A general method to construct recombinant tree approximations for stochastic
volatility models is developed and applied to the Heston model for stock price
dynamics. In this application, the resulting approximation is a four tuple
Markov process. The ?first two components are related to the stock and
volatility processes and take values in a two dimensional Binomial tree. The
other two components of the Markov process are the increments of random walks
with simple values in {-1; +1}.

This paper proposes a Monte Carlo technique for pricing the forward yield to
maturity, when the volatility of the zero-coupon bond is known. We make the
assumption of deterministic default intensity (Hazard Rate Function). We make
no assumption on the volatility of the yield. We actually calculate the initial
value of the forward yield, we calculate the volatility of the yield, and we
write the diffusion of the yield. As direct application we price options on
Constant Maturity Treasury (CMT) in the Hull and White Model for the short
interest rate.

Using tools from spectral analysis, singular and regular perturbation theory,
we develop a systematic method for analytically computing the approximate price
of a derivative-asset. The payoff of the derivative-asset may be
path-dependent. Additionally, the process underlying the derivative may exhibit
killing (i.e. jump to default) as well as combined local/nonlocal stochastic
volatility. The nonlocal component of volatility is multiscale, in the sense
that it is driven by one fast-varying and one slow-varying factor.

We present a new numerical method to price vanilla options quickly in
time-changed Brownian motion models. The method is based on rational function
approximations of the Black-Scholes formula. Detailed numerical results are
given for a number of widely used models. In particular, we use the
variance-gamma model, the CGMY model and the Heston model without correlation
to illustrate our results. Comparison to the standard fast Fourier transform
method with respect to accuracy and speed appears to favour the newly developed
method in the cases considered.

The purpose of this paper is to design an algorithm for the computation of
the counterparty risk which is competitive in regards of a brute force
"Monte-Carlo of Monte-Carlo" method (with nested simulations). This is achieved
using marked branching diffusions describing a Galton-Watson random tree. Such
an algorithm leads at the same time to a computation of the (bilateral)
counterparty risk when we use the default-risky or counterparty-riskless option
values as mark-to-market. Our method is illustrated by various numerical
examples.

We describe, at the microscopic level, the dynamics of N interacting
components where the probability is very small when N is large that a given
component interact more than once, directly or indirectly, up to time t, with
any other component. Due to this fact, we can consider, at the macroscopic
level, the quadratic system of differential equations associated with the
interaction and establish an explicit formula for the solution of this system.
We moreover apply our results to some models of Econophysics.

In this paper, we discuss the application of quasi-Monte Carlo methods to the
Heston model. We base our algorithms on the Broadie-Kaya algorithm, an exact
simulation scheme for the Heston model.

Option contracts are a type of financial derivative that allow investors to
hedge risk and speculate on the variation of an asset's future market price. In
short, an option has a particular payout that is based on the market price for
an asset on a given date in the future. In 1973, Black and Scholes proposed a
valuation model for options that essentially estimates the tail risk of the
asset price under the assumption that the price will fluctuate according to
geometric Brownian motion.

The study of heavy-tailed distributions in economic and financial systems has
been widely addressed since financial time series has become a research
subject.After the eighties, several "highly improbable" market drops were
observed (e.g. the 1987 stock market drop known as "Black Monday" and on even
more recent ones, already in the 21st century) that produce heavy losses that
were unexplainable in a GN environment.

The short-time asymptotic behavior of option prices for a variety of models
with jumps has received much attention in recent years. In the present work, a
novel second-order approximation for ATM option prices under the CGMY L\'evy
model is derived, and then extended to a model with an additional independent
Brownian component. Our results shed light on the connection between both the
volatility of the continuous component and the jump parameters and the behavior
of ATM option prices near expiration.

We extend our studies of a quantum field model defined on a lattice having
the dilation group as a local gauge symmetry. The model is relevant in the
cross-disciplinary area of econophysics. A corresponding proposal by Ilinski
aimed at gauge modeling in non-equilibrium pricing is realized as a numerical
simulation of the one-asset version. The gauge field background enforces
minimal arbitrage, yet allows for statistical fluctuations. The new feature
added to the model is an updating prescription for the simulation that drives
the model market into a self-organized critical state.

In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan options,
and outline the development of new algorithms in this context. We provide a
characterization theorem, a theorem which gives conditions for a martingale to
be surely optimal, and a stability theorem concerning martingales which are
near to be surely optimal in a sense. Guided by these results we develop a
framework of backward algorithms for constructing such a martingale.

In this paper we investigate the effectiveness of Alternating Direction
Implicit (ADI) time discretization schemes in the numerical solution of the
three-dimensional Heston-Hull-White partial differential equation, which is
semidiscretized by applying finite difference schemes on nonuniform spatial
grids. We consider the Heston-Hull-White model with arbitrary correlation
factors, with time-dependent mean-reversion levels, with short and long
maturities, for cases where the Feller condition is satisfied and for cases
where it is not.

We determine the algebra of isovectors for the Black--Scholes equation. As a
consequence, we obtain some previously unknown families of transformations on
the solutions.

In this paper the possibility of computing equilibrium in pure exchange and
production economies by a homotopy method is investigated. The performance of
the algorithm is tested on examples with known equilibria taken from the
literature on general equilibrium models and numerical results are presented.
In computing equilibria, economy will be specified by excess demand function.

We analyze pricing and portfolio optimization problems in defaultable regime
switching markets. We contribute to both of these problems by obtaining novel
characterizations of option prices and optimal portfolio strategies under
regime-switching. Using our option price representation, we develop a novel
efficient method to price claims which may depend on the full path of the
underlying Markov chain. This is done via a change of probability measure and a
short-time asymptotic expansion of the claim' s price in terms of the Laplace
transforms of the symmetric Dirichlet distribution.

Diffusion in a linear potential in the presence of position-dependent killing
is used to mimic a default process. Different assumptions regarding transport
coefficients, initial conditions, and elasticity of the killing measure lead to
diverse models of bankruptcy. One "stylized fact" is fundamental for our
consideration: empirically default is a rather rare event, especially in the
investment grade categories of credit ratings. Hence, the action of killing may
be considered as a small parameter.

The implied volatility surface (IVS) is a fundamental building block in
computational finance. We provide a survey of methodologies for constructing
such surfaces. We also discuss various topics which can influence the
successful construction of IVS in practice: arbitrage-free conditions in both
strike and time, how to perform extrapolation outside the core region, choice
of calibrating functional and selection of numerical optimization algorithms,
volatility surface dynamics and asymptotics.

In this work we detail the application of a fast convolution algorithm
computing high dimensional integrals to the context of multiplicative noise
stochastic processes. The algorithm provides a numerical solution to the
problem of characterizing conditional probability density functions at
arbitrary time, and we applied it successfully to quadratic and piecewise
linear diffusion processes. The ability in reproducing statistical features of
financial return time series, such as thickness of the tails and scaling
properties, makes this processes appealing for option pricing.

We prove limit theorems for the super-replication cost of European options in
a Binomial model with friction. The examples covered are markets with
proportional transaction costs and the illiquid markets. The dual
representation for the super-replication cost in these models are obtained and
used to prove the limit theorems. In particular, the existence of the liquidity
premium for the continuous time limit of the model proposed in [6] is proved.
Hence, this paper extends the previous convergence result of [13] to the
general non-Markovian case.

We discuss utility based pricing and hedging of jump diffusion processes with
emphasis on the practical applicability of the framework. We point out two
difficulties that seem to limit this applicability, namely drift dependence and
essential risk aversion independence. We suggest to solve these by a
re-interpretation of the framework. This leads to the notion of an implied
drift. We also present a heuristic derivation of the marginal indifference
price and the marginal optimal hedge that might be useful in numerical
computations.

We present a numerical approach for solving the free boundary problem for the
Black-Scholes equation for pricing American style of floating strike Asian
options. A fixed domain transformation of the free boundary problem into a
parabolic equation defined on a fixed spatial domain is performed. As a result
a nonlinear time-dependent term is involved in the resulting equation. Two new
numerical algorithms are proposed. In the first algorithm a predictor-corrector
scheme is used. The second one is based on the Newton method.

In this paper we consider a jump-diffusion dynamic whose parameters are
driven by a continuous time and stationary Markov Chain on a finite state space
as a model for the underlying of European contingent claims. For this class of
processes we firstly outline the Fourier transform method both in log-price and
log-strike to efficiently calculate the value of various types of options and
as a concrete example of application, we present some numerical results within
a two-state regime switching version of the Merton jump-diffusion model.

A new standpoint on financial time series, without the use of any
mathematical model and of probabilistic tools, yields not only a rigorous
approach of trends and volatility, but also efficient calculations which were
already successfully applied in automatic control and in signal processing. It
is based on a theorem due to P. Cartier and Y. Perrin, which was published in
1995. The above results are employed for sketching a dynamical portfolio and
strategy management, without any global optimization technique. Numerous
computer simulations are presented.

In a previous paper it was shown that a Markov-functional model with
log-normally distributed rates in the terminal measure displays nonanalytic
behaviour as a function of the volatility, which is similar to a phase
transition in condensed matter physics. More precisely, certain expectation
values have discontinuous derivatives with respect to the volatility at a
certain critical value of the volatility. Here we discuss the implications of
these results for the pricing of interest rates derivatives.

We examine whether hedging effectiveness is affected by asymmetry in the
return distribution by applying tail specific metrics to compare the hedging
effectiveness of short and long hedgers using crude oil futures contracts. The
metrics used include Lower Partial Moments (LPM), Value at Risk (VaR) and
Conditional Value at Risk (CVAR). Comparisons are applied to a number of
hedging strategies including OLS and both Symmetric and Asymmetric GARCH
models.

One of the most popular copulas for modeling dependence structures is
t-copula. Recently the grouped t-copula was generalized to allow each group to
have one member only, so that a priori grouping is not required and the
dependence modeling is more flexible. This paper describes a Markov chain Monte
Carlo (MCMC) method under the Bayesian inference framework for estimating and
choosing t-copula models. Using historical data of foreign exchange (FX) rates
as a case study, we found that Bayesian model choice criteria overwhelmingly
favor the generalized t-copula.

This paper analyzes Least Squares Monte Carlo (LSM) algorithm, which is
proposed by Longstaff and Schwartz (2001) for pricing American style
securities. This algorithm is based on the projection of the value of
continuation onto a certain set of basis functions via the least squares
problem. We analyze the stability of the algorithm when the number of exercise
dates increases and prove that if the underlying process for the stock price is
continuous then the regression problem is ill-conditioned for small values of
parameter t, time.

The Cartier-Perrin theorem, which was published in 1995 and is expressed in
the language of nonstandard analysis, permits, for the first time perhaps, a
clear-cut mathematical definition of the volatility of a financial asset. It
yields as a byproduct a new understanding of the means of returns, of the beta
coefficient, and of the Sharpe and Treynor ratios. New estimation techniques
from automatic control and signal processing, which were already successfully
applied in quantitative finance, lead to several computer experiments with some
quite convincing forecasts.

In this paper we analyze American style of floating strike Asian call options
belonging to the class of financial derivatives whose payoff diagram depends
not only on the underlying asset price but also on the path average of
underlying asset prices over some predetermined time interval. The mathematical
model for the option price leads to a free boundary problem for a parabolic
partial differential equation.

We introduce a new probabilistic method for solving a class of impulse
control problems based on their representations as Backward Stochastic
Differential Equations (BSDEs for short) with constrained jumps. As an example,
our method is used for pricing Swing options. We deal with the jump constraint
by a penalization procedure and apply a discrete-time backward scheme to the
resulting penalized BSDE with jumps.

This paper deals with stability in the numerical solution of the prominent
Heston partial differential equation from mathematical finance. We study the
well-known central second-order finite difference discretization, which leads
to large semi-discrete systems with non-normal matrices A. By employing the
logarithmic spectral norm we prove practical, rigorous stability bounds. Our
theoretical stability results are illustrated by ample numerical experiments.

The optimal dividend problem by De Finetti (1957) has been recently
generalized to the spectrally negative L\'evy model where the implementation of
optimal strategies draws upon the computation of scale functions and their
derivatives. This paper proposes a phase-type fitting approximation of the
optimal strategy. We consider spectrally negative L\'evy processes with
phase-type jumps as well as meromorphic L\'evy processes (Kuznetsov et al.,
2010a), and use their scale functions to approximate the scale function for a
general spectrally negative L\'evy process.

In frictionless markets, utility maximization problems are typically solved
either by stochastic control or by martingale methods. Beginning with the
seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713],
stochastic control theory has also been used to solve various problems of this
type in the presence of proportional transaction costs. Martingale methods, on
the other hand, have so far only been used to derive general structural
results.

We explore a simple lattice field model intended to describe statistical
properties of high frequency financial markets. The model is relevant in the
cross-disciplinary area of econophysics. Its signature feature is the emergence
of a self-organized critical state. This implies scale invariance of the model,
without tuning parameters. Prominent results of our simulation are time series
of gains, prices, volatility, and gains frequency distributions, which all
compare favorably to features of historical market data.

Motivated by applications to bond markets, we propose a multivariate
framework for discrete time financial markets with proportional transaction
costs and a countable infinite number of tradable assets. We show that the
no-arbitrage of second kind property (NA2 in short), introduced by \cite{ras09}
for finite dimensional markets, allows to provide a closure property for the
set of attainable claims in a very natural way, under a suitable efficient
friction condition.

The Heston model stands out from the class of stochastic volatility (SV)
models mainly for two reasons. Firstly, the process for the volatility is
non-negative and mean-reverting, which is what we observe in the markets.
Secondly, there exists a fast and easily implemented semi-analytical solution
for European options. In this article we adapt the original work of Heston
(1993) to a foreign exchange (FX) setting. We discuss the computational aspects
of using the semi-analytical formulas, performing Monte Carlo simulations,
checking the Feller condition, and option pricing with FFT.

Computational aspects of the optimal consumption and investment with the
partially observed stochastic volatility of the asset prices are considered.
The new quantization approach to filtering - density quantization - is
introduced which reduces the original infinite dimensional state space of the
problem to the finite quantization set. The density quantization is embedded
into the numerical algorithm to solve the dynamic programming equation related
to the portfolio optimization.

Motivated by the pricing of lookback options in exponential L\'evy models, we
study the difference between the continuous and discrete supremum of L\'evy
processes. In particular, we extend the results of Broadie et al. (1999) to
jump-diffusion models. We also derive bounds for general exponential L\'evy
models.

Filiz et al. (2008) proposed a model for the pattern of defaults seen among a
group of firms at the end of a given time period. The ingredients in the model
are a graph, where the vertices correspond to the firms and the edges describe
the network of interdependencies between the firms, a parameter for each vertex
that captures the individual propensity of that firm to default, and a
parameter for each edge that captures the joint propensity of the two connected
firms to default.

Estimation of the operational risk capital under the Loss Distribution
Approach requires evaluation of aggregate (compound) loss distributions which
is one of the classic problems in risk theory. Closed-form solutions are not
available for the distributions typically used in operational risk. However
with modern computer processing power, these distributions can be calculated
virtually exactly using numerical methods. This paper reviews numerical
algorithms that can be successfully used to calculate the aggregate loss
distributions.

We derive the exact solution of a one-dimensional Markov functional model
with log-normally distributed interest rates in discrete time. The model is
shown to have two distinct limiting states, corresponding to small and
asymptotically large volatilities, respectively. These volatility regimes are
separated by a phase transition at some critical value of the volatility.

We adress the maximization problem of expected utility from terminal wealth.
The special feature of this paper is that we consider a financial market where
the price process of risky assets can have a default time. Using dynamic
programming, we characterize the value function with a backward stochastic
differential equation and the optimal portfolio policies. We separately treat
the cases of exponential, power and logarithmic utility.

The aim of this work is to provide fast and accurate approximation schemes
for the Monte-Carlo pricing of derivatives in the L\'evy LIBOR model of
Eberlein and \"Ozkan (2005). Standard methods can be applied to solve the
stochastic differential equations of the successive LIBOR rates but the methods
are generally slow. We propose an alternative approximation scheme based on
Picard iterations. Our approach is similar in accuracy to the full numerical
solution, but with the feature that each rate is, unlike the standard method,
evolved independently of the other rates in the term structure.

We start by showing that the finite-time absolute ruin probability in the
classical risk model with constant interest force can be expressed in terms of
the transition probability of a positive Ornstein-Uhlenbeck type process, say
X. Our methodology applies to the case when the dynamics of the aggregate
claims process is a subordinator. From this expression, we easily deduce
necessary and sufficient conditions for the infinite-time absolute ruin to
occur.

Arora, Barak, Brunnermeier, and Ge showed that taking computational
complexity into account, a dishonest seller could increase the lemon costs of a
family of financial derivatives dramatically. We show that if the seller is
required to construct derivatives of a certain form, then this phenomenon
disappears. In particular, we define and construct pseudorandom derivative
families, for which lemon placement only slightly affects the values of the
derivatives. Our constructions use randomness extractors and expander graphs.
We study our derivatives in a more general setting than Arora et al.

The paper generalizes the construction by stochastic flows of consistent
utilities processes introduced by M. Mrad and N. El Karoui (2010). The market
is incomplete and securities are modeled as locally bounded positive
semimartingales. Making minimal assumptions and convex constraints on
test-portfolios, we construct by composing two stochastic flows of
homeomorphisms, all the consistent stochastic utilities whose the optimal
wealth process is a given admissible portfolio, strictly increasing in initial
capital. Proofs are essentially based on change of variables techniques.

This paper investigates the use of multiple directions of stratification as a
variance reduction technique for Monte Carlo simulations of path-dependent
options driven by Gaussian vectors. The precision of the method depends on the
choice of the directions of stratification and the allocation rule within each
strata.

The intention of this paper is to estimate a Bayesian distribution-free chain
ladder (DFCL) model using approximate Bayesian computation (ABC) methodology.
We demonstrate how to estimate quantities of interest in claims reserving and
compare the estimates to those obtained from classical and credibility
approaches. In this context, a novel numerical procedure utilising Markov chain
Monte Carlo (MCMC), ABC and a Bayesian bootstrap procedure was developed in a
truly distribution-free setting.

We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely
efficient calculation of correlation Risk of option prices computed with Monte
Carlo simulations. A key point in the construction is the use of binning to
simultaneously achieve computational efficiency and accurate confidence
intervals. We illustrate the method for a copula-based Monte Carlo computation
of claims written on a basket of underlying assets, and we test it numerically
for Portfolio Default Options.

We show that shortfall risks of American options in a sequence of multinomial
approximations of the multidimensional Black--Scholes (BS) market converge to
the corresponding quantities for similar American options in the
multidimensional BS market with path dependent payoffs. In comparison to
previous papers we consider the multi assets case for which we use the weak
convergence approach.

We study shortfall risk minimization for American options with path dependent
payoffs under proportional transaction costs in the Black--Scholes (BS) model.
We show that for this case the shortfall risk is a limit of similar terms in an
appropriate sequence of binomial models. We also prove that in the continuous
time BS model for a given initial capital there exists a portfolio strategy
which minimizes the shortfall risk. In the absence of transactions costs
(complete markets) similar limit theorems were obtained in Dolinsky and Kifer
(2008, 2010) for game options.

The log-periodic power law (LPPL) is a model of asset prices during
endogenous bubbles. If the on-going development of a bubble is suspected, asset
prices can be fit numerically to the LPPL law. The best solutions can then
indicate whether a bubble is in progress and, if so, the bubble critical time
(i.e., when the bubble is expected to burst). Consequently, the LPPL model is
useful only if the data can be fit to the model with algorithms that are
accurate and computationally efficient.

In mathematical finance a popular approach for pricing options under some
Levy model is to consider underlying that follows a Poisson jump diffusion
process. As it is well known this results in a partial integro-differential
equation (PIDE) that usually does not allow an analytical solution while
numerical solution brings some problems. In this paper we elaborate a new
approach on how to transform the PIDE to some class of so-called
pseudo-parabolic equations which are known in mathematics but are relatively
new for mathematical finance.

An efficient adaptive direct numerical integration (DNI) algorithm is
developed for computing high quantiles and conditional Value at Risk (CVaR) of
compound distributions using characteristic functions. A key innovation of the
numerical scheme is an effective tail integration approximation that reduces
the truncation errors significantly with little extra effort. High precision
results of the 0.999 quantile and CVaR were obtained for compound losses with
heavy tails and a very wide range of loss frequencies using the DNI, Fast
Fourier Transform (FFT) and Monte Carlo (MC) methods.

Most previous contributions to BSDEs, and the related theories of nonlinear
expectation and dynamic risk measures, have been in the framework of continuous
time diffusions or jump diffusions. Using solutions of BSDEs on spaces related
to finite state, continuous time Markov chains, we develop a theory of
nonlinear expectations in the spirit of [Dynamically consistent nonlinear
evaluations and expectations (2005) Shandong Univ.]. We prove basic properties
of these expectations and show their applications to dynamic risk measures on
such spaces.

The hybrid Monte Carlo (HMC) algorithm is applied for the Bayesian inference
of the stochastic volatility (SV) model. We use the HMC algorithm for the
Markov chain Monte Carlo updates of volatility variables of the SV model. First
we compute parameters of the SV model by using the artificial financial data
and compare the results from the HMC algorithm with those from the Metropolis
algorithm. We find that the HMC algorithm decorrelates the volatility variables
faster than the Metropolis algorithm.

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal.,
to appear] for integrating semi-discrete advection-diffusion PDEs, is extended
for application to some of the important equations of mathematical finance.
Using estimates for the numerical range of the spatial operator, optimal
contour parameters are derived theoretically and tested numerically. Test
examples presented are the Black-Scholes PDE in one space dimension and the
Heston PDE in two dimensions. In the latter case efficiency is compared to ADI
splitting schemes for solving this problem.

We analyze the regularity of the optimal exercise boundary for the American
Put option when the underlying asset pays a discrete dividend at a known time
$t_d$ during the lifetime of the option. The ex-dividend asset price process is
assumed to follow Black-Scholes dynamics and the dividend amount is a
deterministic function of the ex-dividend asset price just before the dividend
date. The solution to the associated optimal stopping problem can be
characterised in terms of an optimal exercise boundary which, in contrast to
the case when there are no dividends, is no longer monotone.

We propose a mathematical framework for the study of a family of random
fields - called forward performances - which arise as numerical representation
of certain rational preference relations in mathematical finance. Their spatial
structure corresponds to that of utility functions, while the temporal one
reflects a Nisio-type semigroup property, referred to as self-generation.

Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two risk
measures which are widely used in the practice of risk management. This paper
deals with the problem of computing both VaR and CVaR using stochastic
approximation (with decreasing steps): we propose a first Robbins-Monro
procedure based on Rockaffelar-Uryasev's identity for the CVaR. The convergence
rate of this algorithm to its target satisfies a Gaussian Central Limit
Theorem.

Families of exact solutions are found for a nonlinear modification of the
Black-Scholes equation. This risk-adjusted pricing methodology model (RAPM)
incorporates both transaction costs and the risk from a volatile portfolio.
Using the Lie group analysis we obtain the Lie algebra admitted by the RAPM
equation. It gives us the possibility to describe an optimal system of
subalgebras and correspondingly the set of invariant solutions to the model. On
this way we can describe complete set of possible reductions of the nonlinear
RAPM model.

We consider the exact path sampling of the squared Bessel process and some
other continuous-time Markov processes, such as the CIR model, constant
elasticity of variance diffusion model, and hypergeometric diffusions, which
can all be obtained from a squared Bessel process by using a change of
variable, time and scale transformation, and/or change of measure. All these
diffusions are broadly used in mathematical finance for modelling asset prices,
market indices, and interest rates.

We present a comprehensive study of utility function of the minority game in
its efficient regime. We develop an effective description of state of the game.
For the payoff function $g(x)=\sgn (x)$ we explicitly represent the game as the
Markov process and prove the finitness of number of states. We also demonstrate
boundedness of the utility function. Using these facts we can explain all
interesting observable features of the aggregated demand: appearance of strong
fluctuations, their periodicity and existence of prefered levels.