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This paper offers a starting point for reflection on the similarities and differences of the impact on financial markets of the Great Recession of 2008 and of the Covid-19 pandemic of 2020 in terms of volatility and correlation risk among the most significant financial indexes in Europe. More precisely, the dataset employed includes the daily returns of Ftse100, CaC40, Dax30 and FtseMib40, with reference to the two time periods in which the two major crises manifested their effects on the markets. We use two different methodological approaches: the analysis of the daily conditional variance using various families of GARCH models and the study of the weekly realized volatility using HAR models. Furthermore, the estimation of the dependence structure of the GARCH residuals using copula functions is performed. [ABSTRACT FROM AUTHOR]

In this paper, the longitudinal data analysis is used to interpret the balanced Bühlmann credibility model with correlation risk, and the homogeneous credibility estimator is derived. We obtain the restricted maximum likelihood estimators (RMLE) for the structural parameters involved in the credibility factor and show that they are unbiased. In addition, the linear Bayes method is employed to estimate the structural parameters, and the proposed linear Bayes estimators (LBE) appear to outperform RMLE in terms of the mean squared error matrix (MSEM) criterion. Simulation studies show that the proposed LBE performs well. [ABSTRACT FROM AUTHOR]

This paper studies the optimal reinsurance and investment problem with multiple risky assets and correlation risk. The claim process is described by a Brownian motion with drift. The insurer is allowed to invest in a risk-free asset and multiple risky assets and the instantaneous return rate of each risky asset follows the Ornstein-Uhlenbeck (O-U) model. Moreover, the correlation between risk model and the risky assets' price is taken into account. We first consider the optimal investment problem for the insurer. Subsequently, we assume that the insurer can purchase proportional reinsurance and invest in the financial market. In both cases, the insurer's objective is to maximize the expected exponential utility of the terminal wealth. By applying stochastic control approach, we derive the optimal reinsurance and investment strategies and the corresponding value functions explicitly. Finally, numerical simulations are presented to illustrate the effects of model parameters on the optimal reinsurance and investment strategies. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal proportional reinsurance and investment problem for an insurer whose objective is to maximise the expected exponential utility of terminal wealth. Suppose that the insurer's surplus process follows a Brownian motion with drift. The insurer is allowed to purchase proportional reinsurance and invest in a financial market consisting of a risk-free asset and a risky asset whose price process is described by the constant elasticity of variance (CEV) model. The correlation between risk model and the risky asset's price is considered. By applying dynamic programming approach, we derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation. The asymptotic expansions of the solution to the partial differential equation (PDE) derived from the HJB equation are presented as the parameter appearing in the exponent of the diffusion coefficient tends to 0. We use perturbation theory for partial differential equations (PDEs) to obtain the asymptotic solutions of the optimal reinsurance and investment strategies. Finally, we provide numerical examples and sensitivity analyses to illustrate the effects of model parameters on the optimal reinsurance and investment strategies. [ABSTRACT FROM AUTHOR]

Objective: Modelling dynamic nature of covariance of assets return almost always is a challenging area of finance. Econometrics models just pay attention to variance behavior longitudinally, however, core of numerous finance models need the analysis of the total covariance structure of returns. Method: Among the first models that analyze covariance behavior are multivariate GARCH models which were criticized for the need to estimate a large number of parameters. This paper is aimed to investigate the effect of stock return dynamic correlation structure on systematic risk, idiosyncratic risk and average stock return. To this end, a sample of 148 listed companies in Tehran Stock Exchange is examined during 2003 to 2014. GARCH framework is used for testing this claim. Results: According to the results, securities that were highly correlated with market wide risk factors in the past are likely to have low systematic risk, idiosyncratic risk and average return at present. It can be expected there is significant relationship between idiosyncrstic risk and correlation for lower turnover stock (information transparency proxy) although there is no relationship for smaller firms.

This article introduces a new risk metric of stock returns. It measures the fraction of the non-diversifiable risk relative to the individual risk of a typical stock in the stock market and is shown to be closely related to average correlation between individual stocks. Using the Korean stock market data, we show that the fractional non-diversifiable risk varies over time, and in particular, sharply rises during the financial crisis periods. We find that the relation between risk and return at the aggregate level is negative contemporaneously, but positive with a time lag. As a robustness check, we extend our analysis to the U.S. and the U.K. markets. Our results shed light on resolving the conflicting risk and return relations reported in the literature. [ABSTRACT FROM AUTHOR]

Correlation plays an important role in pricing multi-asset options. In this work we incorporate stochastic correlation into pricing quanto options which is one special and important type of multi-asset option. Motivated by the market observations that the correlations between financial quantities behave like a stochastic process, instead of using a constant correlation, we allow the asset price process and the exchange rate process to be stochastically correlated with a parameter which is driven either by an Ornstein–Uhlenbeck process or a bounded Jacobi process. We derive an exact quanto option pricing formula in the stochastic correlation model of the Ornstein–Uhlenbeck process and a highly accurate approximated pricing formula in the stochastic correlation model of the bounded Jacobi process, where correlation risk has been hedged. The comparison between prices using our pricing formula and the Monte-Carlo method are provided. [ABSTRACT FROM AUTHOR]

This article investigates the optimal investment for insurers with correlation risk, with the variance-covariance matrix among risky financial assets evolving as a stochastic positive definite matrix process. Using the Wishart diffusion matrix process, we formulate the insurer's investment problem as the maximization of the expected constant relative risk-averse utility function subject to stochastic correlation, stochastic volatilities, and Poisson shocks. We obtain the explicit closed-form investment strategy and optimal expected utility through the Hamilton-Jacobi-Bellman framework. A verification theorem is derived to prove the uniform integrability of a tight upper bound for the objective function. The economic implication is that a long-term stable optimal investment policy requires the insurer to maintain a high risk-aversion level when the financial market contains stochastic volatility and/or stochastic correlation. [ABSTRACT FROM AUTHOR]

Guarantees embedded variable annuity contracts exhibit option-like payoff features and the pricing of such instruments naturally leads to risk neutral valuation techniques. This paper considers the pricing of two types of guarantees; namely, the Guaranteed Minimum Maturity Benefit and the Guaranteed Minimum Death Benefit riders written on several underlying assets whose dynamics are given by affine stochastic processes. Within the standard affine framework for the underlying mortality risk, stochastic volatility and correlation risk, we develop the key ingredients to perform the pricing of such guarantees. The model implies that the corresponding characteristic function for the state variables admits a closed form expression. We illustrate the methodology for two possible payoffs for the guarantees leading to prices that can be obtained through numerical integration. Using typical values for the parameters, an implementation of the model is provided and underlines the significant impact of the assets’ correlation structure on the guarantee prices. [ABSTRACT FROM PUBLISHER]

We propose a simple but practical methodology for the quantification of correlation risk in the context of credit derivatives pricing and credit valuation adjustment (CVA), where the correlation between rates and credit is often uncertain or unmodelled. We take the rates model to be Hull–White (normal) and the credit model to be Black–Karasinski (lognormal). We summarise recent work furnishing highly accurate analytic pricing formulae for credit default swaps (CDS) including with defaultable Libor flows, extending this to the situation where they are capped and/or floored. We also consider the pricing of contingent CDS with an interest rate swap underlying. We derive therefrom explicit expressions showing how the dependence of model prices on the uncertain parameter(s) can be captured in analytic formulae that are readily amenable to computation without recourse to Monte Carlo or lattice-based computation. In so doing, we crucially take into account the impact on model calibration of the uncertain (or unmodelled) parameters.

Equity basket correlation can be estimated both using the physical measure from stock prices, and also using the risk neutral measure from option prices. The difference between the two estimates motivates a so-called 'dispersion strategy'. We study the performance of this strategy on the German market and propose several profitability improvement schemes based on implied correlation (IC) forecasts. Modelling IC conceals several challenges. Firstly the number of correlation coefficients would grow with the size of the basket. Secondly, IC is not constant over maturities and strikes. Finally, IC changes over time. We reduce the dimensionality of the problem by assuming equicorrelation. The IC surface (ICS) is then approximated from the implied volatilities of stocks and the implied volatility of the basket. To analyze the dynamics of the ICS we employ a dynamic semiparametric factor model. [ABSTRACT FROM AUTHOR]

This work deals with the stochastic modelling of correlation in finance. It is well known that the correlation between financial products, financial institutions, e.g., plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that the correlation is hardly a deterministic quantity. For example, we illustrate this issue with the analysis of correlation between daily returns time series of S&P Index and Euro/USD exchange rates. The approach of modelling the correlation as a hyperbolic function of a stochastic process has been recently proposed. Here, we review this novel concept and generalize this approach to derive stochastic correlation processes (SCP) from a hyperbolic transformation of the modified Ornstein-Uhlenbeck process. We determine a transition density function of this SCP in closed form which could be used easily to calibrate SCP models to historical data. As an illustrating example of our new approach, we compute the price of a quantity adjusting option (Quanto) and discuss concisely the effect of considering stochastic correlation on pricing the Quanto. [ABSTRACT FROM AUTHOR]

It is well known that the correlation between financial products or financial institutions, e.g. plays an essential role in pricing and evaluation of financial derivatives. Using simply a constant or deterministic correlation may lead to correlation risk, since market observations give evidence that correlation is not a deterministic quantity. In this work, we propose a new approach to model the correlation as a hyperbolic function of a stochastic process. Our general approach provides a stochastic correlation which is much more realistic to model real- world phenomena and could be used in many financial application fields. Furthermore, it is very flexible: any mean-reverting process (with positive and negative values) can be regarded and no additional parameter restrictions appear which simplifies the calibration procedure. As an example, we compute the price of a Quanto applying our new approach. Using our numerical results we discuss concisely the effect of considering stochastic correlation on pricing the Quanto. [ABSTRACT FROM PUBLISHER]

When extreme events happen in the world, the correlation risk among assets rises outstandingly. So how to prevent and hedge the correlation risk in advance becomes an important subject for financial risk management and emergency management. Indeed, the volatility of index is influenced by the market liquidity, stock volatility and the correlation coefficient among stocks, while the stock volatility is not influenced by the correlation coefficient. Therefore, when the extreme events happen, the volatility of index will increase significantly because of the increasing correlation coefficient among assets. As a result, the price of index option will increase substantially. While the price of stock option is only influenced by the stock volatility and is irrelevant to the correlation coefficient, then the difference between stock index option price and the price of stock options contains the "correlation risk premium". Therefore, in theory, the index option can be used to hedge the correlation risk because the index option price contains the "correlation risk premium". Finally, in order to establish a trading strategy to hedge the correlation risk among stocks, we compare the index volatility risk and the stock volatility risk, strip out the correlation risk among stocks, and then hedge the correlation risk. In this paper, we use the Hang Seng Index option and its component stocks for empirical research. The date ranges from March 1, 2007 to August 31, 2011. The empirical result shows that when the extreme events happen, the correlation risk among the Hang Seng Index stocks increases significantly. The correlation trading strategy built in this paper can be used to hedge the correlation risk among stocks in the portfolio and it makes a contribution to control and prevent the correlation risk among stocks in advance, and gives investors a certain amount of revenue. [ABSTRACT FROM AUTHOR]

The Quanto option is a cash-settled, cross-currency derivative in which the underlying asset has a payoff in one country, but the payoff is converted to another currency in which the option is settled. Thus, the correlation between the underlying asset and currency exchange rate plays an important role on pricing such options. Market observations give clear evidence that financial quantities are correlated in a strongly nonlinear way. In this work, instead of assuming a constant correlation, we develop a strategy for pricing the Quanto option under dynamic correlation in a closed formula, including the calibration to market data. By comparing the pricing and hedging strategy with and without dynamic correlation, we study the effect of dynamic correlation on the option pricing and hedging. The numerical results show that the prices of Quanto option under dynamic correlation can be better fitted to the market prices than using simply a constant correlation. [ABSTRACT FROM AUTHOR]

Abstract: The Markowitz mean–variance portfolio selection (MVPS) problem is the building block of modern portfolio theory. Since Markowitz (1952) published his seminal study, there have been numerous extensions to the continuous-time MVPS problem under different market conditions. This paper further enriches the literature by taking account of correlation risk among risky asset returns. Empirical studies reveal that correlations among economic variables change randomly over time and affect hedging and investment demand in different correlation regimes. By incorporating correlation risk into the dynamic MVPS through the Wishart variance–covariance matrix process, this paper derives the explicit closed-form solution to the optimal portfolio policy and determines the market regime in which the optimal policy is stable and well-behaved. This stable market regime is found to be fully characterized by the correlation between market returns and their variance–covariance matrix or, equivalently, the effects of market leverage. [Copyright &y& Elsevier]

The asset allocation strategies don't take into account correlation risks in classical finance theories, but the correlation of markets and assets change all the time in nature, which raise the risks of portfolio. Unlike portfolio choice of traditional finance theories keeping a complete market setup, we reproduce the asymmetric tail dependence of capital market with symmetric Joe-Clayton Copula and solve the portfolio efficient frontiers and strategies when correlation risks exist with CVaR technique. Through the empirical study in Shanghai and Hong Kong market, we discover that ignoring upper tail or lower tail dependence will affect investors to estimate the risks of portfolio, and will make the portfolio suffer extreme negative returns; quantify and control the tail dependence will improve the performance of portfolio. [ABSTRACT FROM AUTHOR]

A new class of foreign equity option pricing model is suggested that not only allows for the volatility but also for the correlation coefficient to vary stochastically over time. A modified Jacobi process is proposed to evaluate risk premium of the stochastic correlation, and a partial differential equation to price the correlation risk for the foreign equity has been set up, whose solution has been compared with the one with constant correlation. Since taking into account the stochastic volatility gives rise to more dimensions that produce more difficulty in numerical implementation of partial differential equation and Monte carlo, we figure out a series solution for pricing options under the correlation risk. [ABSTRACT FROM AUTHOR]

This article gives examples illustrating the static hedging of Collateratized debt obligation (CDO) correlation risk. Changes in correlation result in changes in the relative portioning out of the total expected loss of a reference portfolio to the different tranches. Thus, portfolios with low correlation risk contain a number of CDO tranches whose weights are adjusted, so that the daily changes in the mark-to-market values of the different tranches tend to cancel out. Example portfolios are backtested for immunization against correlation and index spread risks using iTraxx CDO market spreads for the challenging period following the 5 May 2005 Standard and Poor's downgrade of Ford and General Motors. The implementation is carried out by using the static loss-surface model of Walker [CDO Models - Towards the Next Generation: Incomplete Markets and Term Structure, 2005. Available at http://www.physics.utoronto.ca/~qocmp/nextGenDefaultrisk.pdf; CDO Valuation: Term Structure, Tranche Structure, and Loss Distributions, 2006. Available at http://www.physics.utoronto.ca/~qocmp/nextLongB.2006.09.22.pdf] and Torresetti et al. [Implied Expected Tranched Loss Surface from CDO Data, 2006. Available at http://www.damianobrigo.it]. [ABSTRACT FROM AUTHOR]

We summarize recent developments in the credit derivative markets. We show the role of dependence between individual debtors in portfolio derivatives in a study of implied correlation. The risk of changing dependence structures between stock and bond markets becomes evident in an example of capital structure arbitrage. How credit derivatives can introduce new risks is illustrated by the example of “overlay” in basket derivatives. [ABSTRACT FROM AUTHOR]

We propose a simple but practical methodology for the quantification of correlation risk in the context of credit derivatives pricing and credit valuation adjustment (CVA), where the correlation between rates and credit is often uncertain or unmodelled. We take the rates model to be Hull–White (normal) and the credit model to be Black–Karasinski (lognormal). We summarise recent work furnishing highly accurate analytic pricing formulae for credit default swaps (CDS) including with defaultable Libor flows, extending this to the situation where they are capped and/or floored. We also consider the pricing of contingent CDS with an interest rate swap underlying. We derive therefrom explicit expressions showing how the dependence of model prices on the uncertain parameter(s) can be captured in analytic formulae that are readily amenable to computation without recourse to Monte Carlo or lattice-based computation. In so doing, we crucially take into account the impact on model calibration of the uncertain (or unmodelled) parameters. [ABSTRACT FROM AUTHOR]