1,721,682 research outputs found
Backtesting energy portfolio with copula dependence structure
No full textThe energy markets play a crucial role in the economic development of every country. These markets are characterized by high volatilities due to sizeable price fluctuations. The correct development of risk measures to quantify the inherent risk market is then a challenging task for the risk management systems. We consider in this survey a portfolio composed of two energy assets: crude oil and natural gas. We adopt, as a risk measure, the value-at-risk and the expected shortfall. In order to estimate these risk measures efficiently, we model the tails of each marginal with extreme value theory and we adopt a general dependence structure between the two assets given by a t-copula.
The performance of the model is then validated with backtesting technique. To this end, we use a database ranging from years 1997 to 2017. We have then highlighted that the backtesting based on the value-at-risk and the Expected Shortfall passed the most common tests
Electricity derivatives: an application to the futures Italian market
Full text linkSince the liberalization of electricity markets, electricity prices are more volatile and expansion in electricity derivatives trading occurs. Indeed, a well-known feature of electricity prices concerns its high volatility. For this reason, operators use power futures to hedge against unexpected risk deriving from adverse fluctuations of spot prices within the planned delivering period. Indeed, futures contracts permit to fix the price of electricity in advance for the use in the scheduled period. Our paper is devoted specifically to the Italian electricity market. In this respect, we examine empirical data from IDEX, the Energy Derivatives part of the Italian derivatives market IDEM, administered by “Borsa Italiana.” We finally survey the possible connections concerning futures and spot prices and, as a consequence, we deduce information about important indicators whereof the ex-post risk premium and the net convenience yield. For this purpose, we use several regression techniques to determine suitable explanatory variables inherent the Italian market for the ex-post risk premium and the net convenience yield
Optimisation of conditional-VaR in an actuarial model for credit risk assuming a student copula dependence structure
Full text linkIn this paper we present a model for the valuation of the risk of credit portfolios. It uses both traditional tools of credit risk valuations and more recent ones like copula functions and Conditional VaR theory. The model we propose is based on some key assumptions we here summarise: first of all, the risk of default is modelled using the time-until-default of an exposure; moreover the hazard rates are random variables whose values follow gamma distributions coherently with CreditRisk+ proposed by Credit Suisse and others; recovery rates themselves are supposed to be stochastic (following a Beta distribution).
The main aspect of our proposal is the introduction of credit migration in the context of an intensity-based model with copula function dependence structure (we use a Student copula to model correlations between the obligors). This permits to quantify the loss distribution of the portfolio
and to calculate some useful indexes of risk for the probability distribution of the values of the portfolio: expectation, variance, alpha-VaR, and, following Rockafellar & Uryasev, the alpha-conditional VaR (alpha-CVaR) of the portfolio itself.
The final aim of the model is to present a more flexible and realistic approach to valuation and management of the risk of credit portfolios. Infact, in comparison with the traditional approaches,
we remove some restrictive assumptions and try to generalize the valuation scheme (i.e. CreditMetrics considers constant hazard rates while CreditRisk+ takes into account constant recovery rates with no credit migrations).
We conclude the article with a large numerical example in order to test the model
Pricing index linked policies with basket cliquet options embedded using a copula approach
No full textIn this paper we present a model for the pricing of an index linked insurance contract with a basket cliquet option embedded.
The model moves from the seminal and widely accepted model of Brennan & Schwartz but uses a copula approach to describe the dependence between the two stochastic indexes composing the underlying basket.
The pricing is made via Monte Carlo stochastic simulation; some useful algorithms are described.
An application and a comparative static analysis are presente
Pricing credit derivatives with a copula-based actuarial model for credit risk
No full textCredit derivatives are financial contracts whose pay-off are contingent on the creditworthiness of some counterparts. As was pointed out in some recent works (Mashal & Naldi (2002), Meneguzzo & Vecchiato (2002)), they have become in recent years the main tool for transferring and hedging credit risk. The most complicated of such instruments are the multinames ones. Indeed, these instruments are not quoted (market prices are not available). Besides, we do not posses closed forms for their pricing: we must necessarily set up a Monte Carlo simulation procedure. The key to perform this task consists in modelling correctly multiple defaults. A dependence structure using copulas methods was first set up by Li (2000). In this paper, Li considers time-until-default for each obligor and model their dependence structure through a Student t-copula. Other papers which take into account a copula dependence structure are due to Cherubini & Luciano (2002, 2004), Galiani (2003), Gregory & Laurent (2002), Li describes a default for a single obligor through the so-called survival function S(t) " Pr T # t! which represents the probability that this counterpart attains age t and is the time-until-default.
Li also assumes that the hazard rate function is constant, . This means that the survival time is exponentially distributed with constant parameter . Other features of this model are the following: credit
migrations at the end of the time horizon were not taken into account and recovery rates in default situations are assumed deterministic.
This model has been resumed by Mashal & Naldi with the intent to price particular multinames credit derivatives such as nth-to-default baskets. Their model is a hybrid of the well-known structural and reduced form approaches for modelling defaults.
After simulating a large number of multivariate times-until-default, one deduces pay-off for our derivative. Finally, the pricing is estimated using
standard risk-neutral pricing technology (by assuming complete markets and no-arbitrage hypothesis). The credit risk model for the underlying portfolio, already developed in Masala, Menzietti & Micocci (2004), follows a general credit risk framework: hazard rates are random variables whose values follow gamma distributions coherently with Credit Risk Plus (1997), Micocci (2000), Burgisser, Kurth & Wagner (2001) and Menzietti (2002); recovery rates themselves are supposed to be stochastic as in Gupton, Finger & Bathia (1997), and following a Beta distribution, moreover credit migrations are allowed. This feature becomes very important when we treat credit derivatives whose payoff depends on credit spread. The paper is structured as follows. Section 2 presents the model for default and credit migration; the section is divided in subsections facing the problems of time-until-default, the hazard rate function and the recovery rates, the credit migration and the exposure valuation, the loss distribution. Section 3 introduces some basket credit derivatives with numerical applications. Section 4 concludes
Economic capital management for insurance companies using conditional value at risk and a copula approach
No full textThe loss ratio (LR) for insurance companies is defined as the ratio of incurred claims and earned premiums for a specified class of insurance (CoI). The company may estimate then its capital requirement for that particular CoI by using Value at Risk (VaR) or conditional VaR (CVaR) of the loss ratio distribution at a specified probability value. The overall objective of the company is to evaluate the aggregate capital requirement through a weighted sum of marginal capital requirements for all the classes of insurance. Nevertheless, this procedure may tend to over-estimate the aggregate capital requirement because it does not take into consideration the real dependence amongst the different classes of insurance. In other words, perfect dependence does not allow considering diversification effects. In this paper, we present a model which permits to take into consideration real correlations of the several CoIs. Thanks to copula functions, we are able to generate (by Monte Carlo simulations) correlated loss ratios with known marginal distributions. This approach is described through a numerical example that used data collected from some of the most important Italian non life insurance companies. We show then the diversification benefit thus obtained. We conclude the paper building an efficient frontier on the plane LR - CVaR; the efficient frontier may be considered a useful tool to manage the global company risk
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