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3,600 |
On three filtering problems arising in mathematical finance
|
q-fin.CP
|
Three situations in which filtering theory is used in mathematical finance
are illustrated at different levels of detail. The three problems originate
from the following different works: 1) On estimating the stochastic volatility
model from observed bilateral exchange rate news, by R. Mahieu, and P.
Schotman; 2) A state space approach to estimate multi-factors CIR models of the
term structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3)
Risk-minimizing hedging strategies under partial observation in pricing
financial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In the
first problem we propose to use a recent nonlinear filtering technique based on
geometry to estimate the volatility time series from observed bilateral
exchange rates. The model used here is the stochastic volatility model. The
filters that we propose are known as projection filters, and a brief derivation
of such filters is given. The second problem is introduced in detail, and a
possible use of different filtering techniques is hinted at. In fact the
filters used for this problem in 2) and part of the literature can be
interpreted as projection filters and we will make some remarks on how more
general and possibly more suitable projection filters can be constructed. The
third problem is only presented shortly.
|
finance
|
3,601 |
A method of moments approach to pricing double barrier contracts driven by a general class of jump diffusions
|
q-fin.CP
|
We present the method of moments approach to pricing barrier-type options
when the underlying is modelled by a general class of jump diffusions. By
general principles the option prices are linked to certain infinite dimensional
linear programming problems. Subsequently approximating those systems by finite
dimensional linear programming problems, upper and lower bounds for the prices
of such options are found. As numerical illustration we apply the method to the
valuation of several barrier-type options (double barrier knockout option,
American corridor and double no touch) under a number of different models,
including a case with deterministic interest rates, and compare with Monte
Carlo simulation results. In all cases we find tight bounds with short
execution times. Theoretical convergence results are also provided.
|
finance
|
3,602 |
Optimal systems of subalgebras for a nonlinear Black-Scholes equation
|
q-fin.CP
|
The main object of our study is a four dimensional Lie algebra which
describes the symmetry properties of a nonlinear Black-Scholes model. This
model implements a feedback effect which is typical for an illiquid market. The
structure of the Lie algebra depends on one parameter, i.e. we have to do with
a one-parametric family of algebras. We provide a classification of these
algebras using Patera--Winternitz method. Optimal systems of one-, two- and
three- dimensional subalgebras are described for the family of symmetry
algebras of the nonlinear Black-Scholes equation. The optimal systems give us
the possibility to describe a complete set of invariant solutions to the
equation.
|
finance
|
3,603 |
Bayesian inference with an adaptive proposal density for GARCH models
|
q-fin.CP
|
We perform the Bayesian inference of a GARCH model by the Metropolis-Hastings
algorithm with an adaptive proposal density. The adaptive proposal density is
assumed to be the Student's t-distribution and the distribution parameters are
evaluated by using the data sampled during the simulation. We apply the method
for the QGARCH model which is one of asymmetric GARCH models and make empirical
studies for for Nikkei 225, DAX and Hang indexes. We find that autocorrelation
times from our method are very small, thus the method is very efficient for
generating uncorrelated Monte Carlo data. The results from the QGARCH model
show that all the three indexes show the leverage effect, i.e. the volatility
is high after negative observations.
|
finance
|
3,604 |
Markov Chain Monte Carlo on Asymmetric GARCH Model Using the Adaptive Construction Scheme
|
q-fin.CP
|
We perform Markov chain Monte Carlo simulations for a Bayesian inference of
the GJR-GARCH model which is one of asymmetric GARCH models. The adaptive
construction scheme is used for the construction of the proposal density in the
Metropolis-Hastings algorithm and the parameters of the proposal density are
determined adaptively by using the data sampled by the Markov chain Monte Carlo
simulation. We study the performance of the scheme with the artificial
GJR-GARCH data. We find that the adaptive construction scheme samples GJR-GARCH
parameters effectively and conclude that the Metropolis-Hastings algorithm with
the adaptive construction scheme is an efficient method to the Bayesian
inference of the GJR-GARCH model.
|
finance
|
3,605 |
Defaultable bonds with an infinite number of Levy factors
|
q-fin.CP
|
A market with defaultable bonds where the bond dynamics is in a
Heath-Jarrow-Morton setting and the forward rates are driven by an infinite
number of Levy factors is considered. The setting includes rating migrations
driven by a Markov chain. All basic types of recovery are investigated. We
formulate necessary and sufficient conditions (generalized HJM conditions)
under which the market is arbitrage free. Connections with consistency
conditions are discussed.
|
finance
|
3,606 |
On the Performance of Delta Hedging Strategies in Exponential Lévy Models
|
q-fin.CP
|
We consider the performance of non-optimal hedging strategies in exponential
L\'evy models. Given that both the payoff of the contingent claim and the
hedging strategy admit suitable integral representations, we use the Laplace
transform approach of Hubalek et al. (2006) to derive semi-explicit formulas
for the resulting mean squared hedging error in terms of the cumulant
generating function of the underlying L\'evy process. In two numerical
examples, we apply these results to compare the efficiency of the Black-Scholes
hedge and the model delta to the mean-variance optimal hedge in a normal
inverse Gaussian and a diffusion-extended CGMY L\'evy model.
|
finance
|
3,607 |
Appraisal of a contour integral method for the Black-Scholes and Heston equations
|
q-fin.CP
|
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. In the examples it is found that
the contour integral method is superior for the range of medium to high
accuracy requirements. Further improvements to the current implementation of
the contour integral method are suggested.
|
finance
|
3,608 |
Simulation de trajectoires de processus continus
|
q-fin.CP
|
Continuous time stochastic processes are useful models especially for
financial and insurance purposes. The numerical simulation of such models is
dependant of the time discrete discretization, of the parametric estimation and
of the choice of a random number generator. The aim of this paper is to provide
the tools for the practical implementation of diffusion processes simulation,
particularly for insurance contexts.
|
finance
|
3,609 |
Sequential optimizing investing strategy with neural networks
|
q-fin.CP
|
In this paper we propose an investing strategy based on neural network models
combined with ideas from game-theoretic probability of Shafer and Vovk. Our
proposed strategy uses parameter values of a neural network with the best
performance until the previous round (trading day) for deciding the investment
in the current round. We compare performance of our proposed strategy with
various strategies including a strategy based on supervised neural network
models and show that our procedure is competitive with other strategies.
|
finance
|
3,610 |
Basket Options Valuation for a Local Volatility Jump-Diffusion Model with the Asymptotic Expansion Method
|
q-fin.CP
|
In this paper we discuss the basket options valuation for a jump-diffusion
model. The underlying asset prices follow some correlated local volatility
diffusion processes with systematic jumps. We derive a forward partial integral
differential equation (PIDE) for general stochastic processes and use the
asymptotic expansion method to approximate the conditional expectation of the
stochastic variance associated with the basket value process. The numerical
tests show that the suggested method is fast and accurate in comparison with
the Monte Carlo and other methods in most cases.
|
finance
|
3,611 |
Computational LPPL Fit to Financial Bubbles
|
q-fin.CP
|
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 this paper, we address primarily the
computational efficiency and secondarily the precision of the LPPL non-linear
least-square fit. Specifically, we present a parallel Levenberg-Marquardt
algorithm (LMA) for LPPL least-square fit that sped up computation of more than
a factor of four over a sequential LMA on historical and synthetic price
series. Additionally, we isolate a linear sub-structure of the LPPL
least-square fit that can be paired with an exact computation of the Jacobian,
give new settings for the Levenberg-Marquardt damping factor, and describe a
heuristic method to choose initial solutions.
|
finance
|
3,612 |
Indifference of Defaultable Bonds with Stochastic Intensity models
|
q-fin.CP
|
The utility-based pricing of defaultable bonds in the case of stochastic
intensity models of default risk is discussed. The Hamilton-Jacobi- Bellman
(HJB) equations for the value functions is derived. A finite difference method
is used to solve this problem. The yield-spreads for both buyer and seller are
extracted. The behaviour of the spread curve given the default intensity is
analyzed. Finally the impacts of the risk aversion and the correlation
coefficient are discussed.
|
finance
|
3,613 |
Dynamics on/in financial markets: dynamical decoupling and stylized facts
|
q-fin.CP
|
Stylized facts can be regarded as constraints for any modeling attempt of
price dynamics on a financial market, in that an empirically reasonable model
has to reproduce these stylized facts at least qualitatively. The dynamics of
market prices is modeled on a macro-level as the result of the dynamic coupling
of two dynamical components. The degree of their dynamical decoupling is shown
to have a significant impact on the stochastic properties of return trials such
as the return distribution, volatility clustering, and the multifractal
behavior of time scales of asset returns. Particularly we observe a cross over
in the return distribution from a Gaussian-like to a Levy-like shape when the
degree of decoupling increases. In parallel, the larger the degree of
decoupling is the more pronounced is volatility clustering. These findings
suggest that the considerations of time in an economic system, in general, and
the coupling of constituting processes is essential for understanding the
behavior of a financial market.
|
finance
|
3,614 |
Fast Correlation Greeks by Adjoint Algorithmic Differentiation
|
q-fin.CP
|
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. For any number of underlying assets or names in
a portfolio, the sensitivities of the option price with respect to all the
pairwise correlations is obtained at a computational cost which is at most 4
times the cost of calculating the option value itself. For typical
applications, this results in computational savings of several order of
magnitudes with respect to standard methods.
|
finance
|
3,615 |
Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems
|
q-fin.CP
|
In this paper we prove that there exists a smooth classical solution to the
HJB equation for a large class of constrained problems with utility functions
that are not necessarily differentiable or strictly concave. The value function
is smooth if admissible controls satisfy an integrability condition or if it is
continuous on the closure of its domain. The key idea is to work on the dual
control problem and the dual HJB equation. We construct a smooth, strictly
convex solution to the dual HJB equation and show that its conjugate function
is a smooth, strictly concave solution to the primal HJB equation satisfying
the terminal and boundary conditions.
|
finance
|
3,616 |
Analysis of the sensitivity to discrete dividends : A new approach for pricing vanillas
|
q-fin.CP
|
The incorporation of a dividend yield in the classical option pricing model
of Black- Scholes results in a minor modification of the Black-Scholes formula,
since the lognormal dynamic of the underlying asset is preserved. However,
market makers prefer to work with cash dividends with fixed value instead of a
dividend yield. Since there is no closed-form solution for the price of a
European Call in this case, many methods have been proposed in the literature
to approximate it. Here, we present a new approach. We derive an exact analytic
formula for the sensitivity to dividends of an European option. We use this
result to elaborate a proxy which possesses the same Taylor expansion around 0
with respect to the dividends as the exact price. The obtained approximation is
very fast to compute (the same complexity than the usual Black-Scholes formula)
and numerical tests show the extreme accuracy of the method for all practical
cases.
|
finance
|
3,617 |
Numerical methods for optimal insurance demand under marked point processes shocks
|
q-fin.CP
|
This paper deals with numerical solutions of maximizing expected utility from
terminal wealth under a non-bankruptcy constraint. The wealth process is
subject to shocks produced by a general marked point process. The problem of
the agent is to derive the optimal insurance strategy which allows "lowering"
the level of the shocks. This optimization problem is related to a suitable
dual stochastic control problem in which the delicate boundary constraints
disappear. In Mnif \cite{mnif10}, the dual value function is characterized as
the unique viscosity solution of the corresponding Hamilton Jacobi Bellman
Variational Inequality (HJBVI in short). We characterize the optimal insurance
strategy by the solution of the variational inequality which we solve
numerically by using an algorithm based on policy iterations.
|
finance
|
3,618 |
Constrained NonSmooth Utility Maximization on the Positive Real Line
|
q-fin.CP
|
We maximize the expected utility of terminal wealth in an incomplete market
where there are cone constraints on the investor's portfolio process and the
utility function is not assumed to be strictly concave or differentiable. We
establish the existence of the optimal solutions to the primal and dual
problems and their dual relationship. We simplify the present proofs in this
area and extend the existing duality theory to the constrained nonsmooth
setting.
|
finance
|
3,619 |
Stability of central finite difference schemes for the Heston PDE
|
q-fin.CP
|
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.
|
finance
|
3,620 |
Swing Options Valuation: a BSDE with Constrained Jumps Approach
|
q-fin.CP
|
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. We study the convergence of this numerical
method, with respect to the main approximation parameters: the jump intensity
$\lambda$, the penalization parameter $p > 0$ and the time step. In particular,
we obtain a convergence rate of the error due to penalization of order
$(\lambda p)^{\alpha - \frac{1}{2}}, \forall \alpha \in \left(0,
\frac{1}{2}\right)$. Combining this approach with Monte Carlo techniques, we
then work out the valuation problem of (normalized) Swing options in the Black
and Scholes framework. We present numerical tests and compare our results with
a classical iteration method.
|
finance
|
3,621 |
The computation of Greeks with multilevel Monte Carlo
|
q-fin.CP
|
We study the use of the multilevel Monte Carlo technique in the context of
the calculation of Greeks. The pathwise sensitivity analysis differentiates the
path evolution and reduces the payoff's smoothness. This leads to new
challenges: the inapplicability of pathwise sensitivities to non-Lipschitz
payoffs often makes the use of naive algorithms impossible. These challenges
can be addressed in three different ways: payoff smoothing using conditional
expectations of the payoff before maturity; approximating the previous
technique with path splitting for the final timestep; using of a hybrid
combination of pathwise sensitivity and the Likelihood Ratio Method. We
investigate the strengths and weaknesses of these alternatives in different
multilevel Monte Carlo settings.
|
finance
|
3,622 |
Bayesian Model Choice of Grouped t-copula
|
q-fin.CP
|
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. In addition, all the criteria also agree on the
second most likely model and these inferences are all consistent with classical
likelihood ratio tests. Finally, we demonstrate the impact of model choice on
the conditional Value-at-Risk for portfolios of six major FX rates.
|
finance
|
3,623 |
Defaultable Bonds via HKA
|
q-fin.CP
|
To construct a no-arbitrage defaultable bond market, we work on the state
price density framework. Using the heat kernel approach (HKA for short) with
the killing of a Markov process, we construct a single defaultable bond market
that enables an explicit expression of a defaultable bond and credit spread
under quadratic Gaussian settings. Some simulation results show that the model
is not only tractable but realistic.
|
finance
|
3,624 |
Exact Simulation of the 3/2 Model
|
q-fin.CP
|
This paper discusses the exact simulation of the stock price process
underlying the 3/2 model. Using a result derived by Craddock and Lennox using
Lie Symmetry Analysis, we adapt the Broadie-Kaya algorithm for the simulation
of affine processes to the 3/2 model. We also discuss variance reduction
techniques and find that conditional Monte Carlo techniques combined with
quasi-Monte Carlo point sets result in significant variance reductions.
|
finance
|
3,625 |
Analytic results and weighted Monte Carlo simulations for CDO pricing
|
q-fin.CP
|
We explore the possibilities of importance sampling in the Monte Carlo
pricing of a structured credit derivative referred to as Collateralized Debt
Obligation (CDO). Modeling a CDO contract is challenging, since it depends on a
pool of (typically about 100) assets, Monte Carlo simulations are often the
only feasible approach to pricing. Variance reduction techniques are therefore
of great importance. This paper presents an exact analytic solution using
Laplace-transform and MC importance sampling results for an easily tractable
intensity-based model of the CDO, namely the compound Poissonian. Furthermore
analytic formulae are derived for the reweighting efficiency. The computational
gain is appealing, nevertheless, even in this basic scheme, a phase transition
can be found, rendering some parameter regimes out of reach. A
model-independent transform approach is also presented for CDO pricing.
|
finance
|
3,626 |
Comparison of Two Numerical Methods for Computation of American Type of the Floating Strike Asian Option
|
q-fin.CP
|
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. Computational
experiments, confirming the accuracy of the algorithms are presented and
discussed.
|
finance
|
3,627 |
Pricing of average strike Asian call option using numerical PDE methods
|
q-fin.CP
|
In this paper, a standard PDE for the pricing of arithmetic average strike
Asian call option is presented. A Crank-Nicolson Implicit Method and a Higher
Order Compact finite difference scheme for this pricing problem is derived.
Both these schemes were implemented for various values of risk free rate and
volatility. The option prices for the same set of values of risk free rate and
volatility was also computed using Monte Carlo simulation. The comparative
results of the two numerical PDE methods shows close match with the Monte Carlo
results, with the Higher Order Compact scheme exhibiting a better match. To the
best of our knowledge, this is the first work to use the numerical PDE approach
for pricing Asian call options with average strike.
|
finance
|
3,628 |
Duality and Convergence for Binomial Markets with Friction
|
q-fin.CP
|
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. Moreover, the special case of small transaction
costs yields, in the continuous limit, the $G$-expectation of Peng as earlier
proved by Kusuoka in [14].
|
finance
|
3,629 |
Multilevel Monte Carlo method for jump-diffusion SDEs
|
q-fin.CP
|
We investigate the extension of the multilevel Monte Carlo path simulation
method to jump-diffusion SDEs. We consider models with finite rate activity,
using a jump-adapted discretisation in which the jump times are computed and
added to the standard uniform dis- cretisation times. The key component in
multilevel analysis is the calculation of an expected payoff difference between
a coarse path simulation and a fine path simulation with twice as many
timesteps. If the Poisson jump rate is constant, the jump times are the same on
both paths and the multilevel extension is relatively straightforward, but the
implementation is more complex in the case of state-dependent jump rates for
which the jump times naturally differ.
|
finance
|
3,630 |
Multiplicative noise, fast convolution, and pricing
|
q-fin.CP
|
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. Since exact
analytical results are missing, we exploit the fast convolution as a numerical
method alternative to the Monte Carlo simulation both in objective and risk
neutral settings. In numerical sections we document how fast convolution
outperforms Monte Carlo both in velocity and efficiency terms.
|
finance
|
3,631 |
Adjoints and Automatic (Algorithmic) Differentiation in Computational Finance
|
q-fin.CP
|
Two of the most important areas in computational finance: Greeks and,
respectively, calibration, are based on efficient and accurate computation of a
large number of sensitivities. This paper gives an overview of adjoint and
automatic differentiation (AD), also known as algorithmic differentiation,
techniques to calculate these sensitivities. When compared to finite difference
approximation, this approach can potentially reduce the computational cost by
several orders of magnitude, with sensitivities accurate up to machine
precision. Examples and a literature survey are also provided.
|
finance
|
3,632 |
Fast resolution of a single factor Heath-Jarrow-Morton model with stochastic volatility
|
q-fin.CP
|
This paper considers the single factor Heath-Jarrow-Morton model for the
interest rate curve with stochastic volatility. Its natural formulation,
described in terms of stochastic differential equations, is solved through
Monte Carlo simulations, that usually involve rather large computation time,
inefficient from a practical (financial) perspective. This model turns to be
Markovian in three dimensions and therefore it can be mapped into a 3D partial
differential equations problem. We propose an optimized numerical method to
solve the 3D PDE model in both low computation time and reasonable accuracy, a
fundamental criterion for practical purposes. The spatial and temporal
discretization are performed using finite-difference and Crank-Nicholson
schemes respectively, and the computational efficiency is largely increased
performing a scale analysis and using Alternating Direction Implicit schemes.
Several numerical considerations such as convergence criteria or computation
time are analyzed and discussed.
|
finance
|
3,633 |
Arbitrage-free Self-organizing Markets with GARCH Properties: Generating them in the Lab with a Lattice Model
|
q-fin.CP
|
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. Taking advantage of some
flexibility of the updating prescription, stylized features and dynamical
behaviors of real-world markets are reproduced in some detail.
|
finance
|
3,634 |
Quasi-Monte Carlo methods for the Heston model
|
q-fin.CP
|
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. As the joint transition densities are
not available in closed-form, the Linear Transformation method due to Imai and
Tan, a popular and widely applicable method to improve the effectiveness of
quasi-Monte Carlo methods, cannot be employed in the context of path-dependent
options when the underlying price process follows the Heston model.
Consequently, we tailor quasi-Monte Carlo methods directly to the Heston model.
The contributions of the paper are threefold: We firstly show how to apply
quasi-Monte Carlo methods in the context of the Heston model and the SVJ model,
secondly that quasi-Monte Carlo methods improve on Monte Carlo methods, and
thirdly how to improve the effectiveness of quasi-Monte Carlo methods by using
bridge constructions tailored to the Heston and SVJ models. Finally, we provide
some extensions for computing greeks, barrier options, multidimensional and
multi-asset pricing, and the 3/2 model.
|
finance
|
3,635 |
Counterparty Risk Valuation: A Marked Branching Diffusion Approach
|
q-fin.CP
|
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.
|
finance
|
3,636 |
Fast computation of vanilla prices in time-changed models and implied volatilities using rational approximations
|
q-fin.CP
|
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. We present error estimates for the option
prices. Additionally, we use this method to derive a procedure to compute, for
a given set of arbitrage-free European call option prices, the corresponding
Black-Scholes implied volatility surface. To achieve this, rational function
approximations of the inverse of the Black-Scholes formula are used. We are
thus able to work out implied volatilities more efficiently than one can by the
use of other common methods. Error estimates are presented for a wide range of
parameters.
|
finance
|
3,637 |
The potential approach in practice
|
q-fin.CP
|
The potential approach is a general and simple method for modelling interest
rates, foreign exchange rates, and in principle other types of financial
assets. This paper takes data on some liquid interest rate derivatives, and
fits potential models using a small finite-state Markov chain as the base
Markov process.
|
finance
|
3,638 |
Interlinkages and structural changes in cross-border liabilities: a network approach
|
q-fin.CP
|
We study the international interbank market through a geometrical and a
topological analysis of empirical data. The geometrical analysis of the time
series of cross-country liabilities shows that the systematic information of
the interbank international market is contained in a space of small dimension,
from which a topological characterization could be conveniently carried out.
Weighted and complete networks of financial linkages across countries are
developed, for which continuous clustering, degree centrality and closeness
centrality are computed. The behavior of these topological coefficients reveals
an important modification acting in the financial linkages in the period
1997-2011. Here we show that, besides the generalized clustering increase,
there is a persistent increment in the degree of connectivity and in the
closeness centrality of some countries. These countries seem to correspond to
critical locations where tax policies might provide opportunities to shift
debts. Such critical locations highlight the role that specific countries play
in the network structure and helps to situates the turbulent period that has
been characterizing the global financial system since the Summer 2007 as the
counterpart of a larger structural change going on for a more than one decade.
|
finance
|
3,639 |
A New Kind of Finance
|
q-fin.CP
|
Finance has benefited from the Wolfram's NKS approach but it can and will
benefit even more in the future, and the gains from the influence may actually
be concentrated among practitioners who unintentionally employ those principles
as a group.
|
finance
|
3,640 |
Multilevel Monte Carlo methods for applications in finance
|
q-fin.CP
|
Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.
|
finance
|
3,641 |
An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model
|
q-fin.CP
|
This paper derives a new semi closed-form approximation formula for pricing
an up-and-out barrier option under a certain type of stochastic volatility
model including SABR model by applying a rigorous asymptotic expansion method
developed by Kato, Takahashi and Yamada (2012). We also demonstrate the
validity of our approximation method through numerical examples.
|
finance
|
3,642 |
Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation
|
q-fin.CP
|
The multilevel Monte Carlo path simulation method introduced by Giles ({\it
Operations Research}, 56(3):607-617, 2008) exploits strong convergence
properties to improve the computational complexity by combining simulations
with different levels of resolution. In this paper we analyse its efficiency
when using the Milstein discretisation; this has an improved order of strong
convergence compared to the standard Euler-Maruyama method, and it is proved
that this leads to an improved order of convergence of the variance of the
multilevel estimator. Numerical results are also given for basket options to
illustrate the relevance of the analysis.
|
finance
|
3,643 |
An extension of Paulsen-Gjessing's risk model with stochastic return on investments
|
q-fin.CP
|
We consider in this paper a general two-sided jump-diffusion risk model that
allows for risky investments as well as for correlation between the two
Brownian motions driving insurance risk and investment return. We first
introduce the model and then find the integro-differential equations satisfied
by the Gerber-Shiu functions as well as the expected discounted penalty
functions at ruin caused by a claim or by oscillation; We also study the
dividend problem for the threshold and barrier strategies, the moments and
moment-generating function of the total discounted dividends until ruin are
discussed. Some examples are given for special cases.
|
finance
|
3,644 |
The first passage time problem for mixed-exponential jump processes with applications in insurance and finance
|
q-fin.CP
|
This paper stidies the first passage times to constant boundaries for
mixed-exponential jump diffusion processes. Explicit solutions of the Laplace
transforms of the distribution of the first passage times, the joint
distribution of the first passage times and undershoot (overshoot) are
obtained. As applications, we present explicit expression of the Gerber-Shiu
functions for surplus processes with two-sided jumps, present the analytical
solutions for popular path-dependent options such as lookback and barrier
options in terms of Laplace transforms and give a closed-form expression on the
price of the zero-coupon bond under a structural credit risk model with jumps.
|
finance
|
3,645 |
Pricing American options via multi-level approximation methods
|
q-fin.CP
|
In this article we propose a novel approach to reduce the computational
complexity of various approximation methods for pricing discrete time American
options. Given a sequence of continuation values estimates corresponding to
different levels of spatial approximation and time discretization, we propose a
multi-level low biased estimate for the price of an American option. It turns
out that the resulting complexity gain can be rather high and can even reach
the order (\varepsilon^{-1}) with (\varepsilon) denoting the desired precision.
The performance of the proposed multilevel algorithm is illustrated by a
numerical example of pricing Bermudan max-call options.
|
finance
|
3,646 |
Pricing approximations and error estimates for local Lévy-type models with default
|
q-fin.CP
|
We find approximate solutions of partial integro-differential equations,
which arise in financial models when defaultable assets are described by
general scalar L\'evy-type stochastic processes. We derive rigorous error
bounds for the approximate solutions. We also provide numerical examples
illustrating the usefulness and versatility of our methods in a variety of
financial settings.
|
finance
|
3,647 |
Pricing TARN Using a Finite Difference Method
|
q-fin.CP
|
Typically options with a path dependent payoff, such as Target Accumulation
Redemption Note (TARN), are evaluated by a Monte Carlo method. This paper
describes a finite difference scheme for pricing a TARN option. Key steps in
the proposed scheme involve tracking of multiple one-dimensional finite
difference solutions, application of jump conditions at each cash flow exchange
date, and a cubic spline interpolation of results after each jump. Since a
finite difference scheme for TARN has significantly different features from a
typical finite difference scheme for options with a path independent payoff, we
give a step by step description on the implementation of the scheme, which is
not available in the literature. The advantages of the proposed finite
difference scheme over the Monte Carlo method are illustrated by examples with
three different knockout types. In the case of constant or time dependent
volatility models (where Monte Carlo requires simulation at cash flow dates
only), the finite difference method can be faster by an order of magnitude than
the Monte Carlo method to achieve the same accuracy in price. Finite difference
method can be even more efficient in comparison with Monte Carlo in the case of
local volatility model where Monte Carlo requires significantly larger number
of time steps. In terms of robust and accurate estimation of Greeks, the
advantage of the finite difference method will be even more pronounced.
|
finance
|
3,648 |
A robust tree method for pricing American options with CIR stochastic interest rate
|
q-fin.CP
|
We propose a robust and stable lattice method which permits to obtain very
accurate American option prices in presence of CIR stochastic interest rate
without any numerical restriction on its parameters. Numerical results show the
reliability and the accuracy of the proposed method.
|
finance
|
3,649 |
Monte Carlo approximation to optimal investment
|
q-fin.CP
|
This paper sets up a methodology for approximately solving optimal investment
problems using duality methods combined with Monte Carlo simulations. In
particular, we show how to tackle high dimensional problems in incomplete
markets, where traditional methods fail due to the curse of dimensionality.
|
finance
|
3,650 |
CORN: Correlation-Driven Nonparametric Learning Approach for Portfolio Selection -- an Online Appendix
|
q-fin.CP
|
This appendix proves CORN's universal consistency. One of Bin's PhD thesis
examiner (Special thanks to Vladimir Vovk from Royal Holloway, University of
London) suggested that CORN is universal and provided sketch proof of Lemma
1.6, which is the key of this proof. Based on the proof in Gy\"prfi et al.
[2006], we thus prove CORN's universal consistency. Note that the notations in
this appendix follows Gy\"orfi et al. [2006].
|
finance
|
3,651 |
Explicit implied volatilities for multifactor local-stochastic volatility models
|
q-fin.CP
|
We consider an asset whose risk-neutral dynamics are described by a general
class of local-stochastic volatility models and derive a family of asymptotic
expansions for European-style option prices and implied volatilities. Our
implied volatility expansions are explicit; they do not require any special
functions nor do they require numerical integration. To illustrate the accuracy
and versatility of our method, we implement it under five different model
dynamics: CEV local volatility, quadratic local volatility, Heston stochastic
volatility, $3/2$ stochastic volatility, and SABR local-stochastic volatility.
|
finance
|
3,652 |
On Modeling Economic Default Time: A Reduced-Form Model Approach
|
q-fin.CP
|
In the aftermath of the global financial crisis, much attention has been paid
to investigating the appropriateness of the current practice of default risk
modeling in banking, finance and insurance industries. A recent empirical study
by Guo et al.(2008) shows that the time difference between the economic and
recorded default dates has a significant impact on recovery rate estimates. Guo
et al.(2011) develop a theoretical structural firm asset value model for a firm
default process that embeds the distinction of these two default times. To be
more consistent with the practice, in this paper, we assume the market
participants cannot observe the firm asset value directly and developed a
reduced-form model to characterize the economic and recorded default times. We
derive the probability distribution of these two default times. The numerical
study on the difference between these two shows that our proposed model can
both capture the features and fit the empirical data.
|
finance
|
3,653 |
A note on Keen's model: The limits of Schumpeter's "Creative Destruction"
|
q-fin.CP
|
This paper presents a general solution for a recent model by Keen for
endogenous money creation. The solution provides an analytic framework that
explains all significant dynamical features of Keen's model and their
parametric dependence, including an exact result for both the period and
subsidence rate of the Great Moderation. It emerges that Keen's model has just
two possible long term solutions: stable growth or terminal collapse. While
collapse can come about immediately from economies that are nonviable by virtue
of unsuitable parameters or initial conditions, in general the collapse is
preceded by an interval of exponential growth. In first approximation, the
duration of that exponential growth is half a period of a sinusoidal
oscillation. The period is determined by reciprocal of the imaginary part of
one root of a certain quintic polynomial. The real part of the same root
determines the rate of growth of the economy. The coefficients of that
polynomial depend in a complicated way upon the numerous parameters in the
problem and so, therefore, the pattern of roots. For a favorable choice of
parameters, the salient root is purely real. This is the circumstance that
admits the second possible long term solution, that of indefinite stable
growth, i.e. an infinite period.
|
finance
|
3,654 |
A hybrid approach for the implementation of the Heston model
|
q-fin.CP
|
We propose a hybrid tree-finite difference method in order to approximate the
Heston model. We prove the convergence by embedding the procedure in a
bivariate Markov chain and we study the convergence of European and American
option prices. We finally provide numerical experiments that give accurate
option prices in the Heston model, showing the reliability and the efficiency
of the algorithm.
|
finance
|
3,655 |
Over-the-counter market models with several assets
|
q-fin.CP
|
We study two classes of over-the-counter markets specified by systems of
ODE's, in the spirit of Duffie-Garleanu-Pedersen, Econometrica, 2005. We first
compute the steady states for many of these ODE's. Then we obtain the prices at
which investors trade with each other at these steady states. Finally, we study
the stability of the solutions of these ODE's.
|
finance
|
3,656 |
A Taylor series approach to pricing and implied vol for LSV models
|
q-fin.CP
|
Using classical Taylor series techniques, we develop a unified approach to
pricing and implied volatility for European-style options in a general
local-stochastic volatility setting. Our price approximations require only a
normal CDF and our implied volatility approximations are fully explicit (ie,
they require no special functions, no infinite series and no numerical
integration). As such, approximate prices can be computed as efficiently as
Black-Scholes prices, and approximate implied volatilities can be computed
nearly instantaneously.
|
finance
|
3,657 |
Fast Convergence of Regress-Later Estimates in Least Squares Monte Carlo
|
q-fin.CP
|
Many problems in financial engineering involve the estimation of unknown
conditional expectations across a time interval. Often Least Squares Monte
Carlo techniques are used for the estimation. One method that can be combined
with Least Squares Monte Carlo is the "Regress-Later" method. Unlike
conventional methods where the value function is regressed on a set of basis
functions valued at the beginning of the interval, the "Regress-Later" method
regresses the value function on a set of basis functions valued at the end of
the interval. The conditional expectation across the interval is then computed
exactly for each basis function. We provide sufficient conditions under which
we derive the convergence rate of Regress-Later estimators. Importantly, our
results hold on non-compact sets. We show that the Regress-Later method is
capable of converging significantly faster than conventional methods and
provide an explicit example. Achieving faster convergence speed provides a
strong motivation for using Regress-Later methods in estimating conditional
expectations across time.
|
finance
|
3,658 |
Asymptotic expansion for characteristic function in Heston stochastic volatility model with fast mean-reverting correction
|
q-fin.CP
|
In this note, we derive the characteristic function expansion for logarithm
of the underlying asset price in corrected Heston model as proposed by Fouque
and Lorig.
|
finance
|
3,659 |
Exact simulation pricing with Gamma processes and their extensions
|
q-fin.CP
|
Exact path simulation of the underlying state variable is of great practical
importance in simulating prices of financial derivatives or their sensitivities
when there are no analytical solutions for their pricing formulas. However, in
general, the complex dependence structure inherent in most nontrivial
stochastic volatility (SV) models makes exact simulation difficult. In this
paper, we present a nontrivial SV model that parallels the notable Heston SV
model in the sense of admitting exact path simulation as studied by Broadie and
Kaya. The instantaneous volatility process of the proposed model is driven by a
Gamma process. Extensions to the model including superposition of independent
instantaneous volatility processes are studied. Numerical results show that the
proposed model outperforms the Heston model and two other L\'evy driven SV
models in terms of model fit to the real option data. The ability to exactly
simulate some of the path-dependent derivative prices is emphasized. Moreover,
this is the first instance where an infinite-activity volatility process can be
applied exactly in such pricing contexts.
|
finance
|
3,660 |
A central limit theorem for Latin hypercube sampling with dependence and application to exotic basket option pricing
|
q-fin.CP
|
We consider the problem of estimating $\mathbb{E} [f(U^1, \ldots, U^d)]$,
where $(U^1, \ldots, U^d)$ denotes a random vector with uniformly distributed
marginals. In general, Latin hypercube sampling (LHS) is a powerful tool for
solving this kind of high-dimensional numerical integration problem. In the
case of dependent components of the random vector $(U^1, \ldots, U^d)$ one can
achieve more accurate results by using Latin hypercube sampling with dependence
(LHSD). We state a central limit theorem for the $d$-dimensional LHSD
estimator, by this means generalising a result of Packham and Schmidt.
Furthermore we give conditions on the function $f$ and the distribution of
$(U^1, \ldots, U^d)$ under which a reduction of variance can be achieved.
Finally we compare the effectiveness of Monte Carlo and LHSD estimators
numerically in exotic basket option pricing problems.
|
finance
|
3,661 |
Pricing of vanilla and first generation exotic options in the local stochastic volatility framework: survey and new results
|
q-fin.CP
|
Stochastic volatility (SV) and local stochastic volatility (LSV) processes
can be used to model the evolution of various financial variables such as FX
rates, stock prices, and so on. Considerable efforts have been devoted to
pricing derivatives written on underliers governed by such processes. Many
issues remain, though, including the efficacy of the standard alternating
direction implicit (ADI) numerical methods for solving SV and LSV pricing
problems. In general, the amount of required computations for these methods is
very substantial. In this paper we address some of these issues and propose a
viable alternative to the standard ADI methods based on Galerkin-Ritz ideas. We
also discuss various approaches to solving the corresponding pricing problems
in a semi-analytical fashion. We use the fact that in the zero correlation case
some of the pricing problems can be solved analytically, and develop a
closed-form series expansion in powers of correlation. We perform a thorough
benchmarking of various numerical solutions by using analytical and
semi-analytical solutions derived in the paper.
|
finance
|
3,662 |
Computation of the "Enrichment" of a Value Functions of an Optimization Problem on Cumulated Transaction-Costs through a Generalized Lax-Hopf Formula
|
q-fin.CP
|
The Lax-Hopf formula simplifies the value function of an intertemporal
optimization (infinite dimensional) problem associated with a convex
transaction-cost function which depends only on the transactions (velocities)
of a commodity evolution: it states that the value function is equal to the
marginal fonction of a finite dimensional problem with respect to durations and
average ransactions, much simpler to solve. The average velocity of the value
function on a investment temporal window is regarded as an enrichment,
proportional to the profit and inversely proportional to the investment
duration. At optimum, the Lax-Hopf formula implies that the enrichment is equal
to the cost of the average transaction on the investment temporal window. In
this study, we generalize the Lax-Hopf formula when the transaction-cost
function depends also on time and commodity, for reducing the infinite
dimensional problem to a finite dimensional problem. For that purpose, we
introduce the moderated ansaction-cost function which depends only on the
duration and on a commodity. Here again, the generalized Lax-Hopf formula
reduces the computation of the value function to the marginal fonction of an
optimization problem on durations and commodities involving the moderated
transaction cost function. At optimum, the enrichment of the value function is
still equal to the moderated transition cost-function of average transaction.
|
finance
|
3,663 |
Pricing of basket options I
|
q-fin.CP
|
Pricing of high-dimensional options is a deep problem of the Theoretical
Financial Mathematics. In this article we present a new class of L\'{e}vy
driven models of stock markets. In our opinion, any market model should be
based on a transparent and intuitively easily acceptable concept. In our case
this is a linear system of stochastic equations. Our market model is based on
the principle of inheritance, i.e. for the particular choice of parameters it
coincides with known models. Also, the model proposed is effectively
numerically realizable. For the class of models under cosideration, we give an
explicit representations of characteristic functions. This allows us us to
construct a sequence of approximation formulas to price basket options. We show
that our approximation formulas have almost optimal rate of convergence in the
sense of respective n-widths.
|
finance
|
3,664 |
Efficient tree methods for pricing digital barrier options
|
q-fin.CP
|
We propose an efficient lattice procedure which permits to obtain European
and American option prices under the Black and Scholes model for digital
options with barrier features. Numerical results show the accuracy of the
proposed method.
|
finance
|
3,665 |
Estimate nothing
|
q-fin.CP
|
In the econometrics of financial time series, it is customary to take some
parametric model for the data, and then estimate the parameters from historical
data. This approach suffers from several problems. Firstly, how is estimation
error to be quantified, and then taken into account when making statements
about the future behaviour of the observed time series? Secondly, decisions may
be taken today committing to future actions over some quite long horizon, as in
the trading of derivatives; if the model is re-estimated at some intermediate
time, our earlier decisions would need to be revised - but the derivative has
already been traded at the earlier price. Thirdly, the exact form of the
parametric model to be used is generally taken as given at the outset; other
competitor models might possibly work better in some circumstances, but the
methodology does not allow them to be factored into the inference. What we
propose here is a very simple (Bayesian) alternative approach to inference and
action in financial econometrics which deals decisively with all these issues.
The key feature is that nothing is being estimated.
|
finance
|
3,666 |
Accelerating Implicit Finite Difference Schemes Using a Hardware Optimized Tridiagonal Solver for FPGAs
|
q-fin.CP
|
We present a design and implementation of the Thomas algorithm optimized for
hardware acceleration on an FPGA, the Thomas Core. The hardware-based algorithm
combined with the custom data flow and low level parallelism available in an
FPGA reduces the overall complexity from 8N down to 5N serial arithmetic
operations, and almost halves the overall latency by parallelizing the two
costly divisions. Combining this with a data streaming interface, we reduce
memory overheads to 2 N-length vectors per N-tridiagonal system to be solved.
The Thomas Core allows for multiple independent tridiagonal systems to be
continuously solved in parallel, providing an efficient and scalable
accelerator for many numerical computations. Finally we present applications
for derivatives pricing problems using implicit finite difference schemes on an
FPGA accelerated system and we investigate the use and limitations of
fixed-point arithmetic in our algorithm.
|
finance
|
3,667 |
The role of information in a two-traders market
|
q-fin.CP
|
In a very simple stock market, made by only two \emph{initially equivalent}
traders, we discuss how the information can affect the performance of the
traders. More in detail, we first consider how the portfolios of the traders
evolve in time when the market is \emph{closed}. After that, we discuss two
models in which an interaction with the outer world is allowed. We show that,
in this case, the two traders behave differently, depending on \textbf{i)} the
amount of information which they receive from outside; and \textbf{ii)}the
quality of this information.
|
finance
|
3,668 |
High-Order Splitting Methods for Forward PDEs and PIDEs
|
q-fin.CP
|
This paper is dedicated to the construction of high-order (in both space and
time) finite-difference schemes for both forward and backward PDEs and PIDEs,
such that option prices obtained by solving both the forward and backward
equations are consistent. This approach is partly inspired by Andreasen & Huge,
2011 who reported a pair of consistent finite-difference schemes of first-order
approximation in time for an uncorrelated local stochastic volatility model. We
extend their approach by constructing schemes that are second-order in both
space and time and that apply to models with jumps and discrete dividends.
Taking correlation into account in our approach is also not an issue.
|
finance
|
3,669 |
Multilevel Monte Carlo For Exponential Lévy Models
|
q-fin.CP
|
We apply multilevel Monte Carlo for option pricing problems using exponential
L\'{e}vy models with a uniform timestep discretisation to monitor the running
maximum required for lookback and barrier options. The numerical results
demonstrate the computational efficiency of this approach. We derive estimates
of the convergence rate for the error introduced by the discrete monitoring of
the running supremum of a broad class of L\'{e}vy processes. We use these to
obtain upper bounds on the multilevel Monte Carlo variance convergence rate for
the Variance Gamma, NIG and $\alpha$-stable processes used in the numerical
experiments. We also show numerical results and analysis of a trapezoidal
approximation for Asian options.
|
finance
|
3,670 |
The acceptance-rejection method for low-discrepancy sequences
|
q-fin.CP
|
Generation of pseudorandom numbers from different probability distributions
has been studied extensively in the Monte Carlo simulation literature. Two
standard generation techniques are the acceptance-rejection and inverse
transformation methods. An alternative approach to Monte Carlo simulation is
the quasi-Monte Carlo method, which uses low-discrepancy sequences, instead of
pseudorandom numbers, in simulation. Low-discrepancy sequences from different
distributions can be obtained by the inverse transformation method, just like
for pseudorandom numbers. In this paper, we will present an
acceptance-rejection algorithm for low-discrepancy sequences. We will prove a
convergence result, and present error bounds. We will then use this
acceptance-rejection algorithm to develop quasi-Monte Carlo versions of some
well known algorithms to generate beta and gamma distributions, and investigate
the efficiency of these algorithms numerically. We will also consider the
simulation of the variance gamma model, a model used in computational finance,
where the generation of these probability distributions are needed. Our results
show that the acceptance-rejection technique can result in significant
improvements in computing time over the inverse transformation method in the
context of low-discrepancy sequences.
|
finance
|
3,671 |
Asymptotics for $d$-dimensional Lévy-type processes
|
q-fin.CP
|
We consider a general d-dimensional Levy-type process with killing. Combining
the classical Dyson series approach with a novel polynomial expansion of the
generator A(t) of the Levy-type process, we derive a family of asymptotic
approximations for transition densities and European-style options prices.
Examples of stochastic volatility models with jumps are provided in order to
illustrate the numerical accuracy of our approach. The methods described in
this paper extend the results from Corielli et al. (2010), Pagliarani and
Pascucci (2013) and Lorig et al. (2013a) for Markov diffusions to Markov
processes with jumps.
|
finance
|
3,672 |
Macroprudential oversight, risk communication and visualization
|
q-fin.CP
|
This paper discusses the role of risk communication in macroprudential
oversight and of visualization in risk communication. Beyond the soar in data
availability and precision, the transition from firm-centric to system-wide
supervision imposes vast data needs. Moreover, except for internal
communication as in any organization, broad and effective external
communication of timely information related to systemic risks is a key mandate
of macroprudential supervisors, further stressing the importance of simple
representations of complex data. This paper focuses on the background and
theory of information visualization and visual analytics, as well as techniques
within these fields, as potential means for risk communication. We define the
task of visualization in risk communication, discuss the structure of
macroprudential data, and review visualization techniques applied to systemic
risk. We conclude that two essential, yet rare, features for supporting the
analysis of big data and communication of risks are analytical visualizations
and interactive interfaces. For visualizing the so-called macroprudential data
cube, we provide the VisRisk platform with three modules: plots, maps and
networks. While VisRisk is herein illustrated with five web-based interactive
visualizations of systemic risk indicators and models, the platform enables and
is open to the visualization of any data from the macroprudential data cube.
|
finance
|
3,673 |
Leveraged {ETF} implied volatilities from {ETF} dynamics
|
q-fin.CP
|
The growth of the exhange-traded fund (ETF) industry has given rise to the
trading of options written on ETFs and their leveraged counterparts {(LETFs)}.
We study the relationship between the ETF and LETF implied volatility surfaces
when the underlying ETF is modeled by a general class of local-stochastic
volatility models. A closed-form approximation for prices is derived for
European-style options whose payoff depends on the terminal value of the ETF
and/or LETF. Rigorous error bounds for this pricing approximation are
established. A closed-form approximation for implied volatilities is also
derived. We also discuss a scaling procedure for comparing implied volatilities
across leverage ratios. The implied volatility expansions and scalings are
tested in three well-known settings: CEV, Heston and SABR.
|
finance
|
3,674 |
Valuation of Barrier Options using Sequential Monte Carlo
|
q-fin.CP
|
Sequential Monte Carlo (SMC) methods have successfully been used in many
applications in engineering, statistics and physics. However, these are seldom
used in financial option pricing literature and practice. This paper presents
SMC method for pricing barrier options with continuous and discrete monitoring
of the barrier condition. Under the SMC method, simulated asset values rejected
due to barrier condition are re-sampled from asset samples that do not breach
the barrier condition improving the efficiency of the option price estimator;
while under the standard Monte Carlo many simulated asset paths can be rejected
by the barrier condition making it harder to estimate option price accurately.
We compare SMC with the standard Monte Carlo method and demonstrate that the
extra effort to implement SMC when compared with the standard Monte Carlo is
very little while improvement in price estimate can be significant. Both
methods result in unbiased estimators for the price converging to the true
value as $1/\sqrt{M}$, where $M$ is the number of simulations (asset paths).
However, the variance of SMC estimator is smaller and does not grow with the
number of time steps when compared to the standard Monte Carlo. In this paper
we demonstrate that SMC can successfully be used for pricing barrier options.
SMC can also be used for pricing other exotic options and also for cases with
many underlying assets and additional stochastic factors such as stochastic
volatility; we provide general formulas and references.
|
finance
|
3,675 |
Splitting and Matrix Exponential approach for jump-diffusion models with Inverse Normal Gaussian, Hyperbolic and Meixner jumps
|
q-fin.CP
|
This paper is a further extension of the method proposed in Itkin, 2014 as
applied to another set of jump-diffusion models: Inverse Normal Gaussian,
Hyperbolic and Meixner. To solve the corresponding PIDEs we accomplish few
steps. First, a second-order operator splitting on financial processes
(diffusion and jumps) is applied to these PIDEs. To solve the diffusion
equation, we use standard finite-difference methods. For the jump part, we
transform the jump integral into a pseudo-differential operator and construct
its second order approximation on a grid which supersets the grid that we used
for the diffusion part. The proposed schemes are unconditionally stable in time
and preserve positivity of the solution which is computed either via a matrix
exponential, or via P'ade approximation of the matrix exponent. Various
numerical experiments are provided to justify these results.
|
finance
|
3,676 |
Multilevel path simulation for weak approximation schemes
|
q-fin.CP
|
In this paper we discuss the possibility of using multilevel Monte Carlo
(MLMC) methods for weak approximation schemes. It turns out that by means of a
simple coupling between consecutive time discretisation levels, one can achieve
the same complexity gain as under the presence of a strong convergence. We
exemplify this general idea in the case of weak Euler scheme for L\'evy driven
stochastic differential equations, and show that, given a weak convergence of
order $\alpha\geq 1/2,$ the complexity of the corresponding "weak" MLMC
estimate is of order $\varepsilon^{-2}\log ^{2}(\varepsilon).$ The numerical
performance of the new "weak" MLMC method is illustrated by several numerical
examples.
|
finance
|
3,677 |
High Performance Financial Simulation Using Randomized Quasi-Monte Carlo Methods
|
q-fin.CP
|
GPU computing has become popular in computational finance and many financial
institutions are moving their CPU based applications to the GPU platform. Since
most Monte Carlo algorithms are embarrassingly parallel, they benefit greatly
from parallel implementations, and consequently Monte Carlo has become a focal
point in GPU computing. GPU speed-up examples reported in the literature often
involve Monte Carlo algorithms, and there are software tools commercially
available that help migrate Monte Carlo financial pricing models to GPU.
We present a survey of Monte Carlo and randomized quasi-Monte Carlo methods,
and discuss existing (quasi) Monte Carlo sequences in GPU libraries. We discuss
specific features of GPU architecture relevant for developing efficient (quasi)
Monte Carlo methods. We introduce a recent randomized quasi-Monte Carlo method,
and compare it with some of the existing implementations on GPU, when they are
used in pricing caplets in the LIBOR market model and mortgage backed
securities.
|
finance
|
3,678 |
Approximating the zero-coupon bond price in a general one-factor model with constant coefficients
|
q-fin.CP
|
We consider a general one-factor short rate model, in which the instantaneous
interest rate is driven by a univariate diffusion with time independent drift
and volatility. We construct recursive formula for the coefficients of the
Taylor expansion of the bond price and its logarithm around $\tau=0$, where
$\tau$ is time to maturity. We provide numerical examples of convergence of the
partial sums of the series and compare them with the known exact values in the
case of Cox-Ingersoll-Ross and Dothan model.
|
finance
|
3,679 |
Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation
|
q-fin.CP
|
There is a vast literature on numerical valuation of exotic options using
Monte Carlo, binomial and trinomial trees, and finite difference methods. When
transition density of the underlying asset or its moments are known in closed
form, it can be convenient and more efficient to utilize direct integration
methods to calculate the required option price expectations in a backward
time-stepping algorithm. This paper presents a simple, robust and efficient
algorithm that can be applied for pricing many exotic options by computing the
expectations using Gauss-Hermite integration quadrature applied on a cubic
spline interpolation. The algorithm is fully explicit but does not suffer the
inherent instability of the explicit finite difference counterpart. A `free'
bonus of the algorithm is that it already contains the function for fast and
accurate interpolation of multiple solutions required by many discretely
monitored path dependent options. For illustrations, we present examples of
pricing a series of American options with either Bermudan or continuous
exercise features, and a series of exotic path-dependent options of target
accumulation redemption note (TARN). Results of the new method are compared
with Monte Carlo and finite difference methods, including some of the most
advanced or best known finite difference algorithms in the literature. The
comparison shows that, despite its simplicity, the new method can rival with
some of the best finite difference algorithms in accuracy and at the same time
it is significantly faster. Virtually the same algorithm can be applied to
price other path-dependent financial contracts such as Asian options and
variable annuities.
|
finance
|
3,680 |
Analysis of Spin Financial Market by GARCH Model
|
q-fin.CP
|
A spin model is used for simulations of financial markets. To determine
return volatility in the spin financial market we use the GARCH model often
used for volatility estimation in empirical finance. We apply the Bayesian
inference performed by the Markov Chain Monte Carlo method to the parameter
estimation of the GARCH model. It is found that volatility determined by the
GARCH model exhibits "volatility clustering" also observed in the real
financial markets. Using volatility determined by the GARCH model we examine
the mixture-of-distribution hypothesis (MDH) suggested for the asset return
dynamics. We find that the returns standardized by volatility are approximately
standard normal random variables. Moreover we find that the absolute
standardized returns show no significant autocorrelation. These findings are
consistent with the view of the MDH for the return dynamics.
|
finance
|
3,681 |
Perturbation analysis of a nonlinear equation arising in the Schaefer-Schwartz model of interest rates
|
q-fin.CP
|
We deal with the interest rate model proposed by Schaefer and Schwartz, which
models the long rate and the spread, defined as the difference between the
short and the long rates. The approximate analytical formula for the bond
prices suggested by the authors requires a computation of a certain constant,
defined via a nonlinear equation and an integral of a solution to a system of
ordinary differential equations. In this paper we use perturbation methods to
compute this constant. Coefficients of its expansion are given in a closed form
and can be constructed to arbitrary order. However, our numerical results show
that a very good accuracy is achieved already after using a small number of
terms.
|
finance
|
3,682 |
Exact solution of a generalized version of the Black-Scholes equation
|
q-fin.CP
|
We analyze a generalized version of the Black-Scholes equation depending on a
parameter $a\!\in \!(-\infty,0)$. It satisfies the martingale condition and
coincides with the Black-Scholes equation in the limit case $a\nearrow 0$. We
show that the generalized equation is exactly solvable in terms of Hermite
polynomials and numerically compare its solution with the solution of the
Black-Scholes equation.
|
finance
|
3,683 |
Valuation of Variable Annuities with Guaranteed Minimum Withdrawal and Death Benefits via Stochastic Control Optimization
|
q-fin.CP
|
In this paper we present a numerical valuation of variable annuities with
combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum
Death Benefit (GMDB) under optimal policyholder behaviour solved as an optimal
stochastic control problem. This product simultaneously deals with financial
risk, mortality risk and human behaviour. We assume that market is complete in
financial risk and mortality risk is completely diversified by selling enough
policies and thus the annuity price can be expressed as appropriate
expectation. The computing engine employed to solve the optimal stochastic
control problem is based on a robust and efficient Gauss-Hermite quadrature
method with cubic spline. We present results for three different types of death
benefit and show that, under the optimal policyholder behaviour, adding the
premium for the death benefit on top of the GMWB can be problematic for
contracts with long maturities if the continuous fee structure is kept, which
is ordinarily assumed for a GMWB contract. In fact for some long maturities it
can be shown that the fee cannot be charged as any proportion of the account
value -- there is no solution to match the initial premium with the fair
annuity price. On the other hand, the extra fee due to adding the death benefit
can be charged upfront or in periodic instalment of fixed amount, and it is
cheaper than buying a separate life insurance.
|
finance
|
3,684 |
Convergence of an Euler scheme for a hybrid stochastic-local volatility model with stochastic rates in foreign exchange markets
|
q-fin.CP
|
We study the Heston-Cox-Ingersoll-Ross++ stochastic-local volatility model in
the context of foreign exchange markets and propose a Monte Carlo simulation
scheme which combines the full truncation Euler scheme for the stochastic
volatility component and the stochastic domestic and foreign short interest
rates with the log-Euler scheme for the exchange rate. We establish the
exponential integrability of full truncation Euler approximations for the
Cox-Ingersoll-Ross process and find a lower bound on the explosion time of
these exponential moments. Under a full correlation structure and a realistic
set of assumptions on the so-called leverage function, we prove the strong
convergence of the exchange rate approximations and deduce the convergence of
Monte Carlo estimators for a number of vanilla and path-dependent options.
Then, we perform a series of numerical experiments for an autocallable barrier
dual currency note.
|
finance
|
3,685 |
Valuation Algorithms for Structural Models of Financial Interconnectedness
|
q-fin.CP
|
Much research in systemic risk is focused on default contagion. While this
demands an understanding of valuation, fewer articles specifically deal with
the existence, the uniqueness, and the computation of equilibrium prices in
structural models of interconnected financial systems. However, beyond
contagion research, these topics are also essential for risk-neutral pricing.
In this article, we therefore study and compare valuation algorithms in the
standard model of debt and equity cross-ownership which has crystallized in the
work of several authors over the past one and a half decades. Since known
algorithms have potentially infinite runtime, we develop a class of new
algorithms, which find exact solutions in finitely many calculation steps. A
simulation study for a range of financial system designs allows us to derive
conclusions about the efficiency of different numerical methods under different
system parameters.
|
finance
|
3,686 |
Liquidity costs: a new numerical methodology and an empirical study
|
q-fin.CP
|
We consider rate swaps which pay a fixed rate against a floating rate in
presence of bid-ask spread costs. Even for simple models of bid-ask spread
costs, there is no explicit strategy optimizing an expected function of the
hedging error. We here propose an efficient algorithm based on the stochastic
gradient method to compute an approximate optimal strategy without solving a
stochastic control problem. We validate our algorithm by numerical experiments.
We also develop several variants of the algorithm and discuss their
performances in terms of the numerical parameters and the liquidity cost.
|
finance
|
3,687 |
A hybrid tree/finite-difference approach for Heston-Hull-White type models
|
q-fin.CP
|
We study a hybrid tree-finite difference method which permits to obtain
efficient and accurate European and American option prices in the Heston
Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we
provide a new simulation scheme to be used for Monte Carlo evaluations.
Numerical results show the reliability and the efficiency of the proposed
methods
|
finance
|
3,688 |
Anomalous volatility scaling in high frequency financial data
|
q-fin.CP
|
Volatility of intra-day stock market indices computed at various time
horizons exhibits a scaling behaviour that differs from what would be expected
from fractional Brownian motion (fBm). We investigate this anomalous scaling by
using empirical mode decomposition (EMD), a method which separates time series
into a set of cyclical components at different time-scales. By applying the EMD
to fBm, we retrieve a scaling law that relates the variance of the components
to a power law of the oscillating period. In contrast, when analysing 22
different stock market indices, we observe deviations from the fBm and Brownian
motion scaling behaviour. We discuss and quantify these deviations, associating
them to the characteristics of financial markets, with larger deviations
corresponding to less developed markets.
|
finance
|
3,689 |
IMEX schemes for a Parabolic-ODE system of European Options with Liquidity Shocks
|
q-fin.CP
|
The coupled system, where one is a degenerate parabolic equation and the
other has not a diffusion term arises in the modeling of European options with
liquidity shocks. Two implicit-explicit (IMEX) schemes that preserve the
positivity of the differential problem solution are constructed and analyzed.
Numerical experiments confirm the theoretical results and illustrate the high
accuracy and efficiency of the schemes in combination with Richardson
extrapolation
|
finance
|
3,690 |
Application of Operator Splitting Methods in Finance
|
q-fin.CP
|
Financial derivatives pricing aims to find the fair value of a financial
contract on an underlying asset. Here we consider option pricing in the partial
differential equations framework. The contemporary models lead to
one-dimensional or multidimensional parabolic problems of the
convection-diffusion type and generalizations thereof. An overview of various
operator splitting methods is presented for the efficient numerical solution of
these problems.
Splitting schemes of the Alternating Direction Implicit (ADI) type are
discussed for multidimensional problems, e.g. given by stochastic volatility
(SV) models. For jump models Implicit-Explicit (IMEX) methods are considered
which efficiently treat the nonlocal jump operator. For American options an
easy-to-implement operator splitting method is described for the resulting
linear complementarity problems.
Numerical experiments are presented to illustrate the actual stability and
convergence of the splitting schemes. Here European and American put options
are considered under four asset price models: the classical Black-Scholes
model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV
model with jumps.
|
finance
|
3,691 |
Estimating the Algorithmic Complexity of Stock Markets
|
q-fin.CP
|
Randomness and regularities in Finance are usually treated in probabilistic
terms. In this paper, we develop a completely different approach in using a
non-probabilistic framework based on the algorithmic information theory
initially developed by Kolmogorov (1965). We present some elements of this
theory and show why it is particularly relevant to Finance, and potentially to
other sub-fields of Economics as well. We develop a generic method to estimate
the Kolmogorov complexity of numeric series. This approach is based on an
iterative "regularity erasing procedure" implemented to use lossless
compression algorithms on financial data. Examples are provided with both
simulated and real-world financial time series. The contributions of this
article are twofold. The first one is methodological : we show that some
structural regularities, invisible with classical statistical tests, can be
detected by this algorithmic method. The second one consists in illustrations
on the daily Dow-Jones Index suggesting that beyond several well-known
regularities, hidden structure may in this index remain to be identified.
|
finance
|
3,692 |
Chebyshev Interpolation for Parametric Option Pricing
|
q-fin.CP
|
Recurrent tasks such as pricing, calibration and risk assessment need to be
executed accurately and in real-time. Simultaneously we observe an increase in
model sophistication on the one hand and growing demands on the quality of risk
management on the other. To address the resulting computational challenges, it
is natural to exploit the recurrent nature of these tasks. We concentrate on
Parametric Option Pricing (POP) and show that polynomial interpolation in the
parameter space promises to reduce run-times while maintaining accuracy. The
attractive properties of Chebyshev interpolation and its tensorized extension
enable us to identify criteria for (sub)exponential convergence and explicit
error bounds. We show that these results apply to a variety of European
(basket) options and affine asset models. Numerical experiments confirm our
findings. Exploring the potential of the method further, we empirically
investigate the efficiency of the Chebyshev method for multivariate and
path-dependent options.
|
finance
|
3,693 |
Approximations of Bond and Swaption Prices in a Black-Karasiński Model
|
q-fin.CP
|
We derive semi-analytic approximation formulae for bond and swaption prices
in a Black-Karasi\'{n}ski interest rate model. Approximations are obtained
using a novel technique based on the Karhunen-Lo\`{e}ve expansion. Formulas are
easily computable and prove to be very accurate in numerical tests. This makes
them useful for numerically efficient calibration of the model.
|
finance
|
3,694 |
Numerical analysis on local risk-minimization forexponential Lévy models
|
q-fin.CP
|
We illustrate how to compute local risk minimization (LRM) of call options
for exponential L\'evy models. We have previously obtained a representation of
LRM for call options; here we transform it into a form that allows use of the
fast Fourier transform method suggested by Carr & Madan. In particular, we
consider Merton jump-diffusion models and variance gamma models as concrete
applications.
|
finance
|
3,695 |
Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio
|
q-fin.CP
|
We study the finite horizon Merton portfolio optimization problem in a
general local-stochastic volatility setting. Using model coefficient expansion
techniques, we derive approximations for the both the value function and the
optimal investment strategy. We also analyze the `implied Sharpe ratio' and
derive a series approximation for this quantity. The zeroth-order approximation
of the value function and optimal investment strategy correspond to those
obtained by Merton (1969) when the risky asset follows a geometric Brownian
motion. The first-order correction of the value function can, for general
utility functions, be expressed as a differential operator acting on the
zeroth-order term. For power utility functions, higher order terms can also be
computed as a differential operator acting on the zeroth-order term. We give a
rigorous accuracy bound for the higher order approximations in this case in
pure stochastic volatility models. A number of examples are provided in order
to demonstrate numerically the accuracy of our approximations.
|
finance
|
3,696 |
Nonparametric and arbitrage-free construction of call surfaces using l1-recovery
|
q-fin.CP
|
This paper is devoted to the application of an $l_1$ -minimisation technique
to construct an arbitrage-free call-option surface. We propose a
nononparametric approach to obtaining model-free call option surfaces that are
perfectly consistent with market quotes and free of static arbitrage. The
approach is inspired from the compressed-sensing framework that is used in
signal processing to deal with under-sampled signals. We address the problem of
fitting the call-option surface to sparse option data. To illustrate the
methodology, we proceed to the construction of the whole call-price surface of
the S\&P500 options, taking into account the arbitrage possibilities in the
time direction. The resulting object is a surface free of both butterfly and
calendar-spread arbitrage that matches the original market points. We then move
on to an FX application, namely the HKD/USD call-option surface.
|
finance
|
3,697 |
Double-jump stochastic volatility model for VIX: evidence from VVIX
|
q-fin.CP
|
The paper studies the continuous-time dynamics of VIX with stochastic
volatility and jumps in VIX and volatility. Built on the general parametric
affine model with stochastic volatility and jump in logarithm of VIX, we derive
a linear relation between the stochastic volatility factor and VVIX index. We
detect the existence of co-jump of VIX and VVIX and put forward a double-jump
stochastic volatility model for VIX through its joint property with VVIX. With
VVIX index as a proxy for the stochastic volatility, we use MCMC method to
estimate the dynamics of VIX. Comparing nested models on VIX, we show the jump
in VIX and the volatility factor is statistically significant. The jump
intensity is also statedependent. We analyze the impact of jump factor on the
VIX dynamics.
|
finance
|
3,698 |
A State-Space Estimation of the Lee-Carter Mortality Model and Implications for Annuity Pricing
|
q-fin.CP
|
In this article we investigate a state-space representation of the Lee-Carter
model which is a benchmark stochastic mortality model for forecasting
age-specific death rates. Existing relevant literature focuses mainly on
mortality forecasting or pricing of longevity derivatives, while the full
implications and methods of using the state-space representation of the
Lee-Carter model in pricing retirement income products is yet to be examined.
The main contribution of this article is twofold. First, we provide a rigorous
and detailed derivation of the posterior distributions of the parameters and
the latent process of the Lee-Carter model via Gibbs sampling. Our assumption
for priors is slightly more general than the current literature in this area.
Moreover, we suggest a new form of identification constraint not yet utilised
in the actuarial literature that proves to be a more convenient approach for
estimating the model under the state-space framework. Second, by exploiting the
posterior distribution of the latent process and parameters, we examine the
pricing range of annuities, taking into account the stochastic nature of the
dynamics of the mortality rates. In this way we aim to capture the impact of
longevity risk on the pricing of annuities. The outcome of our study
demonstrates that an annuity price can be more than 4% under-valued when
different assumptions are made on determining the survival curve constructed
from the distribution of the forecasted death rates. Given that a typical
annuity portfolio consists of a large number of policies with maturities which
span decades, we conclude that the impact of longevity risk on the accurate
pricing of annuities is a significant issue to be further researched. In
addition, we find that mis-pricing is increasingly more pronounced for older
ages as well as for annuity policies having a longer maturity.
|
finance
|
3,699 |
New Analytical Solutions of a Modified Black-Scholes Equation with the European Put Option
|
q-fin.CP
|
Using Maple, we compute some analytical solutions of a modified Black-Scholes
equation, recently proposed, in the case of the European put option. We show
that the modified Black-Scholes equation with the European put option is
exactly solvable in terms of associated Laguerre polynomials. We make some
numerical experiments with the analytical solutions and we compare our results
with the results derived from numerical experiments using the standard
Black-Scholes equation.
|
finance
|
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