COUNTERPARTY CREDIT RISK IN A CLEARING NETWORK

2020 ◽  
Vol 23 (06) ◽  
pp. 2050035
Author(s):  
ALEXANDER VON FELBERT
Keyword(s):  
Network Model ◽  
Random Variable ◽  
Mathematical Finance ◽  
Closed Form Expression ◽  
Characteristic Functions ◽  
Counterparty Risk ◽  
Hilbert Transforms ◽  
Market Participants ◽  
Counterparty Credit Risk ◽  
General Answer

In this paper, we offer a network model that derives the expected counterparty risk of an arbitrary market after netting in a closed-form expression. Graph theory is used to represent market participants and their relationship among each other. We apply the powerful theory of characteristic functions (c.f.) and Hilbert transforms to determine the expected counterparty risk. The latter concept is used to express the c.f. of the random variable (r.v.) [Formula: see text] in terms of the c.f. of the r.v. [Formula: see text]. This paper applies this concept for the first time in mathematical finance in order to generalize results of Duffie & Zhu (2011), in several ways. The introduced network model is 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 risk to clear via a central counterparty or classically bilateral between the two involved counterparties.

scholarly journals Distributions Escaping to Infinity and the Limiting Power of the Cliff-Ord Test for Autocorrelation

2012 ◽  
Vol 2012 ◽  
pp. 1-39 ◽  
Author(s):  
Kairat T. Mynbaev
Keyword(s):  
Linear Regression ◽  
Regression Model ◽  
Linear Regression Model ◽  
Random Variables ◽  
Random Variable ◽  
Closed Form Expression ◽  
Characteristic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Form Expression

We consider a family of proper random variables which converges to an improper random variable. The limit in distribution is found and applied to obtain a closed-form expression for the limiting power of the Cliff-Ord test for autocorrelation. The applications include the theory of characteristic functions of proper random variables, the theory of almost periodic functions, and the test for spatial correlation in a linear regression model.

Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

2021 ◽  
Vol 48 (3) ◽  
pp. 91-96
Author(s):  
Shigeo Shioda
Keyword(s):  
Distribution Function ◽  
Probability Density Function ◽  
Closed Form ◽  
Probability Density ◽  
Infinite Number ◽  
Stable Distribution ◽  
Random Variable ◽  
Cauchy Distribution ◽  
Closed Form Expression ◽  
Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

On the Distribution of Sum of Independent Positive Binomial Variables

1970 ◽  
Vol 13 (1) ◽  
pp. 151-152 ◽  
Author(s):  
J. C. Ahuja
Keyword(s):  
Power Series ◽  
Inversion Formula ◽  
Probability Function ◽  
Random Variables ◽  
Random Variable ◽  
Characteristic Functions ◽  
Binomial Probability ◽  
Generalized Power Series ◽  
Alternative Derivation ◽  
Image Position

Let X1, X2, …, Xn be n independent and identically distributed random variables having the positive binomial probability function1where 0 < p < 1, and T = {1, 2, …, N}. Define their sum as Y=X1 + X2 + … +Xn. The distribution of the random variable Y has been obtained by Malik [2] using the inversion formula for characteristic functions. It appears that his result needs some correction. The purpose of this note is to give an alternative derivation of the distribution of Y by applying one of the results, established by Patil [3], for the generalized power series distribution.

A CLOSED-FORM GARCH VALUATION MODEL FOR POWER EXCHANGE OPTIONS WITH COUNTERPARTY RISK

2019 ◽  
Vol 34 (2) ◽  
pp. 279-296
Author(s):  
Xingchun Wang ◽  
Guangli Xu ◽  
Dan Li
Keyword(s):  
Closed Form ◽  
Default Risk ◽  
Reduced Form ◽  
Characteristic Functions ◽  
Counterparty Risk ◽  
Total Risk ◽  
Valuation Model ◽  
Power Exchange ◽  
Measure Change ◽  
Exchange Options

AbstractIn this paper, a discrete-time framework is proposed to value power exchange options with counterparty default risk, where counterparty risk is considered in a reduced-form setting and the variance processes of the underlying assets are captured by GARCH processes. In addition, the proposed model allows for the correlation between the intensity of default and the variances of the underlying assets by breaking down the total risk into systematic and idiosyncratic components. By dint of measure-change techniques and characteristic functions, we obtain the closed-form pricing formula for the value of power exchange options with counterparty default risk. Finally, numerical results are presented to show the power exchange option values.

scholarly journals Dam rain and cumulative gain

2008 ◽  
Vol 464 (2095) ◽  
pp. 1801-1822 ◽  
Author(s):  
Dorje C Brody ◽  
Lane P Hughston ◽  
Andrea Macrina
Keyword(s):  
Cash Flow ◽  
Random Variable ◽  
Closed Form Expression ◽  
Time Interval ◽  
Defined Benefit ◽  
Relevant Market ◽  
Financial Contract ◽  
Aggregate Claims ◽  
Credit Portfolio ◽  
The Given

We consider a financial contract that delivers a single cash flow given by the terminal value of a cumulative gains process. The problem of modelling such an asset and associated derivatives is important, for example, in the determination of optimal insurance claims reserve policies, and in the pricing of reinsurance contracts. In the insurance setting, aggregate claims play the role of cumulative gains, and the terminal cash flow represents the totality of the claims payable for the given accounting period. A similar example arises when we consider the accumulation of losses in a credit portfolio, and value a contract that pays an amount equal to the totality of the losses over a given time interval. An expression for the value process of such an asset is derived as follows. We fix a probability space, together with a pricing measure, and model the terminal cash flow by a random variable; next, we model the cumulative gains process by the product of the terminal cash flow and an independent gamma bridge; finally, we take the filtration to be that generated by the cumulative gains process. An explicit expression for the value process is obtained by taking the discounted expectation of the future cash flow, conditional on the relevant market information. The price of an Arrow–Debreu security on the cumulative gains process is determined, and is used to obtain a closed-form expression for the price of a European-style option on the value of the asset at the given intermediate time. The results obtained make use of remarkable properties of the gamma bridge process, and are applicable to a wide variety of financial products based on cumulative gains processes such as aggregate claims, credit portfolio losses, defined benefit pension schemes, emissions and rainfall.

scholarly journals Estimation for Wrapped Zero Inflated Poisson and Wrapped Poisson Distributions

2016 ◽  
Vol 5 (3) ◽  
pp. 1 ◽  
Author(s):  
Aerambamoorthy Thavaneswaran ◽  
Saumen Mandal ◽  
Dharini Pathmanathan
Keyword(s):  
Closed Form ◽  
Information Matrix ◽  
Score Function ◽  
Function Approach ◽  
Closed Form Expression ◽  
Characteristic Functions ◽  
Model Parameters ◽  
Estimating Function ◽  
Method Of Moment ◽  
Poisson Distributions

There has been a growing interest in discrete circular models such as wrapped zero inflated Poisson and wrapped Poisson distributions and the trigonometric moments (see Brobbey et al., 2016 and Girija et al., 2014). Also, characteristic functions of stable processes have been used to study the estimation of the model parameters using estimating function approach (see Thavaneswaran et al., 2013). One difficulty in estimating the circular mean and the resultant mean length parameter of wrapped Poisson (WP) or wrapped zero inflated Poisson (WZIP) is that neither the likelihood of WP/WZIP random variable nor the score function is available in closed form, which leads one to use either trigonometric method of moment estimation (TMME) or an estimating function approach. In this paper, we study the estimation of WZIP distribution and WP distribution using estimating functions and obtain the closed form expression of the information matrix. We also derive the asymptotic distribution of the tangent of the mean direction for both the WZIP and WP distributions.

Capacity of Optical Wireless System over Log-Normal Channels with Spatial Diversity in Presence of Atmospheric Losses

2018 ◽  
Vol 39 (3) ◽  
pp. 349-357 ◽  
Author(s):  
Rahul Kaushik ◽  
Vineet Khandelwal ◽  
R.C. Jain
Keyword(s):  
Atmospheric Turbulence ◽  
Channel Capacity ◽  
Random Variable ◽  
Weather Conditions ◽  
Spatial Diversity ◽  
Closed Form Expression ◽  
Atmospheric Conditions ◽  
Optical Wireless ◽  
Diversity Techniques ◽  
Log Normal

Abstract In this paper, average channel capacity of optical wireless communication system is evaluated under the combined effect of geometrical loss, attenuation due to weather conditions and weak atmospheric turbulence using a simple closed form expression. Fading induced due to atmospheric turbulence is modeled by log-normal distribution. Considering the fact that the sum of log-normal random variables can be well approximated by another log-normal random variable, the proposed expression has been utilized to compute the channel capacity for spatial diversity reception employing maximum ratio combining and equal gain combining over uncorrelated turbulence-induced fading conditions. It is shown that spatial diversity is an effective technique to mitigate the impairments caused by various atmospheric conditions such as haze, rain and fog. The quantitative improvement in channel capacity achieved by using diversity techniques is investigated and compared. Accuracy of the results is validated with exact results computed using Monte Carlo simulation.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

A Note on Nonparametric Estimation of the CTE

2009 ◽  
Vol 39 (2) ◽  
pp. 717-734 ◽  
Author(s):  
Bangwon Ko ◽  
Ralph P. Russo ◽  
Nariankadu D. Shyamalkumar
Keyword(s):  
Order Term ◽  
Random Variable ◽  
Small Samples ◽  
Closed Form Expression ◽  
Form Expression ◽  
Continuous Random Variable ◽  
Underlying Distribution ◽  
First Order ◽  
Bootstrap Approach ◽  
Conditional Tail Expectation

AbstractThe α-level Conditional Tail Expectation (CTE) of a continuous random variable X is defined as its conditional expectation given the event {X > qα} where qα represents its α-level quantile. It is well known that the empirical CTE (the average of the n (1 – α) largest order statistics in a sample of size n) is a negatively biased estimator of the CTE. This bias vanishes as the sample size increases, but in small samples can be significant. In this article it is shown that an unbiased nonparametric estimator of the CTE does not exist. In addition, the asymptotic behavior of the bias of the empirical CTE is studied, and a closed form expression for its first order term is derived. This expression facilitates the study of the behavior of the empirical CTE with respect to the underlying distribution, and suggests an alternative (to the bootstrap) approach to bias correction. The performance of the resulting estimator is assessed via simulation.

scholarly journals A Note on Characteristic Functions Which Vanish Identically in an Interval

1962 ◽  
Vol 58 (2) ◽  
pp. 430-432 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Characteristic Function ◽  
Random Variable ◽  
Characteristic Functions ◽  
Unpublished Work

Some years ago, in connexion with some unpublished work in the theory of queues, the question arose as to whether the characteristic function of a non-negative random variable could vanish identically in an interval. The purpose of this note is to show that such a thing is impossible.

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