CDS term structure
derived from fitted survival rate curves. We explain why this measure is better
than the traditionally used Z-spread or Libor OAS and offer simplified hedging
and trading strategies which take advantage of the relative value across the
entire range of maturities of cash and synthetic credit markets.

In this paper we develop structural first passage models (AT1P and SBTV) with
time-varying volatility and characterized by high tractability, moving from the
original work of Brigo and Tarenghi (2004, 2005) [19] [20] and Brigo and Morini
(2006)[15]. The models can be calibrated exactly to credit spreads using
efficient closed-form formulas for default probabilities. Default events are
caused by the value of the firm assets hitting a safety threshold, which
depends on the financial situation of the company and on market conditions. In
AT1P this default barrier is deterministic.

In this paper we develop a tractable structural model with analytical default
probabilities depending on some dynamics parameters, and we show how to
calibrate the model using a chosen number of Credit Default Swap (CDS) market
quotes. We essentially show how to use structural models with a calibration
capability that is typical of the much more tractable credit-spread based
intensity models. We apply the structural model to a concrete calibration case
and observe what happens to the calibrated dynamics when the CDS-implied credit
quality deteriorates as the firm approaches default.

In equity and foreign exchange markets the risk-neutral dynamics of the
underlying asset are commonly represented by a stochastic volatility model with
jumps. In this paper we consider a dense subclass of such models and develop
analytically tractable formulae for the prices of a range of first-generation
exotic derivatives. We provide closed form formulae for the Fourier transforms
of vanilla and forward starting options as well as the formula for the slope of
the implied volatility smile for large strikes. A simple explicit approximation
formula for the variance swap price is given.

In this paper, we give a numerical method for pricing long maturity, path
dependent options by using the Markov property for each underlying asset. This
enables us to approximate a path dependent option by using some kinds of plain
vanillas. We give some examples whose underlying assets behave as some popular
Levy processes. Moreover, we give some payoffs and functions used to
approximate them.

The purpose of this paper is introducing rigorous methods and formulas for
bilateral counterparty risk credit valuation adjustments (CVA's) on
interest-rate portfolios. In doing so, we summarize the general arbitrage-free
valuation framework for counterparty risk adjustments in presence of bilateral
default risk, as developed more in detail in Brigo and Capponi (2008),
including the default of the investor.

In this paper we prove an approximate formula expressed in terms of
elementary functions for the implied volatility in the Heston model. The
formula consists of the constant and first order terms in the large maturity
expansion of the implied volatility function. The proof is based on saddlepoint
methods and classical properties of holomorphic functions.

A nonlinear wave alternative for the standard Black-Scholes option-pricing
model is presented. The adaptive-wave model, representing 'controlled Brownian
behavior' of financial markets, is formally defined by adaptive nonlinear
Schr\"odinger (NLS) equations, defining the option-pricing wave function in
terms of the stock price and time. The model includes two parameters:
volatility (playing the role of dispersion frequency coefficient), which can be
either fixed or stochastic, and adaptive market potential that depends on the
interest rate.

In this paper we introduce a class of information-based models for the
pricing of fixed-income securities. We consider a set of continuous- time
information processes that describe the flow of information about market
factors in a monetary economy. The nominal pricing kernel is at any given time
assumed to be given by a function of the values of information processes at
that time.

We review and illustrate how the volatility smile translates into a
probability distribution, the market-implied probability distribution
representing believes priced in. The effects of changes in the smile are
examined. Special attention is given to the effects of slope, which might
appear at first counter-intuitive.

We give a rigorous proof of the representation of implied volatility as a
time-average of weighted expectations of local or stochastic volatility. With
this proof we fix the problem of a circular definition in the original
derivation of Gatheral, who introduced this implied volatility representation
in his book 'The Volatility Surface'.

A financial market model where agents trade using realistic combinations of
buy-and-hold strategies is considered. Minimal assumptions are made on the
discounted asset-price process - in particular, the semimartingale property is
not assumed. Via a natural market viability assumption, namely, absence of
arbitrages of the first kind, we establish that discounted asset-prices have to
be semimartingales.

We present a new approach for studying the problem of optimal hedging of a
European option in a finite and complete discrete-time market model. We
consider partial hedging strategies that maximize the success probability or
minimize the expected shortfall under a cost constraint and show that these
problems can be treated as so called knapsack problems, which are a widely
researched subject in linear programming. This observation gives us better
understanding of the problem of optimal hedging in discrete time.

In this article, we review the construction and properties of some popular
approaches to modeling LIBOR rates. We discuss the following frameworks:
classical LIBOR market models, forward price models and Markov-functional
models. We close with the recently developed affine LIBOR models.

This paper introduces a new semi-parametric approach to the pricing and risk
management of bespoke CDO tranches, with a particular attention to bespokes
that need to be mapped onto more than one reference portfolio. The only user
input in our framework is a multi-factor model (a "prior" model hereafter) for
index portfolios, such as CDX.NA.IG or iTraxx Europe, that are chosen as
benchmark securities for the pricing of a given bespoke CDO. Parameters of the
prior model are fixed, and not tuned to match prices of benchmark index
tranches.

We have embedded the classical theory of stochastic finance into a
differential geometric framework called Geometric Arbitrage Theory and show
that it is possible to:

- Write arbitrage as curvature of a principal fibre bundle.

- Parameterize arbitrage strategies by its holonomy. - Extend the Fundamental
Theorem of Asset Pricing by a differential homotopic characterization for both
complete and not complete arbitrage free markets.

The recent liberalization of the electricity and gas markets has resulted in
the growth of energy exchanges and modelling problems. In this paper, we
modelize jointly gas and electricity spot prices using a mean-reverting model
which fits the correlations structures for the two commodities.

This paper considers a mortgage contract where the borrower pays a fixed
mortgage rate and has the choice of making prepayment. Assume the market
interest follows the CIR model, a free boundary problem is formulated. Here we
focus on the infinite horizon problem. Using variational method, we obtain an
analytical solution to the problem, where the free boundary is implicitly given
by a transcendental algebraic equation.

We derive probabilistic representations for the probability density function
of the arbitrage price of a financial asset and the price of European call and
put options in a linear stochastic volatility model with correlated Brownian
noises. In such models the asset price satisfies a linear SDE with coefficient
of linearity being the volatility process. Examples of such models are
considered, including a log-normal stochastic volatility model. In all examples
a closed formula for the density function is given.

In this paper we provide evidence that financial option markets for equity
indices give rise to non-trivial dependency structures between its
constituents. Thus, if the individual constituent distributions of an equity
index are inferred from the single-stock option markets and combined via a
Gaussian copula, for example, one fails to explain the steepness of the
observed volatility skew of the index. Intuitively, index option prices are
encoding higher correlations in cases where the option is particularly
sensitive to stress scenarios of the market.

In [2] the notion of stickiness for stochastic processes was introduced. It
was also shown that stickiness implies absense of arbitrage in a market with
proportional transaction costs. In this paper, we investigate the notion of
stickiness further. In particular, we give examples of processes that are not
semimartingales but are sticky.