Unnamed: 0
int64 0
41k
| title
stringlengths 4
274
| category
stringlengths 5
18
| summary
stringlengths 22
3.66k
| theme
stringclasses 8
values |
|---|---|---|---|---|
3,900 |
Option Pricing Accuracy for Estimated Heston Models
|
q-fin.MF
|
We consider assets for which price $X_t$ and squared volatility $Y_t$ are
jointly driven by Heston joint stochastic differential equations (SDEs). When
the parameters of these SDEs are estimated from $N$ sub-sampled data $(X_{nT},
Y_{nT})$, estimation errors do impact the classical option pricing PDEs. We
estimate these option pricing errors by combining numerical evaluation of
estimation errors for Heston SDEs parameters with the computation of option
price partial derivatives with respect to these SDEs parameters. This is
achieved by solving six parabolic PDEs with adequate boundary conditions. To
implement this approach, we also develop an estimator $\hat \lambda$ for the
market price of volatility risk, and we study the sensitivity of option pricing
to estimation errors affecting $\hat \lambda$. We illustrate this approach by
fitting Heston SDEs to 252 daily joint observations of the S\&P 500 index and
of its approximate volatility VIX, and by numerical applications to European
options written on the S\&P 500 index.
|
finance
|
3,901 |
Modelling the skew and smile of SPX and DAX index options using the Shifted Log-Normal and SABR stochastic models
|
q-fin.MF
|
We discuss modelling of SPX and DAX index option prices using the Shifted
Log-Normal (SLN) model, (also known as Displaced Diffusion), and the SABR
model. We found out that for SPX options, an example of strongly skewed option
prices, SLN can produce a quite accurate fit. Moreover, for both types of index
options, the SLN model is giving a good fit of near-at-the-forward strikes.
Such a near-at-the-money fit allows us to calculate precisely the skew
parameter without involving directly the 3rd moment of the related probability
distribution. Eventually, we can follow with a procedure in which the skew is
calculated using the SLN model and further smile effects are added as a next
iteration/perturbation. Furthermore, we point out that the SLN trajectories are
exact solutions of the SABR model for rho = +/-1.
|
finance
|
3,902 |
Reconstruction of density functions by sk-splines
|
q-fin.MF
|
Reconstruction of density functions and their characteristic functions by
radial basis functions with scattered data points is a popular topic in the
theory of pricing of basket options. Such functions are usually entire or admit
an analytic extension into an appropriate tube and "bell-shaped" with rapidly
decaying tails. Unfortunately, the domain of such functions is not compact
which creates various technical difficulties. We solve interpolation problem on
an infinite rectangular grid for a wide range of kernel functions and calculate
explicitly their Fourier transform to obtain representations for the respective
density functions.
|
finance
|
3,903 |
Interest rate models and Whittaker functions
|
q-fin.MF
|
I present the technique which can analyse some interest rate models:
Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian
Motion. All these models have the unified structure of Whittaker function. The
main focus of this text is closed-form solutions of the zero-coupon bond value
in these models. In text I emphasize the specific details of mathematical
methods of their determination such as Laplace transform and hypergeometric
functions.
|
finance
|
3,904 |
Intensity Process for a Pure Jump Lévy Structural Model with Incomplete Information
|
q-fin.MF
|
In this paper we discuss a credit risk model with a pure jump L\'evy process
for the asset value and an unobservable random barrier. The default time is the
first time when the asset value falls below the barrier. Using the
indistinguishability of the intensity process and the likelihood process, we
prove the existence of the intensity process of the default time and find its
explicit representation in terms of the distance between the asset value and
its running minimal value. We apply the result to find the instantaneous credit
spread process and illustrate it with a numerical example.
|
finance
|
3,905 |
Valuation and Hedging of Contracts with Funding Costs and Collateralization
|
q-fin.MF
|
The research presented in this work is motivated by recent papers by Brigo et
al. (2011), Burgard and Kjaer (2009), Cr\'epey (2012), Fujii and Takahashi
(2010), Piterbarg (2010) and Pallavicini et al. (2012). Our goal is to provide
a sound theoretical underpinning for some results presented in these papers by
developing a unified framework for the non-linear approach to hedging and
pricing of OTC financial contracts. We introduce a systematic approach to
valuation and hedging in nonlinear markets, that is, in markets where cash
flows of the financial contracts may depend on the hedging strategies. Our
systematic approach allows to identify primary sources of and quantify various
adjustment to valuation and hedging, primarily the funding and liquidity
adjustment and credit risk adjustment. We propose a way to define no-arbitrage
in such nonlinear markets, and we provide conditions that imply absence of
arbitrage in some specific market trading models. Accordingly, we formulate a
concept of no-arbitrage price, and we provide relevant (non-linear) BSDE that
produces the no-arbitrage price in case when the contract's cash flows can be
replicated.
|
finance
|
3,906 |
Explicit investment rules with time-to-build and uncertainty
|
q-fin.MF
|
We establish explicit socially optimal rules for an irreversible investment
deci- sion with time-to-build and uncertainty. Assuming a price sensitive
demand function with a random intercept, we provide comparative statics and
economic interpreta- tions for three models of demand (arithmetic Brownian,
geometric Brownian, and the Cox-Ingersoll-Ross). Committed capacity, that is,
the installed capacity plus the in- vestment in the pipeline, must never drop
below the best predictor of future demand, minus two biases. The discounting
bias takes into account the fact that investment is paid upfront for future
use; the precautionary bias multiplies a type of risk aversion index by the
local volatility. Relying on the analytical forms, we discuss in detail the
economic effects.
|
finance
|
3,907 |
Path Diffusion, Part I
|
q-fin.MF
|
This paper investigates the position (state) distribution of the single step
binomial (multi-nomial) process on a discrete state / time grid under the
assumption that the velocity process rather than the state process is
Markovian. In this model the particle follows a simple multi-step process in
velocity space which also preserves the proper state equation of motion. Many
numerical numerical examples of this process are provided. For a smaller grid
the probability construction converges into a correlated set of probabilities
of hyperbolic functions for each velocity at each state point. It is shown that
the two dimensional process can be transformed into a Telegraph equation and
via transformation into a Klein-Gordon equation if the transition rates are
constant. In the last Section there is an example of multi-dimensional
hyperbolic partial differential equation whose numerical average satisfies
Newton's equation. There is also a momentum measure provided both for the
two-dimensional case as for the multi-dimensional rate matrix.
|
finance
|
3,908 |
Option Pricing in an Imperfect World
|
q-fin.MF
|
In a model with no given probability measure, we consider asset pricing in
the presence of frictions and other imperfections and characterize the property
of coherent pricing, a notion related to (but much weaker than) the no
arbitrage property. We show that prices are coherent if and only if the set of
pricing measures is non empty, i.e. if pricing by expectation is possible. We
then obtain a decomposition of coherent prices highlighting the role of
bubbles. eventually we show that under very weak conditions the coherent
pricing of options allows for a very clear representation from which it is
possible, as in the original work of Breeden and Litzenberger, to extract the
implied probability. Eventually we test this conclusion empirically via a new
non parametric approach.
|
finance
|
3,909 |
Robust pricing and hedging under trading restrictions and the emergence of local martingale models
|
q-fin.MF
|
We consider the pricing of derivatives in a setting with trading
restrictions, but without any probabilistic assumptions on the underlying
model, in discrete and continuous time. In particular, we assume that European
put or call options are traded at certain maturities, and the forward price
implied by these option prices may be strictly decreasing in time. In discrete
time, when call options are traded, the short-selling restrictions ensure no
arbitrage, and we show that classical duality holds between the smallest
super-replication price and the supremum over expectations of the payoff over
all supermartingale measures. More surprisingly in the case where the only
vanilla options are put options, we show that there is a duality gap. Embedding
the discrete time model into a continuous time setup, we make a connection with
(strict) local-martingale models, and derive framework and results often seen
in the literature on financial bubbles. This connection suggests a certain
natural interpretation of many existing results in the literature on financial
bubbles.
|
finance
|
3,910 |
Optimal Hybrid Dividend Strategy Under The Markovian Regime-Switching Economy
|
q-fin.MF
|
In this paper, we consider the optimal dividend problem for a company. We
describe the surplus process of the company by a diffusion model with regime
switching. The aim of the company is to choose a dividend policy to maximize
the expected total discounted payments until ruin. In this article, we consider
a hybrid dividend strategy, that is, the company is allowed to conduct
continuous dividend strategy as well as impulsive dividend strategy. In
addition, we consider the change of economy, which is characterized by a
markovian regime-switching, and under the setting of two regimes, we solve the
problem and obtain the analytical solution for the value function.
|
finance
|
3,911 |
Utility indifference pricing and hedging for structured contracts in energy markets
|
q-fin.MF
|
In this paper we study the pricing and hedging of structured products in
energy markets, such as swing and virtual gas storage, using the exponential
utility indifference pricing approach in a general incomplete multivariate
market model driven by finitely many stochastic factors. The buyer of such
contracts is allowed to trade in the forward market in order to hedge the risk
of his position. We fully characterize the buyer's utility indifference price
of a given product in terms of continuous viscosity solutions of suitable
nonlinear PDEs. This gives a way to identify reasonable candidates for the
optimal exercise strategy for the structured product as well as for the
corresponding hedging strategy. Moreover, in a model with two correlated
assets, one traded and one nontraded, we obtain a representation of the price
as the value function of an auxiliary simpler optimization problem under a risk
neutral probability, that can be viewed as a perturbation of the minimal
entropy martingale measure. Finally, numerical results are provided.
|
finance
|
3,912 |
Long Term Optimal Investment in Matrix Valued Factor Models
|
q-fin.MF
|
Long term optimal investment problems are studied in a factor model with
matrix valued state variables. Explicit parameter restrictions are obtained
under which, for an isoelastic investor, the finite horizon value function and
optimal strategy converge to their long-run counterparts as the investment
horizon approaches infinity. This convergence also yields portfolio turnpikes
for general utilities. By using results on large time behaviour of semi-linear
partial differential equations, our analysis extends affine models, where the
Wishart process drives investment opportunities, to a non-affine setting.
Furthermore, in the affine setting, an example is constructed where the value
function is not exponentially affine, in contrast to models with vector-valued
state variables.
|
finance
|
3,913 |
On Correlated Defaults and Incomplete Information
|
q-fin.MF
|
In this paper, we study a continuous time structural asset value model for
two correlated firms using a two-dimensional Brownian motion. We consider the
situation of incomplete information, where the information set available to the
market participants includes the default time of each firm and the periodic
asset value reports. In this situation, the default time of each firm becomes a
totally inaccessible stopping time to the market participants. The original
structural model is first transformed to a reduced-form model. Then the
conditional distribution of the default time together with the asset value of
each name are derived. We prove the existence of the intensity processes of
default times and also give the explicit form of the intensity processes.
Numerical studies on the intensities of the two correlated names are conducted
for some special cases. We also indicate the possible future research extension
into three names case by considering a special correlation structure.
|
finance
|
3,914 |
Distance to the line in the Heston model
|
q-fin.MF
|
The main object of study in the paper is the distance from a point to a line
in the Riemannian manifold associated with the Heston model. We reduce the
problem of computing such a distance to certain minimization problems for
functions of one variable over finite intervals. One of the main ideas in this
paper is to use a new system of coordinates in the Heston manifold and the
level sets associated with this system. In the case of a vertical line, the
formulas for the distance to the line are rather simple. For slanted lines, the
formulas are more complicated, and a more subtle analysis of the level sets
intersecting the given line is needed. We also find simple formulas for the
Heston distance from a point to a level set. As a natural application, we use
the formulas obtained in the present paper to compute the small maturity limit
of the implied volatility in the correlated Heston model.
|
finance
|
3,915 |
Option pricing in constant elasticity of variance model with liquidity costs
|
q-fin.MF
|
Paper is based on "The cost of illiquidity and its effects on hedging", L. C.
G. Rogers and Surbjeet Singh, 2010. We generalize its thesis to constant
elasticity model, which own previously used Black-Schoels model as a special
case. The Goal of this article is to find optimal hedging strategy of European
call/put option in illiquid environment. We understand illiquidity as a non
linear transaction cost function depending only on rate of change of our
portfolio. In case this function is quadratic, optimal policy is given by
system of 3 PDE. In addition we show, that for small $\epsilon$ costs of
selling portfolio in time $T$ be important ($O(\epsilon)$) and shouldn't be
neglected in Value function ($o(\epsilon^k)$- our result).
|
finance
|
3,916 |
Multi-asset consumption-investment problems with infinite transaction costs
|
q-fin.MF
|
The subject of this paper is an optimal consumption/optimal portfolio problem
with transaction costs and with multiple risky assets.
In our model the transaction costs take a special form in that transaction
costs on purchases of one of the risky assets (the endowed asset) are infinite,
and transaction costs involving the other risky assets are zero. Effectively,
the endowed asset can only be sold. In general, multi-asset optional
consumption/optimal portfolio problems are very challenging, but the extra
structure we introduce allows us to make significant progress towards an
analytical solution.
For an agent with CRRA utility we completely characterise the different types
of optimal behaviours. These include always selling the entire holdings of the
endowed asset immediately, selling the endowed asset whenever the ratio of the
value of the holdings of the endowed asset to other wealth gets above a
critical ratio, and selling the endowed asset only when other wealth is zero.
This characterisation is in terms of solutions of a boundary crossing problem
for a first order ODE. The technical contribution is to show that the problem
of solving the HJB equation, which is a second order, non-linear PDE subject to
smooth fit at an unknown free boundary, can be reduced to this much simpler
problem involving an explicit first order ODE. This technical contribution is
at the heart of our analytical and numerical results, and allows us to prove
monotonicity of the critical exercise threshold and the certainty equivalent
value in the model parameters.
|
finance
|
3,917 |
Generalized Dynkin game of switching type representation for defaultable claims in presence of contingent CSA
|
q-fin.MF
|
We study the solution's existence for a generalized Dynkin game of switching
type which is shown to be the natural representation for general defaultable
OTC contract with contingent CSA. This is a theoretical counterparty risk
mitigation mechanism that allows the counterparty of a general OTC contract to
switch from zero to full/perfect collateralization and switch back whenever she
wants until contract maturity paying some switching costs and taking into
account the running costs that emerge over time. In this paper we allow for the
strategic interaction between the counterparties of the underlying contract,
which makes the problem solution much more tough. We are motivated in this
research by the importance to show the economic sense - in terms of optimal
contract design - of a contingent counterparty risk mitigation mechanism like
our one. In particular, we show that the existence of the solution and the game
Nash equilibrium is connected with the solution of a system of non-linear
reflected BSDE which remains an open problem. We then provide the basic ideas
to numerically search the game equilibrium via an iterative optimal stopping
approach and we show the existence of the solution for our problem under strong
condition, in the so called symmetric case.
|
finance
|
3,918 |
Rationality parameter for exercising American put
|
q-fin.MF
|
The main result of this paper is a probabilistic proof of the penalty method
for approximating the price of an American put in the Black-Scholes market. The
method gives a parametrized family of partial differential equations, and by
varying the parameter the corresponding solutions converge to the price of an
American put. For each PDE the parameter may be interpreted as a rationality
parameter of the holder of the option. The method may be extended to other
valuation situations given as an optimal stopping problem with no explicit
solution. The method may also be used for valuations where actors do not behave
completely rationally but instead have randomness affecting their choices. The
rationality parameter is a measure for this randomness.
|
finance
|
3,919 |
Ross Recovery with Recurrent and Transient Processes
|
q-fin.MF
|
Recently, Ross showed that it is possible to recover an objective measure
from a risk-neutral measure. His model assumes that there is a finite-state
Markov process X that drives the economy in discrete time. Many authors
extended his model to a continuous-time setting with a Markov diffusion process
X with state space R. Unfortunately, the continuous-time model fails to recover
an objective measure from a risk-neutral measure. We determine under which
information recovery is possible in the continuous-time model. It was proven
that if X is recurrent under the objective measure, then recovery is possible.
In this article, when X is transient under the objective measure, we
investigate what information is sufficient to recover.
|
finance
|
3,920 |
Arbitrage theory without a numéraire
|
q-fin.MF
|
This note develops an arbitrage theory for a discrete-time market model
without the assumption of the existence of a num\'eraire asset. Fundamental
theorems of asset pricing are stated and proven in this context. The
distinction between the notions of investment-consumption arbitrage and
pure-investment arbitrage provide a discrete-time analogue of the distinction
between the notions of absolute arbitrage and relative arbitrage in the
continuous-time theory. Applications to the modelling of bubbles is discussed.
|
finance
|
3,921 |
Banach geometry of arbitrage free markets
|
q-fin.MF
|
The article presents a description of geometry of Banach structures forming
mathematical base of markets arbitrage absence type phenomena. In this
connection the role of reflexive subspaces (replacing classically considered
finite-dimensional subspaces) and plasterable cones is uncovered.
|
finance
|
3,922 |
Visualisation of financial time series by linear principal component analysis and nonlinear principal component analysis
|
q-fin.MF
|
In this dissertation, the main goal is visualisation of financial time
series. We expect that visualisation of financial time series will be a useful
auxiliary for technical analysis. Firstly, we review the technical analysis
methods and test our trading rules, which are built by the essential concepts
of technical analysis. Next, we compare the quality of linear principal
component analysis and nonlinear principal component analysis in financial
market visualisation. We compare different methods of data preprocessing for
visualisation purposes. Using visualisation, we demonstrate the difference
between normal and crisis time period. Thus, the visualisation of financial
market can be a tool to support technical analysis.
|
finance
|
3,923 |
Positive Eigenfunctions of Markovian Pricing Operators: Hansen-Scheinkman Factorization, Ross Recovery and Long-Term Pricing
|
q-fin.MF
|
This paper develops a spectral theory of Markovian asset pricing models where
the underlying economic uncertainty follows a continuous-time Markov process X
with a general state space (Borel right process (BRP)) and the stochastic
discount factor (SDF) is a positive semimartingale multiplicative functional of
X. A key result is the uniqueness theorem for a positive eigenfunction of the
pricing operator such that X is recurrent under a new probability measure
associated with this eigenfunction (recurrent eigenfunction). As economic
applications, we prove uniqueness of the Hansen and Scheinkman (2009)
factorization of the Markovian SDF corresponding to the recurrent
eigenfunction, extend the Recovery Theorem of Ross (2015) from discrete time,
finite state irreducible Markov chains to recurrent BRPs, and obtain the long
maturity asymptotics of the pricing operator. When an asset pricing model is
specified by given risk-neutral probabilities together with a short rate
function of the Markovian state, we give sufficient conditions for existence of
a recurrent eigenfunction and provide explicit examples in a number of
important financial models, including affine and quadratic diffusion models and
an affine model with jumps. These examples show that the recurrence assumption,
in addition to fixing uniqueness, rules out unstable economic dynamics, such as
the short rate asymptotically going to infinity or to a zero lower bound trap
without possibility of escaping.
|
finance
|
3,924 |
Incorporating Views on Market Dynamics in Options Hedging
|
q-fin.MF
|
We examine the possibility of incorporating information or views of market
movements during the holding period of a portfolio, in the hedging of European
options with respect to the underlying. Given a fixed holding period interval,
we explore whether it is possible to adjust the number of shares needed to
effectively hedge our position to account for views on market dynamics from
present until the end of our interval, to account for the time-dependence of
the options' sensitivity to the underlying. We derive an analytical expression
for the number of shares needed by adjusting the standard Black-Scholes-Merton
$\Delta$ quantity, in the case of an arbitrary process for implied volatility,
and we present numerical results.
|
finance
|
3,925 |
The Intrinsic Bounds on the Risk Premium of Markovian Pricing Kernels
|
q-fin.MF
|
The risk premium is one of main concepts in mathematical finance. It is a
measure of the trade-offs investors make between return and risk and is defined
by the excess return relative to the risk-free interest rate that is earned
from an asset per one unit of risk. The purpose of this article is to determine
upper and lower bounds on the risk premium of an asset based on the market
prices of options. One of the key assumptions to achieve this goal is that the
market is Markovian. Under this assumption, we can transform the problem of
finding the bounds into a second-order differential equation. We then obtain
upper and lower bounds on the risk premium by analyzing the differential
equation.
|
finance
|
3,926 |
Dynamic Defaultable Term Structure Modelling beyond the Intensity Paradigm
|
q-fin.MF
|
The two main approaches in credit risk are the structural approach pioneered
in Merton (1974) and the reduced-form framework proposed in Jarrow & Turnbull
(1995) and in Artzner & Delbaen (1995). The goal of this article is to provide
a unified view on both approaches. This is achieved by studying reduced-form
approaches under weak assumptions. In particular we do not assume the global
existence of a default intensity and allow default at fixed or predictable
times with positive probability, such as coupon payment dates.
In this generalized framework we study dynamic term structures prone to
default risk following the forward-rate approach proposed in
Heath-Jarrow-Morton (1992). It turns out, that previously considered models
lead to arbitrage possibilities when default may happen at a predictable time
with positive probability. A suitable generalization of the forward-rate
approach contains an additional stochastic integral with atoms at predictable
times and necessary and sufficient conditions for a suitable no-arbitrage
condition (NAFL) are given. In the view of efficient implementations we develop
a new class of affine models which do not satisfy the standard assumption of
stochastic continuity.
The chosen approach is intimately related to the theory of enlargement of
filtrations, to which we provide a small example by means of filtering theory
where the Azema supermartingale contains upward and downward jumps, both at
predictable and totally inaccessible stopping times.
|
finance
|
3,927 |
Optimal Starting-Stopping and Switching of a CIR Process with Fixed Costs
|
q-fin.MF
|
This paper analyzes the problem of starting and stopping a Cox-Ingersoll-Ross
(CIR) process with fixed costs. In addition, we also study a related optimal
switching problem that involves an infinite sequence of starts and stops. We
establish the conditions under which the starting-stopping and switching
problems admit the same optimal starting and/or stopping strategies. We
rigorously prove that the optimal starting and stopping strategies are of
threshold type, and give the analytical expressions for the value functions in
terms of confluent hypergeometric functions. Numerical examples are provided to
illustrate the dependence of timing strategies on model parameters and
transaction costs.
|
finance
|
3,928 |
Asymptotic behaviour of the fractional Heston model
|
q-fin.MF
|
We consider the fractional Heston model originally proposed by Comte, Coutin
and Renault. Inspired by recent ground-breaking work on rough volatility, which
showed that models with volatility driven by fractional Brownian motion with
short memory allows for better calibration of the volatility surface and more
robust estimation of time series of historical volatility, we provide a
characterisation of the short- and long-maturity asymptotics of the implied
volatility smile. Our analysis reveals that the short-memory property precisely
provides a jump-type behaviour of the smile for short maturities, thereby
fixing the well-known standard inability of classical stochastic volatility
models to fit the short-end of the volatility smile.
|
finance
|
3,929 |
Existence and Uniqueness of a Steady State for an OTC Market with Several Assets
|
q-fin.MF
|
We introduce and study a class of over-the-counter market models specified by
systems of Ordinary Differential Equations (ODE's), in the spirit of Duffie-
G^arleanu-Pedersen [6]. The key innovation is allowing for multiple assets. We
show the existence and uniqueness of a steady state for these ODE's.
|
finance
|
3,930 |
Reserve-Dependent Surrender
|
q-fin.MF
|
We study the modelling and valuation of surrender and other behavioural
options in life insurance and pension. We place ourselves in between the two
extremes of completely arbitrary intervention and optimal intervention by the
policyholder. We present a method that is based on differential equations and
that can be used to approximate contract values when policyholders exhibit
optimal behaviour. This presentation includes a specification of sufficient
conditions for both consistency of the model and convergence of the contract
values. When not going to the limit in the approximation we obtain a technique
for balancing off arbitrary and optimal behaviour in a simple, intuitive way.
This leads to our suggestions for intervention models where one single
parameter reflects the extent of rationality among policyholders. In a series
of numerical examples we illustrate the impact of the rationality parameter on
the contract values.
|
finance
|
3,931 |
A BSDE approach to fair bilateral pricing under endogenous collateralization
|
q-fin.MF
|
Our previous results are extended to the case of the margin account, which
may depend on the contract's value for the hedger and/or the counterparty. The
present work generalizes also the papers by Bergman (1995), Mercurio (2013) and
Piterbarg (2010). Using the comparison theorems for BSDEs, we derive
inequalities for the unilateral prices and we give the range for its fair
bilateral prices. We also establish results yielding the link to the market
model with a single interest rate. In the case where the collateral amount is
negotiated between the counterparties, so that it depends on their respective
unilateral values, the backward stochastic viability property studied by
Buckdahn et al. (2000) is used to derive the bounds on fair bilateral prices.
|
finance
|
3,932 |
Indifference prices and implied volatilities
|
q-fin.MF
|
We consider a general local-stochastic volatility model and an investor with
exponential utility. For a European-style contingent claim, whose payoff may
depend on either a traded or non-traded asset, we derive an explicit
approximation for both the buyer's and seller's indifference price. For
European calls on a traded asset, we translate indifference prices into an
explicit approximation of the buyer's and seller's implied volatility surface.
For European claims on a non-traded asset, we establish rigorous error bounds
for the indifference price approximation. Finally, we implement our
indifference price and implied volatility approximations in two examples.
|
finance
|
3,933 |
Fundamental theorem of asset pricing: a strengthened version and $p$-summable markets
|
q-fin.MF
|
In the article a strenthened version of the 'Fundamental Theorem of asset
Pricing' for one-period market model is proven. The principal role in this
result play total and nonanihilating cones.
|
finance
|
3,934 |
Optimal switching for pairs trading rule: a viscosity solutions approach
|
q-fin.MF
|
This paper studies the problem of determining the optimal cut-off for pairs
trading rules. We consider two correlated assets whose spread is modelled by a
mean-reverting process with stochastic volatility, and the optimal pair trading
rule is formulated as an optimal switching problem between three regimes: flat
position (no holding stocks), long one short the other and short one long the
other. A fixed commission cost is charged with each transaction. We use a
viscosity solutions approach to prove the existence and the explicit
characterization of cut-off points via the resolution of quasi-algebraic
equations. We illustrate our results by numerical simulations.
|
finance
|
3,935 |
On financial applications of the two-parameter Poisson-Dirichlet distribution
|
q-fin.MF
|
Capital distribution curve is defined as log-log plot of normalized stock
capitalizations ranked in descending order. The curve displays remarkable
stability over periods of time.
Theory of exchangeable distributions on set partitions, developed for
purposes of mathematical genetics and recently applied in non-parametric
Bayesian statistics, provides probabilistic-combinatorial approach for analysis
and modeling of the capital distribution curve. Framework of the two-parameter
Poisson-Dirichlet distribution contains rich set of methods and tools,
including infinite-dimensional diffusion process.
The purpose of this note is to introduce framework of exchangeable
distributions on partitions in the financial context. In particular, it is
shown that averaged samples from the Poisson-Dirichlet distribution provide
approximation to the capital distribution curves in equity markets. This
suggests that the two-parameter model can be employed for modelling evolution
of market weights and prices fluctuating in stochastic equilibrium.
|
finance
|
3,936 |
Non-concave utility maximisation on the positive real axis in discrete time
|
q-fin.MF
|
We treat a discrete-time asset allocation problem in an arbitrage-free,
generically incomplete financial market, where the investor has a possibly
non-concave utility function and wealth is restricted to remain non-negative.
Under easily verifiable conditions, we establish the existence of optimal
portfolios.
|
finance
|
3,937 |
Effect of Volatility Clustering on Indifference Pricing of Options by Convex Risk Measures
|
q-fin.MF
|
In this article, we look at the effect of volatility clustering on the risk
indifference price of options described by Sircar and Sturm in their paper
(Sircar, R., & Sturm, S. (2012). From smile asymptotics to market risk
measures. Mathematical Finance. Advance online publication.
doi:10.1111/mafi.12015). The indifference price in their article is obtained by
using dynamic convex risk measures given by backward stochastic differential
equations. Volatility clustering is modelled by a fast mean-reverting
volatility in a stochastic volatility model for stock price. Asymptotics of the
indifference price of options and their corresponding implied volatility are
obtained in this article, as the mean-reversion time approaches zero.
Correction terms to the asymptotic option price and implied volatility are also
obtained.
|
finance
|
3,938 |
Short-time at-the-money skew and rough fractional volatility
|
q-fin.MF
|
The Black-Scholes implied volatility skew at the money of SPX options is
known to obey a power law with respect to the time-to-maturity. We construct a
model of the underlying asset price process which is dynamically consistent to
the power law. The volatility process of the model is driven by a fractional
Brownian motion with Hurst parameter less than half. The fractional Brownian
motion is correlated with a Brownian motion which drives the asset price
process. We derive an asymptotic expansion of the implied volatility as the
time-to-maturity tends to zero. For this purpose we introduce a new approach to
validate such an expansion, which enables us to treat more general models than
in the literature. The local-stochastic volatility model is treated as well
under an essentially minimal regularity condition in order to show such a
standard model cannot be dynamically consistent to the power law.
|
finance
|
3,939 |
Convex duality with transaction costs
|
q-fin.MF
|
Convex duality for two two different super--replication problems in a
continuous time financial market with proportional transaction cost is proved.
In this market, static hedging in a finite number of options, in addition to
usual dynamic hedging with the underlying stock, are allowed. The first one the
problems considered is the model--independent hedging that requires the
super--replication to hold for every continuous path. In the second one the
market model is given through a probability measure P and the inequalities are
understood P almost surely. The main result, using the convex duality, proves
that the two super--replication problems have the same value provided that P
satisfies the conditional full support property. Hence, the transaction costs
prevents one from using the structure of a specific model to reduce the
super--replication cost.
|
finance
|
3,940 |
Archimedean-based Marshall-Olkin Distributions and Related Copula Functions
|
q-fin.MF
|
A new class of bivariate distributions is introduced that extends the
Generalized Marshall-Olkin distributions of Li and Pellerey (2011). Their
dependence structure is studied through the analysis of the copula functions
that they induce. These copulas, that include as special cases the Generalized
Marshall-Olkin copulas and the Scale Mixture of Marshall-Olkin copulas (see Li,
2009),are obtained through suitable distortions of bivariate Archimedean
copulas: this induces asymmetry, and the corresponding Kendall's tau as well as
the tail dependence parameters are studied.
|
finance
|
3,941 |
The pricing of lookback options and binomial approximation
|
q-fin.MF
|
Refining a discrete model of Cheuk and Vorst we obtain a closed formula for
the price of a European lookback option at any time between emission and
maturity. We derive an asymptotic expansion of the price as the number of
periods tends to infinity, thereby solving a problem posed by Lin and Palmer.
We prove, in particular, that the price in the discrete model tends to the
price in the continuous Black-Scholes model. Our results are based on an
asymptotic expansion of the binomial cumulative distribution function that
improves several recent results in the literature.
|
finance
|
3,942 |
Consistent Recalibration of Yield Curve Models
|
q-fin.MF
|
The analytical tractability of affine (short rate) models, such as the
Vasicek and the Cox-Ingersoll-Ross models, has made them a popular choice for
modelling the dynamics of interest rates. However, in order to account properly
for the dynamics of real data, these models need to exhibit time-dependent or
even stochastic parameters. This in turn breaks their tractability, and
modelling and simulating becomes an arduous task. We introduce a new class of
Heath-Jarrow-Morton (HJM) models that both fit the dynamics of real market data
and remain tractable. We call these models consistent recalibration (CRC)
models. These CRC models appear as limits of concatenations of forward rate
increments, each belonging to a Hull-White extended affine factor model with
possibly different parameters. That is, we construct HJM models from "tangent"
affine models. We develop a theory for a continuous path version of such models
and discuss their numerical implementations within the Vasicek and
Cox-Ingersoll-Ross frameworks.
|
finance
|
3,943 |
Extreme-Strike Asymptotics for General Gaussian Stochastic Volatility Models
|
q-fin.MF
|
We consider a stochastic volatility asset price model in which the volatility
is the absolute value of a continuous Gaussian process with arbitrary
prescribed mean and covariance. By exhibiting a Karhunen-Lo\`{e}ve expansion
for the integrated variance, and using sharp estimates of the density of a
general second-chaos variable, we derive asymptotics for the asset price
density for large or small values of the variable, and study the wing behavior
of the implied volatility in these models. Our main result provides explicit
expressions for the first five terms in the expansion of the implied
volatility. The expressions for the leading three terms are simple, and based
on three basic spectral-type statistics of the Gaussian process: the top
eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and
the $L^{2}$ norm of the projection of the mean function on the top eigenspace.
The fourth term requires knowledge of all eigen-elements. We present detailed
numerics based on realistic liquidity assumptions in which classical and
long-memory volatility models are calibrated based on our expansion.
|
finance
|
3,944 |
Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments
|
q-fin.MF
|
We develop a multi-curve term structure setup in which the modelling
ingredients are expressed by rational functionals of Markov processes. We
calibrate to LIBOR swaptions data and show that a rational two-factor lognormal
multi-curve model is sufficient to match market data with accuracy. We
elucidate the relationship between the models developed and calibrated under a
risk-neutral measure Q and their consistent equivalence class under the
real-world probability measure P. The consistent P-pricing models are applied
to compute the risk exposures which may be required to comply with regulatory
obligations. In order to compute counterparty-risk valuation adjustments, such
as CVA, we show how positive default intensity processes with rational form can
be derived. We flesh out our study by applying the results to a basis swap
contract.
|
finance
|
3,945 |
Some new results on Dufffie-type OTC markets
|
q-fin.MF
|
The extended Wild sums considered in this article generalize the classi- cal
Wild sums of statistical physics. We first show how to obtain explicit
solutions for the evolution equation of a large system where the interactions
are given by a single, but general, interacting kernel which involves m
components, for a fixed m >= 2. We then show how to retain the explicit
formulas for the case of OTC market models where the dynamics is more directly
described by two (or more) kernels.
|
finance
|
3,946 |
Profitable forecast of prices of stock options on real market data via the solution of an ill-posed problem for the Black-Scholes equation
|
q-fin.MF
|
A new mathematical model for the Black-Scholes equation is proposed to
forecast option prices. This model includes new interval for the price of the
underlying stock as well as new initial and boundary conditions. Conventional
notions of maturity time and strike prices are not used. The Black-Scholes
equation is solved as a parabolic equation with the reversed time, which is an
ill-posed problem. Thus, a regularization method is used to solve it. This idea
is verified on real market data for twenty liquid options. A trading strategy
is proposed. This strategy indicates that our method is profitable on at least
those twenty options. We conjecture that our method might lead to significant
profits of those financial institutions which trade large amounts of options.
We caution, however, that detailed further studies are necessary to verify this
conjecture.
|
finance
|
3,947 |
About the decomposition of pricing formulas under stochastic volatility models
|
q-fin.MF
|
We obtain a decomposition of the call option price for a very general
stochastic volatility diffusion model extending the decomposition obtained by
E. Al\`os in [2] for the Heston model. We realize that a new term arises when
the stock price does not follow an exponential model. The techniques used are
non anticipative. In particular, we see also that equivalent results can be
obtained using Functional It\^o Calculus. Using the same generalizing ideas we
also extend to non exponential models the alternative call option price
decompostion formula obtained in [1] and [3] written in terms of the Malliavin
derivative of the volatility process. Finally, we give a general expression for
the derivative of the implied volatility under both, the anticipative and the
non anticipative case.
|
finance
|
3,948 |
Dynkin Game of Convertible Bonds and Their Optimal Strategy
|
q-fin.MF
|
This paper studies the valuation and optimal strategy of convertible bonds as
a Dynkin game by using the reflected backward stochastic differential equation
method and the variational inequality method. We first reduce such a Dynkin
game to an optimal stopping time problem with state constraint, and then in a
Markovian setting, we investigate the optimal strategy by analyzing the
properties of the corresponding free boundary, including its position,
asymptotics, monotonicity and regularity. We identify situations when call
precedes conversion, and vice versa. Moreover, we show that the irregular
payoff results in the possibly non-monotonic conversion boundary. Surprisingly,
the price of the convertible bond is not necessarily monotonic in time: it may
even increase when time approaches maturity.
|
finance
|
3,949 |
Indifference Pricing and Hedging in a Multiple-Priors Model with Trading Constraints
|
q-fin.MF
|
This paper considers utility indifference valuation of derivatives under
model uncertainty and trading constraints, where the utility is formulated as
an additive stochastic differential utility of both intertemporal consumption
and terminal wealth, and the uncertain prospects are ranked according to a
multiple-priors model of Chen and Epstein (2002). The price is determined by
two optimal stochastic control problems (mixed with optimal stopping time in
the case of American option) of forward-backward stochastic differential
equations. By means of backward stochastic differential equation and partial
differential equation methods, we show that both bid and ask prices are closely
related to the Black-Scholes risk-neutral price with modified dividend rates.
The two prices will actually coincide with each other if there is no trading
constraint or the model uncertainty disappears. Finally, two applications to
European option and American option are discussed.
|
finance
|
3,950 |
Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations
|
q-fin.MF
|
We consider the problem of optimal portfolio selection under forward
investment performance criteria in an incomplete market. The dynamics of the
prices of the traded assets depend on a pair of stochastic factors, namely, a
slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic
volatility). We analyze the associated forward performance SPDE and provide
explicit formulae for the leading order and first order correction terms for
the forward investment process and the optimal feedback portfolios. They both
depend on the investor's initial preferences and the dynamically changing
investment opportunities. The leading order terms resemble their time-monotone
counterparts, but with the appropriate stochastic time changes resulting from
averaging phenomena. The first-order terms compile the reaction of the investor
to both the changes in the market input and his recent performance. Our
analysis is based on an expansion of the underlying ill-posed HJB equation, and
it is justified by means of an appropriate remainder estimate.
|
finance
|
3,951 |
Polynomial term structure models
|
q-fin.MF
|
In this article, we explore a class of tractable interest rate models that
have the property that the price of a zero-coupon bond can be expressed as a
polynomial of a state diffusion process. Our results include a classification
of all such time-homogeneous single-factor models in the spirit of Filipovic's
maximal degree theorem for exponential polynomial models, as well as an
explicit characterisation of the set of feasible parameters in the case when
the factor process is bounded. Extensions to time-inhomogeneous and
multi-factor polynomial models are also considered.
|
finance
|
3,952 |
A Posteriori Error Estimator for a Front-Fixing Finite Difference Scheme for American Options
|
q-fin.MF
|
For the numerical solution of the American option valuation problem, we
provide a script written in MATLAB implementing an explicit finite difference
scheme. Our main contribute is the definition of a posteriori error estimator
for the American options pricing which is based on Richardson's extrapolation
theory. This error estimator allows us to find a suitable grid where the
computed solution, both the option price field variable and the free boundary
position, verify a prefixed error tolerance.
|
finance
|
3,953 |
Network Structure and Counterparty Credit Risk
|
q-fin.MF
|
In this paper we offer a novel type of network model which can capture the
precise structure of a financial market based, for example, on empirical
findings. With the attached stochastic framework it is further possible to
study how an arbitrary network structure and its expected counterparty credit
risk are analytically related to each other. This allows us, for the first
time, to model the precise structure of an arbitrary financial market and to
derive the corresponding expected exposure in a closed-form expression. It
further enables us to draw implications for the study of systemic risk. We
apply the powerful theory of characteristic functions and Hilbert transforms.
The latter concept is used to express the characteristic function (c.f.) of the
random variable (r.v.) $\max(Y, 0)$ in terms of the c.f. of the r.v. $Y$. The
present paper applies this concept for the first time in mathematical finance.
We then characterise Eulerian digraphs as distinguished exposure structures and
show that considering the precise network structures is crucial for the study
of systemic risk. The introduced network model is then applied to study the
features of an over-the-counter and a centrally cleared market. We also give a
more general answer to the question of whether it is more advantageous for the
overall counterparty credit risk to clear via a central counterparty or
classically bilateral between the two involved counterparties. We then show
that the exact market structure is a crucial factor in answering the raised
question.
|
finance
|
3,954 |
On statistical indistinguishability of complete and incomplete discrete time market models
|
q-fin.MF
|
We investigate the possibility of statistical evaluation of the market
completeness for discrete time stock market models. It is known that the market
completeness is not a robust property: small random deviations of the
coefficients convert a complete market model into a incomplete one. The paper
shows that market incompleteness is also non-robust. We show that, for any
incomplete market from a wide class of discrete time models, there exists a
complete market model with arbitrarily close stock prices. This means that
incomplete markets are indistinguishable from the complete markets in the terms
of the market statistics.
|
finance
|
3,955 |
Non-Arbitrage Under Additional Information for Thin Semimartingale Models
|
q-fin.MF
|
This paper completes the two studies undertaken in
\cite{aksamit/choulli/deng/jeanblanc2} and
\cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact
of a random time on the No-Unbounded-Risk-with-Bounded-Profit concept (called
NUPBR hereafter) when the stock price processes are quasi-left-continuous (do
not jump on predictable stopping times). Herein, we focus on the NUPBR for
semimartingales models that live on thin predictable sets only and the
progressive enlargement with a random time. For this flow of information, we
explain how far the NUPBR property is affected when one stops the model by an
arbitrary random time or when one incorporates fully an honest time into the
model. This also generalizes \cite{choulli/deng} to the case when the jump
times are not ordered in anyway. Furthermore, for the current context, we show
how to construct explicitly local martingale deflator under the bigger
filtration from those of the smaller filtration.
|
finance
|
3,956 |
Approximate hedging problem with transaction costs in stochastic volatility markets
|
q-fin.MF
|
This paper studies the problem of option replication in general stochastic
volatility markets with transaction costs, using a new specification for the
volatility adjustment in Leland's algorithm \cite{Leland}. We prove several
limit theorems for the normalized replication error of Leland's strategy, as
well as that of the strategy suggested by L\'epinette. The asymptotic results
obtained not only generalize the existing results, but also enable us to fix
the under-hedging property pointed out by Kabanov and Safarian. We also discuss
possible methods to improve the convergence rate and to reduce the option price
inclusive of transaction costs.
|
finance
|
3,957 |
Approximate hedging with proportional transaction costs in stochastic volatility models with jumps
|
q-fin.MF
|
We study the problem of option replication under constant proportional
transaction costs in models where stochastic volatility and jumps are combined
to capture the market's important features. Assuming some mild condition on the
jump size distribution we show that transaction costs can be approximately
compensated by applying the Leland adjusting volatility principle and the
asymptotic property of the hedging error due to discrete readjustments is
characterized. In particular, the jump risk can be approximately eliminated and
the results established in continuous diffusion models are recovered. The study
also confirms that for the case of constant trading cost rate, the approximate
results established by Kabanov and Safarian (1997)and by Pergamenschikov (2003)
are still valid in jump-diffusion models with deterministic volatility using
the classical Leland parameter in Leland (1986).
|
finance
|
3,958 |
Hedging of defaultable claims in a structural model using a locally risk-minimizing approach
|
q-fin.MF
|
In the context of a locally risk-minimizing approach, the problem of hedging
defaultable claims and their Follmer-Schweizer decompositions are discussed in
a structural model. This is done when the underlying process is a finite
variation Levy process and the claims pay a predetermined payout at maturity,
contingent on no prior default. More precisely, in this particular framework,
the locally risk-minimizing approach is carried out when the underlying process
has jumps, the derivative is linked to a default event, and the probability
measure is not necessarily risk-neutral.
|
finance
|
3,959 |
Small-time asymptotics for Gaussian self-similar stochastic volatility models
|
q-fin.MF
|
We consider the class of self-similar Gaussian stochastic volatility models,
and compute the small-time (near-maturity) asymptotics for the corresponding
asset price density, the call and put pricing functions, and the implied
volatilities. Unlike the well-known model-free behavior for extreme-strike
asymptotics, small-time behaviors of the above depend heavily on the model, and
require a control of the asset price density which is uniform with respect to
the asset price variable, in order to translate into results for call prices
and implied volatilities. Away from the money, we express the asymptotics
explicitly using the volatility process' self-similarity parameter $H$, its
first Karhunen-Loeve eigenvalue at time 1, and the latter's multiplicity.
Several model-free estimators for $H$ result. At the money, a separate study is
required: the asymptotics for small time depend instead on the integrated
variance's moments of orders 1/2 and 3/2, and the estimator for $H$ sees an
affine adjustment, while remaining model-free.
|
finance
|
3,960 |
An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting
|
q-fin.MF
|
We study an optimal multiple stopping problem for call-type payoff driven by
a spectrally negative Levy process. The stopping times are separated by
constant refraction times, and the discount rate can be positive or negative.
The computation involves a distribution of the Levy process at a constant
horizon and hence the solutions in general cannot be attained analytically.
Motivated by the maturity randomization (Canadization) technique by Carr
(1998), we approximate the refraction times by independent, identically
distributed Erlang random variables. In addition, fitting random jumps to
phase-type distributions, our method involves repeated integrations with
respect to the resolvent measure written in terms of the scale function of the
underlying Levy process. We derive a recursive algorithm to compute the value
function in closed form, and sequentially determine the optimal exercise
thresholds. A series of numerical examples are provided to compare our analytic
formula to results from Monte Carlo simulation.
|
finance
|
3,961 |
Good deal bounds with convex constraints
|
q-fin.MF
|
We investigate the structure of good deal bounds, which are subintervals of a
no-arbitrage pricing bound, for financial market models with convex constraints
as an extension of Arai and Fukasawa (2014). The upper and lower bounds of a
good deal bound are naturally described by a convex risk measure. We call such
a risk measure a good deal valuation; and study its properties. We also discuss
superhedging cost and Fundamental Theorem of Asset Pricing for convex
constrained markets.
|
finance
|
3,962 |
An Empirical Approach to Financial Crisis Indicators Based on Random Matrices
|
q-fin.MF
|
The aim of this work is to build financial crisis indicators based on
spectral properties of the dynamics of market data. After choosing an optimal
size for a rolling window, the historical market data in this window is seen
every trading day as a random matrix from which a covariance and a correlation
matrix are obtained. The financial crisis indicators that we have built deal
with the spectral properties of these covariance and correlation matrices and
they are of two kinds. The first one is based on the Hellinger distance,
computed between the distribution of the eigenvalues of the empirical
covariance matrix and the distribution of the eigenvalues of a reference
covariance matrix representing either a calm or agitated market. The idea
behind this first type of indicators is that when the empirical distribution of
the spectrum of the covariance matrix is deviating from the reference in the
sense of Hellinger, then a crisis may be forthcoming. The second type of
indicators is based on the study of the spectral radius and the trace of the
covariance and correlation matrices as a mean to directly study the volatility
and correlations inside the market. The idea behind the second type of
indicators is the fact that large eigenvalues are a sign of dynamic
instability. The predictive power of the financial crisis indicators in this
framework is then demonstrated, in particular by using them as decision-making
tools in a protective-put strategy.
|
finance
|
3,963 |
No-Arbitrage Prices of Cash Flows and Forward Contracts as Choquet Representations
|
q-fin.MF
|
In a market of deterministic cash flows, given as an additive, symmetric
relation of exchangeability on the finite signed Borel measures on the
non-negative real time axis, it is shown that the only arbitrage-free price
functional that fulfills some additional mild requirements is the integral of
the unit zero-coupon bond prices with respect to the payment measures. For
probability measures, this is a Choquet representation, where the Dirac
measures, as unit zero-coupon bonds, are the extreme points. Dropping one of
the requirements, the Lebesgue decomposition is used to construct
counterexamples, where the Choquet price formula does not hold despite of an
arbitrage-free market model. The concept is then extended to deterministic
streams of assets and currencies in general, yielding a valuation principle for
forward markets. Under mild assumptions, it is shown that a foreign cash flow's
worth in local currency is identical to the value of the cash flow in local
currency for which the Radon-Nikodym derivative with respect to the foreign
cash flow is the forward FX rate.
|
finance
|
3,964 |
Optimal Static Quadratic Hedging
|
q-fin.MF
|
We propose a flexible framework for hedging a contingent claim by holding
static positions in vanilla European calls, puts, bonds, and forwards. A
model-free expression is derived for the optimal static hedging strategy that
minimizes the expected squared hedging error subject to a cost constraint. The
optimal hedge involves computing a number of expectations that reflect the
dependence among the contingent claim and the hedging assets. We provide a
general method for approximating these expectations analytically in a general
Markov diffusion market. To illustrate the versatility of our approach, we
present several numerical examples, including hedging path-dependent options
and options written on a correlated asset.
|
finance
|
3,965 |
Model-free Superhedging Duality
|
q-fin.MF
|
In a model free discrete time financial market, we prove the superhedging
duality theorem, where trading is allowed with dynamic and semi-static
strategies. We also show that the initial cost of the cheapest portfolio that
dominates a contingent claim on every possible path $\omega \in \Omega$, might
be strictly greater than the upper bound of the no-arbitrage prices. We
therefore characterize the subset of trajectories on which this duality gap
disappears and prove that it is an analytic set.
|
finance
|
3,966 |
Market shape formation, statistical equilibrium and neutral evolution theory
|
q-fin.MF
|
Mathematical methods of population genetics and framework of exchangeability
provide a Markov chain model for analysis and interpretation of stochastic
behaviour of equity markets, explaining, in particular, market shape formation,
statistical equilibrium and temporal stability of market weights.
|
finance
|
3,967 |
Itô's formula for finite variation Lévy processes: The case of non-smooth functions
|
q-fin.MF
|
Extending It\^o's formula to non-smooth functions is important both in theory
and applications. One of the fairly general extensions of the formula, known as
Meyer-It\^o, applies to one dimensional semimartingales and convex functions.
There are also satisfactory generalizations of It\^o's formula for diffusion
processes where the Meyer-It\^o assumptions are weakened even further. We study
a version of It\^o's formula for multi-dimensional finite variation L\'evy
processes assuming that the underlying function is continuous and admits weak
derivatives. We also discuss some applications of this extension, particularly
in finance.
|
finance
|
3,968 |
Radner equilibrium in incomplete Levy models
|
q-fin.MF
|
We construct continuous-time equilibrium models based on a finite number of
exponential utility investors. The investors' income rates as well as the
stock's dividend rate are governed by discontinuous Levy processes. Our main
result provides the equilibrium (i.e., bond and stock price dynamics) in
closed-form. As an application, we show that the equilibrium Sharpe ratio can
be increased and the equilibrium interest rate can be decreased
(simultaneously) when the investors' income streams cannot be traded.
|
finance
|
3,969 |
Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure
|
q-fin.MF
|
In the problem of optimal investment with utility function defined on
$(0,\infty)$, we formulate sufficient conditions for the dual optimizer to be a
uniformly integrable martingale. Our key requirement consists of the existence
of a martingale measure whose density process satisfies the probabilistic
Muckenhoupt $(A_p)$ condition for the power $p=1/(1-a)$, where $a\in (0,1)$ is
a lower bound on the relative risk-aversion of the utility function. We
construct a counterexample showing that this $(A_p)$ condition is sharp.
|
finance
|
3,970 |
A risk analysis for a system stabilized by a central agent
|
q-fin.MF
|
We formulate and analyze a multi-agent model for the evolution of individual
and systemic risk in which the local agents interact with each other through a
central agent who, in turn, is influenced by the mean field of the local
agents. The central agent is stabilized by a bistable potential, the only
stabilizing force in the system. The local agents derive their stability only
from the central agent. In the mean field limit of a large number of local
agents we show that the systemic risk decreases when the strength of the
interaction of the local agents with the central agent increases. This means
that the probability of transition from one of the two stable quasi-equilibria
to the other one decreases. We also show that the systemic risk increases when
the strength of the interaction of the central agent with the mean field of the
local agents increases. Following the financial interpretation of such models
and their behavior given in our previous paper (Garnier, Papanicolaou and Yang,
SIAM J. Fin. Math. 4, 2013, 151-184), we may interpret the results of this
paper in the following way. From the point of view of systemic risk, and while
keeping the perceived risk of the local agents approximately constant, it is
better to strengthen the interaction of the local agents with the central agent
than the other way around.
|
finance
|
3,971 |
Robust replication of barrier-style claims on price and volatility
|
q-fin.MF
|
We show how to price and replicate a variety of barrier-style claims written
on the $\log$ price $X$ and quadratic variation $\langle X \rangle$ of a risky
asset. Our framework assumes no arbitrage, frictionless markets and zero
interest rates. We model the risky asset as a strictly positive continuous
semimartingale with an independent volatility process. The volatility process
may exhibit jumps and may be non-Markovian. As hedging instruments, we use only
the underlying risky asset, zero-coupon bonds, and European calls and puts with
the same maturity as the barrier-style claim. We consider knock-in, knock-out
and rebate claims in single and double barrier varieties.
|
finance
|
3,972 |
Optimal liquidation of an asset under drift uncertainty
|
q-fin.MF
|
We study a problem of finding an optimal stopping strategy to liquidate an
asset with unknown drift. Taking a Bayesian approach, we model the initial
beliefs of an individual about the drift parameter by allowing an arbitrary
probability distribution to characterise the uncertainty about the drift
parameter. Filtering theory is used to describe the evolution of the posterior
beliefs about the drift once the price process is being observed. An optimal
stopping time is determined as the first passage time of the posterior mean
below a monotone boundary, which can be characterised as the unique solution to
a non-linear integral equation. We also study monotonicity properties with
respect to the prior distribution and the asset volatility.
|
finance
|
3,973 |
Correction to Black-Scholes formula due to fractional stochastic volatility
|
q-fin.MF
|
Empirical studies show that the volatility may exhibit correlations that
decay as a fractional power of the time offset. The paper presents a rigorous
analysis for the case when the stationary stochastic volatility model is
constructed in terms of a fractional Ornstein Uhlenbeck process to have such
correlations. It is shown how the associated implied volatility has a term
structure that is a function of maturity to a fractional power.
|
finance
|
3,974 |
Can You hear the Shape of a Market? Geometric Arbitrage and Spectral Theory
|
q-fin.MF
|
Geometric Arbitrage Theory reformulates a generic asset model possibly
allowing for arbitrage by packaging all assets and their forwards dynamics into
a stochastic principal fibre bundle, with a connection whose parallel transport
encodes discounting and portfolio rebalancing, and whose curvature measures, in
this geometric language, the 'instantaneous arbitrage capability' generated by
the market itself. The cashflow bundle is the vector bundle associated to this
stochastic principal fibre bundle for the natural choice of the vector space
fibre. The cashflow bundle carries a stochastic covariant differentiation
induced by the connection on the principal fibre bundle. The link between
arbitrage theory and spectral theory of the connection Laplacian on the vector
bundle is given by the zero eigenspace resulting in a parametrization of all
risk neutral measures equivalent to the statistical one. This indicates that a
market satisfies the (NFLVR) condition if and only if $0$ is in the discrete
spectrum of the connection Laplacian on the cash flow bundle or of the Dirac
Laplacian of the twisted cash flow bundle with the exterior algebra bundle. We
apply this result by extending Jarrow-Protter-Shimbo theory of asset bubbles
for complete arbitrage free markets to markets not satisfying the (NFLVR).
Moreover, by means of the Atiyah-Singer index theorem, we prove that the Euler
characteristic of the asset nominal space is a topological obstruction to the
the (NFLVR) condition, and, by means of the Bochner-Weitzenb\"ock formula, the
non vanishing of the homology group of the cash flow bundle is revealed to be a
topological obstruction to (NFLVR), too. Asset bubbles are defined, classified
and decomposed for markets allowing arbitrage.
|
finance
|
3,975 |
Optimal Insurance with Rank-Dependent Utility and Increasing Indemnities
|
q-fin.MF
|
Bernard et al. (2015) study an optimal insurance design problem where an
individual's preference is of the rank-dependent utility (RDU) type, and show
that in general an optimal contract covers both large and small losses.
However, their contracts suffer from a problem of moral hazard for paying more
compensation for a smaller loss. This paper addresses this setback by
exogenously imposing the constraint that both the indemnity function and the
insured's retention function be increasing with respect to the loss. We
characterize the optimal solutions via calculus of variations, and then apply
the result to obtain explicitly expressed contracts for problems with Yaari's
dual criterion and general RDU. Finally, we use a numerical example to compare
the results between ours and that of Bernard et al. (2015).
|
finance
|
3,976 |
On the no-arbitrage market and continuity in the Hurst parameter
|
q-fin.MF
|
We consider a market with fractional Brownian motion with stochastic
integrals generated by the Riemann sums. We found that this market is arbitrage
free if admissible strategies that are using observations with an arbitrarily
small delay. Moreover, we found that this approach eliminates the discontinuity
of the stochastic integrals with respect to the Hurst parameter H at H=1/2.
|
finance
|
3,977 |
An example of short-term relative arbitrage
|
q-fin.MF
|
Long-term relative arbitrage exists in markets where the excess growth rate
of the market portfolio is bounded away from zero. Here it is shown that under
a time-homogeneity hypothesis this condition will also imply the existence of
relative arbitrage over arbitrarily short intervals.
|
finance
|
3,978 |
On the Solution of the Multi-asset Black-Scholes model: Correlations, Eigenvalues and Geometry
|
q-fin.MF
|
In this paper, we study the multi-asset Black-Scholes model in terms of the
importance that the correlation parameter space (equivalent to an $N$
dimensional hypercube) has in the solution of the pricing problem. We show that
inside of this hypercube there is a surface, called the Kummer surface
$\Sigma_K$, where the determinant of the correlation matrix $\rho$ is zero, so
the usual formula for the propagator of the $N$ asset Black-Scholes equation is
no longer valid. Worse than that, in some regions outside this surface, the
determinant of $\rho$ becomes negative, so the usual propagator becomes complex
and divergent. Thus the option pricing model is not well defined for these
regions outside $\Sigma_K$. On the Kummer surface instead, the rank of the
$\rho$ matrix is a variable number. By using the Wei-Norman theorem, we compute
the propagator over the variable rank surface $\Sigma_K$ for the general $N$
asset case. We also study in detail the three assets case and its implied
geometry along the Kummer surface.
|
finance
|
3,979 |
Regularity properties in a state-constrained expected utility maximization problem
|
q-fin.MF
|
We consider a stochastic optimal control problem in a market model with
temporary and permanent price impact, which is related to an expected utility
maximization problem under finite fuel constraint. We establish the initial
condition fulfilled by the corresponding value function and show its first
regularity property. Moreover, we can prove the existence and uniqueness of
optimal strategies under rather mild model assumptions. On the one hand, this
result is of independent interest. On the other hand, it will then allow us to
derive further regularity properties of the corresponding value function, in
particular its continuity and partial differentiability. As a consequence of
the continuity of the value function, we will prove the dynamic programming
principle without appealing to the classical measurable selection arguments.
|
finance
|
3,980 |
Hedging with Temporary Price Impact
|
q-fin.MF
|
We consider the problem of hedging a European contingent claim in a Bachelier
model with transient price impact as proposed by Almgren and Chriss. Following
the approach of Rogers and Singh and Naujokat and Westray, the hedging problem
can be regarded as a cost optimal tracking problem of the frictionless hedging
strategy. We solve this problem explicitly for general predictable target
hedging strategies. It turns out that, rather than towards the current target
position, the optimal policy trades towards a weighted average of expected
future target positions. This generalizes an observation of Garleanu and
Pedersen from their homogenous Markovian optimal investment problem to a
general hedging problem. Our findings complement a number of previous studies
in the literature on optimal strategies in illiquid markets where the
frictionless strategy is confined to diffusions. The consideration of general
predictable reference strategies is made possible by the use of a convex
analysis approach instead of the more common dynamic programming methods.
|
finance
|
3,981 |
Viscosity properties with singularities in a state-constrained expected utility maximization problem
|
q-fin.MF
|
We consider the value function originating from an expected utility
maximization problem with finite fuel constraint and show its close relation to
a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with
singularity. On one hand, we give a so-called verification argument based on
the dynamic programming principle, which allows us to derive conditions under
which a classical solution of the HJB equation coincides with our value
function (provided that it is smooth enough). On the other hand, we establish a
comparison principle, which allows us to characterize our value function as the
unique viscosity solution of the HJB equation.
|
finance
|
3,982 |
Trajectory based models. Evaluation of minmax pricing bounds
|
q-fin.MF
|
The paper studies sub and super-replication price bounds for contingent
claims defined on general trajectory based market models. No prior
probabilistic or topological assumptions are placed on the trajectory space,
trading is assumed to take place at a finite number of occasions but not
bounded in number nor necessarily equally spaced in time. For a given option,
there exists an interval bounding the set of possible fair prices; such
interval exists under more general conditions than the usual no-arbitrage
requirement. The paper develops a backward recursive method to evaluate the
option bounds; the global minmax optimization, defining the price interval, is
reduced to a local minmax optimization via dynamic programming. Trajectory sets
are introduced for which existing non-probabilistic markets models are nested
as a particular case. Several examples are presented, the effect of the
presence of arbitrage on the price bounds is illustrated.
|
finance
|
3,983 |
Foundations for Wash Sales
|
q-fin.MF
|
Consider an ephemeral sale-and-repurchase of a security resulting in the same
position before the sale and after the repurchase. A sale-and-repurchase is a
wash sale if these transactions result in a loss within $\pm 30$ calendar days.
Since a portfolio is essentially the same after a wash sale, any tax advantage
from such a loss is not allowed. That is, after a wash sale a portfolio is
unchanged so any loss captured by the wash sale is deemed to be solely for tax
advantage and not investment purposes.
This paper starts by exploring variations of the birthday problem to model
wash sales. The birthday problem is: Determine the number of independent and
identically distributed random variables required so there is a probability of
at least 1/2 that two or more of these random variables share the same outcome.
This paper gives necessary conditions for wash sales based on variations on the
birthday problem. This allows us to answer questions such as: What is the
likelihood of a wash sale in an unmanaged portfolio where purchases and sales
are independent, uniform, and random? This paper ends by exploring the
Littlewood-Offord problem as it relates capital gains and losses with wash
sales.
|
finance
|
3,984 |
Sensitivity Analysis of Long-Term Cash Flows
|
q-fin.MF
|
In this article, a sensitivity analysis of long-term cash flows with respect
to perturbations in the underlying process is presented. For this purpose, we
employ the martingale extraction through which a pricing operator is
transformed into what is easier to address. The method of Fournie et al. will
be combined with the martingale extraction. We prove that the sensitivity of
long-term cash flows can be represented in a simple form.
|
finance
|
3,985 |
Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE
|
q-fin.MF
|
In an incomplete market, with incompleteness stemming from stochastic factors
imperfectly correlated with the underlying stocks, we derive representations of
homothetic (power, exponential and logarithmic) forward performance processes
in factor-form using ergodic BSDE. We also develop a connection between the
forward processes and infinite horizon BSDE, and, moreover, with risk-sensitive
optimization. In addition, we develop a connection, for large time horizons,
with a family of classical homothetic value function processes with random
endowments.
|
finance
|
3,986 |
Integration with respect to model-free price paths with jumps
|
q-fin.MF
|
For every adapted, c\`agl\`ad process (strategy) $G$ and typical c\`adl\`ag
price paths whose jumps satisfy some mild growth condition we define integral
$G\cdot S$ as a limit of simple integrals.
|
finance
|
3,987 |
On the Existence of Martingale Measures in Jump Diffusion Market Models
|
q-fin.MF
|
In the context of jump-diffusion market models we construct examples that
satisfy the weaker no-arbitrage condition of NA1 (NUPBR), but not NFLVR. We
show that in these examples the only candidate for the density process of an
equivalent local martingale measure is a supermartingale that is not a
martingale, not even a local martingale. This candidate is given by the
supermartingale deflator resulting from the inverse of the discounted growth
optimal portfolio. In particular, we con- sider an example with constraints on
the portfolio that go beyond the standard ones for admissibility.
|
finance
|
3,988 |
A Framework for Analyzing Stochastic Jumps in Finance based on Belief and Knowledge
|
q-fin.MF
|
We introduce a formal language IE that is a variant of the language PAL
developed in [van Benthem 2011] by adding a belief operator and a common belief
operator,specializing to stochastic analysis. A constant symbol in the language
denotes a stochastic process so that we can represent several financial events
as formulae in the language, which is expected to be clues of analyzing the
moments that some stochastic jumps such as financial crises occur based on
knowledge and belief of individuals or those shared within groups of
individuals. In order to represent beliefs, we use sigma-complete Boolean
algebras as generalized sigma-algebras. We use the representation for
constructing a model in which the interpretations of the formulae written in
the language IE reside. The model also uses some new categories for integrating
several components appeared in the theory into one.
|
finance
|
3,989 |
Arbitrage and Hedging in model-independent markets with frictions
|
q-fin.MF
|
We provide a Fundamental Theorem of Asset Pricing and a Superhedging Theorem
for a model independent discrete time financial market with proportional
transaction costs. We consider a probability-free version of the Robust No
Arbitrage condition introduced in Schachermayer ['04] and show that this is
equivalent to the existence of Consistent Price Systems. Moreover, we prove
that the superhedging price for a claim g coincides with the frictionless
superhedging price of g for a suitable process in the bid-ask spread.
|
finance
|
3,990 |
Purely pathwise probability-free Ito integral
|
q-fin.MF
|
This paper gives several simple constructions of the pathwise Ito integral
$\int_0^t\phi d\omega$ for an integrand $\phi$ and a price path $\omega$ as
integrator, with $\phi$ and $\omega$ satisfying various topological and
analytical conditions. The definitions are purely pathwise in that neither
$\phi$ nor $\omega$ are assumed to be paths of stochastic processes, and the
Ito integral exists almost surely in a non-probabilistic financial sense. For
example, one of the results shows the existence of $\int_0^t\phi d\omega$ for a
cadlag integrand $\phi$ and a cadlag integrator $\omega$ with jumps bounded in
a predictable manner.
|
finance
|
3,991 |
Variations on an example of Karatzas and Ruf
|
q-fin.MF
|
Markets composed of stocks with capitalization processes represented by
positive continuous semimartingales are studied under the condition that the
market excess growth rate is bounded away from zero. The following examples of
these markets are given: i) a market with a singular covariance matrix and
instantaneous relative arbitrage; ii) a market with a singular covariance
matrix and no arbitrage; iii) a market with a nonsingular covariance matrix and
no arbitrage; iv) a market with a nonsingular covariance matrix and relative
arbitrage over an arbitrary time horizon.
|
finance
|
3,992 |
A generalized intensity based framework for single-name credit risk
|
q-fin.MF
|
The intensity of a default time is obtained by assuming that the default
indicator process has an absolutely continuous compensator. Here we drop the
assumption of absolute continuity with respect to the Lebesgue measure and only
assume that the compensator is absolutely continuous with respect to a general
$\sigma$-finite measure. This allows for example to incorporate the
Merton-model in the generalized intensity based framework. An extension of the
Black-Cox model is also considered. We propose a class of generalized Merton
models and study absence of arbitrage by a suitable modification of the forward
rate approach of Heath-Jarrow-Morton (1992). Finally, we study affine term
structure models which fit in this class. They exhibit stochastic
discontinuities in contrast to the affine models previously studied in the
literature.
|
finance
|
3,993 |
Calibration and simulation of arbitrage effects in a non-equilibrium quantum Black-Scholes model by using semiclassical methods
|
q-fin.MF
|
An interacting Black-Scholes model for option pricing, where the usual
constant interest rate r is replaced by a stochastic time dependent rate r(t)
of the form r(t)=r+f(t) dW/dt, accounting for market imperfections and prices
non-alignment, was developed in [1]. The white noise amplitude f(t), called
arbitrage bubble, generates a time dependent potential U(t) which changes the
usual equilibrium dynamics of the traditional Black-Scholes model. The purpose
of this article is to tackle the inverse problem, that is, is it possible to
extract the time dependent potential U(t) and its associated bubble shape f(t)
from the real empirical financial data? In order to give an answer to this
question, the interacting Black-Scholes equation must be interpreted as a
quantum Schrodinger equation with hamiltonian operator H=H0+U(t), where H0 is
the equilibrium Black-Scholes hamiltonian and U(t) is the interaction term. If
the U(t) term is small enough, the interaction potential can be thought as a
perturbation, so one can compute the solution of the interacting Black-Scholes
equation in an approximate form by perturbation theory. In [2] by applying the
semi-classical considerations, an approximate solution of the non equilibrium
Black-Scholes equation for an arbitrary bubble shape f(t) was developed. Using
this semi-classical solution and the knowledge about the mispricing of the
financial data, one can determinate an equation, which solutions permit obtain
the functional form of the potential term U(t) and its associated bubble f(t).
In all the studied cases, the non equilibrium model performs a better
estimation of the real data than the usual equilibrium model. It is expected
that this new and simple methodology for calibrating and simulating option
pricing solutions in the presence of market imperfections, could help to
improve option pricing estimations.
|
finance
|
3,994 |
Approximation of forward curve models in commodity markets with arbitrage-free finite dimensional models
|
q-fin.MF
|
In this paper we show how to approximate a Heath-Jarrow-Morton dynamics for
the forward prices in commodity markets with arbitrage-free models which have a
finite dimensional state space. Moreover, we recover a closed form
representation of the forward price dynamics in the approximation models and
derive the rate of convergence uniformly over an interval of time to maturity
to the true dynamics under certain additional smoothness conditions. In the
Markovian case we can strengthen the convergence to be uniform over time as
well. Our results are based on the construction of a convenient Riesz basis on
the state space of the term structure dynamics.
|
finance
|
3,995 |
Symmetry reduction and exact solutions of the non-linear Black--Scholes equation
|
q-fin.MF
|
In this paper, we investigate the non-linear Black--Scholes equation:
$$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$$ and show that
the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an
appropriate point transformation of variables. For the resulting equation, we
study the group-theoretic properties, namely, we find the maximal algebra of
invariance of its in Lie sense, carry out the symmetry reduction and seek for a
number of exact group-invariant solutions of the equation. Using the results
obtained, we get a number of exact solutions of the Black--Scholes equation
under study and apply the ones to resolving several boundary value problems
with appropriate from the economic point of view terminal and boundary
conditions.
|
finance
|
3,996 |
Consistent Re-Calibration of the Discrete-Time Multifactor Vasiček Model
|
q-fin.MF
|
The discrete-time multifactor Vasi\v{c}ek model is a tractable Gaussian spot
rate model. Typically, two- or three-factor versions allow one to capture the
dependence structure between yields with different times to maturity in an
appropriate way. In practice, re-calibration of the model to the prevailing
market conditions leads to model parameters that change over time. Therefore,
the model parameters should be understood as being time-dependent or even
stochastic. Following the consistent re-calibration (CRC) approach, we
construct models as concatenations of yield curve increments of Hull-White
extended multifactor Vasi\v{c}ek models with different parameters. The CRC
approach provides attractive tractable models that preserve the no-arbitrage
premise. As a numerical example, we fit Swiss interest rates using CRC
multifactor Vasi\v{c}ek models.
|
finance
|
3,997 |
Uniform bounds for Black--Scholes implied volatility
|
q-fin.MF
|
In this note, Black--Scholes implied volatility is expressed in terms of
various optimisation problems. From these representations, upper and lower
bounds are derived which hold uniformly across moneyness and call price.
Various symmetries of the Black--Scholes formula are exploited to derive new
bounds from old. These bounds are used to reprove asymptotic formulae for
implied volatility at extreme strikes and/or maturities.
|
finance
|
3,998 |
Speculative Futures Trading under Mean Reversion
|
q-fin.MF
|
This paper studies the problem of trading futures with transaction costs when
the underlying spot price is mean-reverting. Specifically, we model the spot
dynamics by the Ornstein-Uhlenbeck (OU), Cox-Ingersoll-Ross (CIR), or
exponential Ornstein-Uhlenbeck (XOU) model. The futures term structure is
derived and its connection to futures price dynamics is examined. For each
futures contract, we describe the evolution of the roll yield, and compute
explicitly the expected roll yield. For the futures trading problem, we
incorporate the investor's timing option to enter or exit the market, as well
as a chooser option to long or short a futures upon entry. This leads us to
formulate and solve the corresponding optimal double stopping problems to
determine the optimal trading strategies. Numerical results are presented to
illustrate the optimal entry and exit boundaries under different models. We
find that the option to choose between a long or short position induces the
investor to delay market entry, as compared to the case where the investor
pre-commits to go either long or short.
|
finance
|
3,999 |
CoCos under short-term uncertainty
|
q-fin.MF
|
In this paper we analyze an extension of the Jeanblanc and Valchev (2005)
model by considering a short-term uncertainty model with two noises. It is a
combination of the ideas of Duffie and Lando (2001) and Jeanblanc and Valchev
(2005): share quotations of the firm are available at the financial market, and
these can be seen as noisy information about the fundamental value, or the
firm's asset, from which a low level produces the credit event. We assume there
are also reports of the firm, release times, where this short-term uncertainty
disappears. This credit event model is used to describe conversion and default
in a CoCo bond.
|
finance
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.