Results 1  10
of
66
A jump to default extended CEV model: An application of Bessel processes
, 2005
"... We consider the problem of developing a ßexible and analytically tractable framework which uniÞes the valuation of corporate liabilities, credit derivatives, and equity derivatives. Theory and empirical evidence suggest that default indicators such as credit default swap (CDS) spreads and corporate ..."
Abstract

Cited by 41 (3 self)
 Add to MetaCart
(Show Context)
We consider the problem of developing a ßexible and analytically tractable framework which uniÞes the valuation of corporate liabilities, credit derivatives, and equity derivatives. Theory and empirical evidence suggest that default indicators such as credit default swap (CDS) spreads and corporate bond yields are positively related to historical volatility and implied volatilities of equity options. Theory and empirical evidence also suggest that a stocks realized volatility is negatively related to its price (leverage effect) and that implied volatilities are decreasing in the options strike price (skew). We propose a parsimonious reducedform model of default which captures all of these fundamental relationships. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a Constant Elasticity of Variance (CEV) speciÞcation for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our
2001), ‘Pricing and hedging pathdependent options under the CEV process
 Management Science
"... Much of the work on pathdependent options assumes that the underlying asset pricefollows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the socalled constant ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
(Show Context)
Much of the work on pathdependent options assumes that the underlying asset pricefollows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the socalled constant elasticity of variance (CEV) diffusion model where the volatility is a function of the underlying asset price. We derive analytical formulae for the prices of important types of pathdependent options under this assumption. We demonstrate that the prices of options, which depend on extrema, such as barrier and lookback options, can be much more sensitive to the specification of the underlying price process than standard call and put options and show that a financial institution that uses the standard geometric Brownian motion assumption is exposed to significant pricing and hedging errors when dealing in pathdependent options.
Asymptotic behavior of the stock pricedistributiondensityandimpliedvolatilityinstochasticvolatilitymodels
 Appl. Math. Optim
"... Abstract We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the Ste ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
(Show Context)
Abstract We study the asymptotic behavior of distribution densities arising in stock price models with stochastic volatility. The main objects of our interest in the present paper are the density of time averages of the squared volatility process and the density of the stock price process in the SteinStein and the Heston model. We find explicit formulas for leading terms in asymptotic expansions of these densities and give error estimates. As an application of our results, sharp asymptotic formulas for the implied volatility in the SteinStein and the Heston model are obtained. Keywords SteinStein model · Heston model · Mixing distribution density · Stock price · Bessel processes · OrnsteinUhlenbeck processes · CIR processes · Asymptotic formulas · Implied volatility 1
On collisions of Brownian particles
 Annals of Applied Probability
, 2010
"... Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the prese ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We examine the behavior of n Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence, and for the presence, of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains. 1.
Bessel processes, the integral of geometric Brownian motion, and Asian options
 Theor. Probab. Appl
, 2004
"... Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been stu ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
Schröder Abstract. This paper is motivated by questions about averages of stochastic processes which originate in mathematical finance, originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the HartmanWatson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the HartmanWatson approach itself: this approach is in general applicable without modifications only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this paper is the developement of three principal ways to overcome these restrictions, in particular by merging stochastics and complex analysis in what seems a novel way, and the discussion of their consequences for the valuation of Asian options proper.
The Skorokhod embedding problem and its offspring
, 2004
"... This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on onedi ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, based on onedimensional potential theory, is made. Some new facts which appeared in a natural way when different solutions were crossexamined, are reported. Azéma and Yor’s and Root’s solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.
TIME DEPENDENT HESTONMODEL
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Excursions and local limit theorems for Bessellike random walks, Elect
 J. Probab
, 2011
"... ar ..."
(Show Context)
THE CALCULATION OF EXPECTATIONS FOR CLASSES OF DIFFUSION PROCESSES BY LIE SYMMETRY METHODS
, 902
"... This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals t of the form Ex(e −λXt− 0 g(Xs) ds) can be reduced to evaluating a single integral of known func ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
(Show Context)
This paper uses Lie symmetry methods to calculate certain expectations for a large class of Itô diffusions. We show that if the problem has sufficient symmetry, then the problem of computing functionals t of the form Ex(e −λXt− 0 g(Xs) ds) can be reduced to evaluating a single integral of known functions. Given a drift f we determine the functions g for which the corresponding functional can be calculated by symmetry. Conversely, given g, we can determine precisely those drifts f for which the transition density and the functional may be computed by symmetry. Many examples are presented to illustrate the method.