Local Risk Minimization Research Articles

A new lattice approach for risk-minimization hedging under generalized autoregressive conditional heteroskedasticity models

This paper explores the calculation of risk-minimization hedging strategies, specifically local and global risk minimization strategies for contingent claims under affine and non-affine GARCH models with the known closed forms of its first four moments across times under the physical measure. A unified and efficient willow tree method is introduced for various GARCH models. Unlike methods that provide option values and hedging ratios solely at the inception time, the proposed willow tree method generates a comprehensive table of option values and hedging ratios at each discrete time step across possible asset prices. Additionally, the method showcases robust performance in hedging at lower frequencies than the underlying asset’s modeling frequency (e.g., weekly or monthly hedging using a daily GARCH model). Lastly, the willow tree method outperforms the Monte Carlo method, offering greater efficiency, accuracy, and flexibility in solving risk-minimization hedging problems.

A Girsanov transformed Clark-Ocone-Haussmann type formula for $$L^1$$-pure jump additive processes and its application to portfolio optimization

AbstractWe derive a Clark-Ocone-Haussmann (COH) type formula under a change of measure for $$ L^1 $$ L 1 -canonical additive processes, providing a tool for representing financial derivatives under a risk-neutral probability measure. COH formulas are fundamental in stochastic analysis, providing explicit martingale representations of random variables in terms of their Malliavin derivatives. In mathematical finance, the COH formula under a change of measure is crucial for representing financial derivatives under a risk-neutral probability measure. To prove our main results, we use the Malliavin-Skorohod calculus in $$ L^0 $$ L 0 and $$ L^1 $$ L 1 for additive processes, as developed by Di Nunno and Vives (2017). An application of our results is solving the local risk minimization (LRM) problem in financial markets driven by pure jump additive processes. LRM, a prominent hedging approach in incomplete markets, seeks strategies that minimize the conditional variance of the hedging error. By applying our COH formula, we obtain explicit expressions for locally risk-minimizing hedging strategies in terms of Malliavin derivatives under the market model underlying the additive process. These formulas provide practical tools for managing risks in financial market price fluctuations with $$L^1$$ L 1 -additive processes.

Insiders and Their Free Lunches: The Role of Short Positions

Given a stock price process, we analyse the potential of arbitrage by insiders in a context of short-selling prohibitions. We introduce the notion of minimal supermartingale measure, and we analyse its properties in connection to the minimal martingale measure. In particular, we establish conditions when both fail to exist. These correspond to the case when the insider's information set includes some non null events that are perceived as having null probabilities by the uninformed market investors. These results may have different applications, such as in problems related to the local risk-minimisation for insiders whenever strategies are implemented without short selling.

LOCAL RISK MINIMIZATION OF CONTINGENT CLAIMS SIMULTANEOUSLY EXPOSED TO ENDOGENOUS AND EXOGENOUS DEFAULT TIMES

We study the local risk minimization approach for contingent claims that might be simultaneously prone to both endogenous (or structural) and exogenous (or reduced form) default events. The exogenous default time is defined through a hazard rate process that can depend on both the underlying risky asset values and its running infimum process. On the other hand, the endogenous default time could be modeled by a first-passage-time approach. In particular, our framework provides a unification of structural and reduced form credit risk modeling. In our work, the evolution of the underlying risky asset values is modeled by an exponential Lévy process, for example exponential jump-diffusion models. Our aim is to determine locally risk minimizing hedging strategies of the contingent claims that are affected by both structural and reduced form default events, through solutions of either partial differential equations or partial-integro differential equations. Finally, we show that these solutions are numerically implementable, and we provide some numerical examples.

The Use of Subgroup Disproportionality Analyses to Explore the Sensitivity of a Global Database of Individual Case Safety Reports to Known Pharmacogenomic Risk Variants Common in Japan.

IntroductionGenetic variations of enzymes that affect the pharmacokinetics and hence effects of medications differ between ethnicities, resulting in variation in the risk of adverse drug reactions (ADR) between different populations. Previous work has demonstrated that risk-group considerations can be incorporated into approaches of statistical signal detection. It is unknown whether databases of individual case safety reports (ICSRs) are sensitive to pharmacogenomic differences between populations.ObjectiveThe aim of this study was to explore the sensitivity of a global database of ICSRs to known pharmacogenomic risk variants common in Japan.MethodsThe data source was VigiBase, the global database of ICSRs, including all reports entered in the version frozen on 5 January 2020. Subgroup disproportionality analysis was used to compare ICSRs of two subgroups, Japan and rest of world (RoW). Reports for UGT1A1-metabolized irinotecan and the CYP2C19-metabolized drugs voriconazole, escitalopram and clopidogrel were selected for comparison between the subgroups based upon known genetic polymorphisms with high prevalence in Japan. Contrast between the subgroups was quantified by IC delta ({mathrm{IC}}_{Delta }), a robust shrinkage observed-to-expected (OE) ratio on a log scale. Harmonic mean p values (HMP) were calculated for each drug to evaluate whether a list of pre-specified ADRs were collectively significantly over- (or under-)reported as hypothesized. Daily drug dosages were calculated for ICSRs with sufficient information, and dose distributions were compared between Japan and RoW and related to differences in regionally approved doses.ResultsThe predictions of over-reporting patterns for specific ADRs were observed and confirmed in bootstrap HMP analyses (p = 0.004 for irinotecan and p < 0.001 for each of voriconazole, escitalopram and clopidogrel) and compared with similar drugs with different metabolic pathways. The impact of proactive regulatory action, such as recommended dosing and therapeutic drug monitoring (TDM), was also observable within the global database. For irinotecan and escitalopram, there was evidence of use of lower dosages as recommended in the Japanese labels; for voriconazole, there was evidence of use of TDM with an over-reporting of terms related to drug level measurements and an under-reporting of liver toxicity.ConclusionsPharmaco-ethnic vulnerabilities caused by pharmacogenomic differences between populations may contribute to differences in ADR reporting between countries in a global database of ICSRs. Regional analyses within a global database can inform on the effectiveness of local risk minimization measures and should be leveraged to catalyse the conversion of real-world usage into safer use of drugs in ethnically tailored ways.Supplementary InformationThe online version contains supplementary material available at 10.1007/s40264-021-01063-1.

Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

&lt;p style='text-indent:20px;'&gt;This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local risk-minimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.&lt;/p&gt;

Lattice-Based Hedging Schemes Under GARCH Models

This paper proposes an efficient way to implement quadratic hedging schemes for European options when the asset return process follows an asymmetric non-affine GARCH model driven by Gaussian innovations. More specifically, using a lattice approximation for the underlying, we construct locally risk-minimizing (LRM) hedge ratios under both physical and risk-neutral measures, as well as standard delta strategies. We investigate the convergence of option prices and hedges resulting from the LRM strategies relative to the number of intra-daily periods used in the lattice. Several numerical experiments are conducted to assess the sensitivity of the hedge ratios to the equity risk premium and leverage effect parameters, and to compare their performance by computing the corresponding one-period and terminal hedging errors. Our results suggest that the LRM scheme under the physical measure consistently outperforms competing hedging strategies.

Structure Conditions under Progressively Added Information

It has been understood that the “local” existence of the Markowitz optimal portfolio or the solution to the local-risk minimization problem is guaranteed by some specific mathematical structures on...

Weight-Based Multi-view Global and Local Structural Risk Minimization Classifier

Weight-Based Multi-view Global and Local Structural Risk Minimization Classifier

Hedging the Risk of Delayed Data in Defaultable Markets

ABSTRACTWe investigate hedging the risk of delayed data in certain defaultable securities through the local risk minimization approach. From a financial point of view, this indicates that in addition to the risk of default, investors also face incomplete accounting data. In our analysis, the delay is modelled by a random time change, and different levels of information, including the full market’s, management’s, and investors’ information, are distinguished. We obtain semi-explicit solutions for pseudo locally risk minimizing hedging strategies from the perspective of investors where the results are presented according to the solutions of partial differential equations. In obtaining the main results of this paper, we apply a filtration expansion theorem that determines the canonical decomposition of stopped special semimartingales in an enlarged filtration of investors’ information.

Locally Risk-Minimizing Hedging of Counterparty Risk for Portfolio of Credit Derivatives

We discuss dynamic hedging of counterparty risk for a portfolio of credit derivatives by the local risk-minimization approach. We study the problem from the perspective of an investor who, trading with credit default swaps (CDS) referencing the counterparty, wants to protect herself/himself against the loss incurred at the default of the counterparty. We propose a credit risk intensity-based model consisting of interacting default intensities by taking into account direct contagion effects. The portfolio of defaultable claims is of generic type, including CDS portfolios, risky bond portfolios and first-to-default claims with payments allowed to depend on the default state of the reference firms and counterparty. Using the martingale representation of the conditional expectation of the counterparty risk price payment stream under the minimal martingale measure, we recover a closed-form representation for the locally risk minimizing strategy in terms of classical solutions to nonlinear recursive systems of Cauchy problems. We also discuss applications of our framework to the most prominent classes of credit derivatives.

Pricing and hedging equity-indexed annuities via local risk-minimization

ABSTRACTLin et al. (2009) employed the Esscher transform method to price equity-indexed annuities (EIAs) when the dynamic of the market value of a reference asset was driven by a generalized geometric Brownian motion model with regime-switching. Some rare events (release of an unexpected economic figure, major political changes or even a natural disaster in a major economy) can lead to brusque variations in asset prices, and hence we sometimes need to consider jump models. This paper extends the model and analysis in Lin et al. (2009). Specifically, we assume that the financial market has a regime-switching jump-diffusion model, under which we price the point-to-point, the Asian-end, the high water mark and the annual reset EIAs by exploiting the local risk-minimization approach. The effects of the model parameters on the EIAs pricing are illustrated through numerical experiments. Meanwhile, we present the locally risk-minimizing hedging strategies for EIAs.

Pricing and hedging guaranteed minimum withdrawal benefits under a general Lévy framework using the COS method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Lévy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. [Scand. Actuar. J., 2014, 1–20], and Luo and Shevchenko [Int. J. Financ. Eng., 2014, 2, 1–24], to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. [op. cit.]. We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging intra-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Lévy framework. While the second moment and variance have been considered in existing literature, we show that the Value-at-Risk (VaR) may also be of interest as a risk measure to minimise risk in variable annuities portfolios.

Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.

Pricing and Hedging Guaranteed Minimum Withdrawal Benefits under a General LLvy Framework Using the COS Method

This paper extends the Fourier-cosine (COS) method to the pricing and hedging of variable annuities embedded with guaranteed minimum withdrawal benefit (GMWB) riders. The COS method facilitates efficient computation of prices and hedge ratios of the GMWB riders when the underlying fund dynamics evolve under the influence of the general class of Levy processes. Formulae are derived to value the contract at each withdrawal date using a backward recursive dynamic programming algorithm. Numerical comparisons are performed with results presented in Bacinello et al. (2014) and Luo and Shevchenko (2014) to confirm the accuracy of the method. The efficiency of the proposed method is assessed by making comparisons with the approach presented in Bacinello et al. (2014). We find that the COS method presents highly accurate results with notably fast computational times. The valuation framework forms the basis for GMWB hedging. A local risk minimisation approach to hedging inter-withdrawal date risks is developed. A variety of risk measures are considered for minimisation in the general Levy framework. While the second moment and variance have been considered in existing literature, we show that the value-at-risk may also be of interest as a risk measure to minimise risk in variable annuities portfolios.

Hedging and pricing illiquid options with market impacts

In this paper, we consider hedging and pricing of illiquid options on an untradable underlying asset, where an alternative asset is used as a hedging instrument. Particularly, we consider the situation where the trade price of the hedging instrument is subject to market impacts caused by the hedger and the liquidity costs paid as a spread from the mid price. Pricing illiquid options, which often appears in trading of structured products, is a critical issue in practice because of its difficulties in hedging mainly due to untradability of the underlying asset as well as the liquidity costs and market impacts of the hedging instrument. First, by setting the problem under a discrete time model, where the optimal hedging strategy is defined by the local risk-minimization, we present algorithms to obtain the option price along with the hedging strategy by an asymptotic expansion. Moreover, we provide numerical examples. This model enables the estimation of the effect of both the market impacts and the liquidity costs on option prices, which is important in practice.

Option valuation with liquidity risk and jumps

ABSTRACTThis article provides a simple model for pricing and hedging options in the presence of jumps and liquidity costs. In the article, liquidity risk is modelled via a stochastic supply curve function and a jump-diffusion process is approximated by a Markov chain. Local risk minimization incorporating liquidity risk is proposed to price and hedge European options in this discrete-time model. Moreover, an example is provided to implement the modified risk minimization method and to demonstrate the performance of hedging strategies.

Local risk-minimization for Barndorff-Nielsen and Shephard models

We obtain explicit representations of locally risk-minimizing strategies for call and put options in Barndorff-Nielsen and Shephard models, which are Ornstein–Uhlenbeck-type stochastic volatility models. Using Malliavin calculus for Levy processes, Arai and Suzuki (Int. J. Financ. Eng. 2:1550015, 2015) obtained a formula for locally risk-minimizing strategies for Levy markets under many additional conditions. Supposing mild conditions, we make sure that the Barndorff-Nielsen and Shephard models satisfy all the conditions imposed in (Arai and Suzuki in Int. J. Financ. Eng. 2:1550015, 2015). Among others, we investigate the Malliavin differentiability of the density of the minimal martingale measure. Moreover, we introduce some numerical experiments for locally risk-minimizing strategies.

Identifying local smoothness for spatially inhomogeneous functions

We consider a problem of estimating local smoothness of a spatially inhomogeneous function from noisy data under the framework of smoothing splines. Most existing studies related to this problem deal with estimation induced by a single smoothing parameter or partially local smoothing parameters, which may not be efficient to characterize various degrees of smoothness of the underlying function when it is spatially varying. In this paper, we propose a new nonparametric method to estimate local smoothness of the function based on a moving local risk minimization coupled with spatially adaptive smoothing splines. The proposed method provides full information of the local smoothness at every location on the entire data domain, so that it is able to understand the degrees of spatial inhomogeneity of the function. A successful estimate of the local smoothness is useful for identifying abrupt changes of smoothness of the data, performing functional clustering and improving the uniformity of coverage of the confidence intervals of smoothing splines. We further consider a nontrivial extension of the local smoothness of inhomogeneous two-dimensional functions or spatial fields. Empirical performance of the proposed method is evaluated through numerical examples, which demonstrates promising results of the proposed method.

Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models

ABSTRACTThis article focuses on an optimal hedging problem of the vulnerable European contingent claims. The underlying asset of the vulnerable European contingent claims is assumed to be nontradable. The interest rate, the appreciation rate and the volatility of risky assets are modulated by a finite-state continuous-time Markov chain. By using the local risk minimization method, we obtain an explicit closed-form solution for the optimal hedging strategies of the vulnerable European contingent claims. Further, we consider a problem of hedging for a vulnerable European call option. Optimal hedging strategies are obtained. Finally, a numerical example for the optimal hedging strategies of the vulnerable European call option in a two-regime case is provided to illustrate the sensitivities of the hedging strategies.