SEARCH · Searcharxiv

Results for “q-fin.CP”

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 235 records · Page 13Linked to original sources

arXiv2011

Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting

This paper generalizes the framework for arbitrage-free valuation of bilateral counterparty risk to the case where collateral is included, with possible re-hypotecation. We analyze how the payout of claims is modified when collateral margining is included in agreement with current ISDA documentation. We then specialize our analysis to interest-rate swaps as underlying portfolio, and allow for mutual dependences between the default times of the investor and the counterparty and the underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models, including also the effect of interest rate and credit spread volatilities. The impact of re-hypotecation, of collateral margining frequency and of dependencies on the bilateral counterparty risk adjustment is illustrated with a numerical example.

q-fin.RM↗

arXiv2011

Multivariate GARCH estimation via a Bregman-proximal trust-region method

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the "curse of dimensionality". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the dimensionality of the problem. Moreover, these parameters are subjected to convoluted nonlinear constraints necessary to ensure, for instance, the existence of stationary solutions and the positive semidefinite character of the conditional covariance matrices used in the model design. So far, this problem has been addressed in the literature only in low dimensional cases with strong parsimony constraints. In this paper we propose a general formulation of the estimation problem in any dimension and develop a Bregman-proximal trust-region method for its solution. The Bregman-proximal approach allows us to handle the constraints in a very efficient and natural way by staying in the primal space and the Trust-Region mechanism stabilizes and speeds up the scheme. Preliminary computational experiments are presented and confirm the very good performances of the proposed approach.

q-fin.CP↗

arXiv2011

Volatility made observable at last

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

q-fin.CP↗

arXiv2011

Optimal consumption and investment for markets with random coefficients

We consider an optimal investment and consumption problem for a Black-Scholes financial market with stochastic coefficients driven by a diffusion process. We assume that an agent makes consumption and investment decisions based on CRRA utility functions. The dynamical programming approach leads to an investigation of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear partial differential equation (PDE) of the second oder. By using the Feynman - Kac representation we prove uniqueness and smoothness of the solution. Moreover, we study the optimal convergence rate of the iterative numerical schemes for both the value function and the optimal portfolio. We show, that in this case, the optimal convergence rate is super geometrical, i.e. is more rapid than any geometrical one. We apply our results to a stochastic volatility financial market.

q-fin.PM↗

arXiv2011

The computation of Greeks with multilevel Monte Carlo

We study the use of the multilevel Monte Carlo technique in the context of the calculation of Greeks. The pathwise sensitivity analysis differentiates the path evolution and reduces the payoff's smoothness. This leads to new challenges: the inapplicability of pathwise sensitivities to non-Lipschitz payoffs often makes the use of naive algorithms impossible. These challenges can be addressed in three different ways: payoff smoothing using conditional expectations of the payoff before maturity; approximating the previous technique with path splitting for the final timestep; using of a hybrid combination of pathwise sensitivity and the Likelihood Ratio Method. We investigate the strengths and weaknesses of these alternatives in different multilevel Monte Carlo settings.

q-fin.CP↗

arXiv2012

Approximating Functional of Local Martingale Under the Lack of Uniqueness of Black-Scholes PDE

When the underlying stock price is a strict local martingale process under an equivalent local martingale measure, Black-Scholes PDE associated with an European option may have multiple solutions. In this paper, we study an approximation for the smallest hedging price of such an European option. Our results show that a class of rebate barrier options can be used for this approximation. Among of them, a specific rebate option is also provided with a continuous rebate function, which corresponds to the unique classical solution of the associated parabolic PDE. Such a construction makes existing numerical PDE techniques applicable for its computation. An asymptotic convergence rate is also studied when the knocked-out barrier moves to infinity under suitable conditions.

q-fin.PR↗

arXiv2011

Statistical Inference for Time-changed Brownian Motion Credit Risk Models

We consider structural credit modeling in the important special case where the log-leverage ratio of the firm is a time-changed Brownian motion (TCBM) with the time-change taken to be an independent increasing process. Following the approach of Black and Cox, one defines the time of default to be the first passage time for the log-leverage ratio to cross the level zero. Rather than adopt the classical notion of first passage, with its associated numerical challenges, we accept an alternative notion applicable for TCBMs called "first passage of the second kind". We demonstrate how statistical inference can be efficiently implemented in this new class of models. This allows us to compare the performance of two versions of TCBMs, the variance gamma (VG) model and the exponential jump model (EXP), to the Black-Cox model. When applied to a 4.5 year long data set of weekly credit default swap (CDS) quotes for Ford Motor Co, the conclusion is that the two TCBM models, with essentially one extra parameter, can significantly outperform the classic Black-Cox model.

q-fin.ST↗

arXiv2011

On the Stability the Least Squares Monte Carlo

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

q-fin.CP↗

arXiv2011

Weighted Monte Carlo: Calibrating the Smile and Preserving Martingale Condition

Weighted Monte Carlo prices exotic options calibrating the probabilities of previously generated paths by a regular Monte Carlo to fit a set of option premiums. When only vanilla call and put options and forward prices are considered, the Martingale condition might not be preserved. This paper shows that this is indeed the case and overcomes the problem by adding additional synthetic options. A robust, fast and easy-to-implement calibration algorithm is presented. The results are illustrated with a geometric cliquet option which shows how the price impact can be significant.

q-fin.CP↗

arXiv2012

Jump-Diffusion Risk-Sensitive Asset Management II: Jump-Diffusion Factor Model

In this article we extend earlier work on the jump-diffusion risk-sensitive asset management problem [SIAM J. Fin. Math. (2011) 22-54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the HJB equation is a partial integro-differential equation (PIDE). By combining viscosity solutions with a change of notation, a policy improvement argument and classical results on parabolic PDEs we prove that the HJB PIDE admits a unique smooth solution. A verification theorem concludes the resolution of this problem.

q-fin.PM↗

arXiv2011

Representing filtration consistent nonlinear expectations as $g$-expectations in general probability spaces

We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous drivers, where both the martingale and the driver terms are permitted to jump, and the martingale representation is infinite dimensional. To establish this result, we show that this domination condition is sufficient to guarantee that the comparison theorem for BSDEs will hold, and we generalise the nonlinear Doob-Meyer decomposition of Peng to a general context.

math.PR↗

arXiv2012

Inf-convolution of g_Γ-solution and its applications

A risk-neutral method is always used to price and hedge contingent claims in complete market, but another method based on utility maximization or risk minimization is wildly used in more general case. One can find all kinds of special risk measure in literature. In this paper, instead of using market modified risk measure, we use a kind of risk measure induced by g_Γ-solution or the minimal solution of a Constrained Backward Stochastic Differential Equation (CBSDE) directly when constraints on wealth and portfolio process comes to our consideration. Such g_Γ-solution and the risk measure generated by it is well defined on appropriate space under suitable conditions. We adopt the inf-convolution of convex risk measures to solve some optimization problem. A dynamic version risk measures defined through g_Γ-solution and some similar results about optimal problem can be got in our new framework and by our new approach.

math.OC↗

arXiv2011

Defaultable Bonds via HKA

To construct a no-arbitrage defaultable bond market, we work on the state price density framework. Using the heat kernel approach (HKA for short) with the killing of a Markov process, we construct a single defaultable bond market that enables an explicit expression of a defaultable bond and credit spread under quadratic Gaussian settings. Some simulation results show that the model is not only tractable but realistic.

q-fin.CP↗

arXiv2011

Stochastic evolution equations in portfolio credit modelling with applications to exotic credit products

We consider a structural credit model for a large portfolio of credit risky assets where the correlation is due to a market factor. By considering the large portfolio limit of this system we show the existence of a density process for the asset values. This density evolves according to a stochastic partial differential equation and we establish existence and uniqueness for the solution taking values in a suitable function space. The loss function of the portfolio is then a function of the evolution of this density at the default boundary. We develop numerical methods for pricing and calibration of the model to credit indices and consider its performance pre and post credit crunch. Finally, we give further examples illustrating the valuation of exotic credit products, specifically forward starting CDOs.

q-fin.PR↗

arXiv2011

Hedging Effectiveness under Conditions of Asymmetry

We examine whether hedging effectiveness is affected by asymmetry in the return distribution by applying tail specific metrics to compare the hedging effectiveness of short and long hedgers using crude oil futures contracts. The metrics used include Lower Partial Moments (LPM), Value at Risk (VaR) and Conditional Value at Risk (CVAR). Comparisons are applied to a number of hedging strategies including OLS and both Symmetric and Asymmetric GARCH models. Our findings show that asymmetry reduces in-sample hedging performance and that there are significant differences in hedging performance between short and long hedgers. Thus, tail specific performance metrics should be applied in evaluating hedging effectiveness. We also find that the Ordinary Least Squares (OLS) model provides consistently good performance across different measures of hedging effectiveness and estimation methods irrespective of the characteristics of the underlying distribution.

q-fin.CP↗

arXiv2013

Explosive behavior in a log-normal interest rate model

We consider an interest rate model with log-normally distributed rates in the terminal measure in discrete time. Such models are used in financial practice as parametric versions of the Markov functional model, or as approximations to the log-normal Libor market model. We show that the model has two distinct regimes, at high and low volatilities, with different qualitative behavior. The two regimes are separated by a sharp transition, which is similar to a phase transition in condensed matter physics. We study the behavior of the model in the large volatility phase, and discuss the implications of the phase transition for the pricing of interest rate derivatives. In the large volatility phase, certain expectation values and convexity adjustments have an explosive behavior. For sufficiently low volatilities the caplet smile is log-normal to a very good approximation, while in the large volatility phase the model develops a non-trivial caplet skew. The phenomenon discussed here imposes thus an upper limit on the volatilities for which the model behaves as intended.

q-fin.CP↗

arXiv2011

Is a probabilistic modeling really useful in financial engineering? - A-t-on vraiment besoin d'un modèle probabiliste en ingénierie financière ?

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

q-fin.CP↗

arXiv2011

Credit contagion and risk management with multiple non-ordered defaults

The classical reduced-form and filtration expansion framework in credit risk is extended to the case of multiple, non-ordered defaults, assuming that conditional densities of the default times exist. Intensities and pricing formulas are derived, revealing how information driven default contagion arises in these models. We then analyze the impact of ordering the default times before expanding the filtration. While not important for pricing, the effect is significant in the context of risk management, and becomes even more pronounced for highly correlated and asymmetrically distributed defaults. Finally, we provide a general scheme for constructing and simulating the default times, given that a model for the conditional densities has been chosen.

q-fin.RM↗