Risk-minimizing Hedging Strategies Research Articles

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.

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.

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

<p style='text-indent:20px;'>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.</p>

LOCALLY RISK-MINIMIZING HEDGING FOR EUROPEAN CONTINGENT CLAIMS WRITTEN ON NON-TRADABLE ASSETS WITH COMMON JUMP RISK

This article investigates the optimal hedging problem of the European contingent claims written on non-tradable assets. We assume that the risky assets satisfy jump diffusion models with a common jump process which reflects the correlated jump risk. The non-tradable asset and jump risk lead to an incomplete financial market. Hence, the cross-hedging method will be used to reduce the potential risk of the contingent claims seller. First, we obtain an explicit closed-form solution for the locally risk-minimizing hedging strategies of the European contingent claims by using the Föllmer–Schweizer decomposition. Then, we consider the hedging for a European call option as a special case. The value of the European call option under the minimal martingale measure is derived by the Fourier transform method. Next, some semi-closed solution formulae of the locally risk-minimizing hedging strategies for the European call option are obtained. Finally, some numerical examples are provided to illustrate the sensitivities of the optimal hedging strategies. By comparing the optimal hedging strategies when the underlying asset is a non-tradable asset or a tradable asset, we find that the liquidity risk has a significant impact on the optimal hedging strategies.

Pricing and hedging of general rating-sensitive claims in a jump-diffusion market model in the presence of stochastic factors

In this paper we solve a problem of finding a risk minimizing hedging strategy on a general Markovian market with ratings on which prices are influenced by additional factors and rating. The market is described by a system of SDEs driven by a Wiener process and a compensated Poisson random measure. Rating-sensitive claims are considered. We relate the problem of pricing and hedging the contracts described by a general cash-flow process to solving Cauchy/Dirichlet problems and subsequently to solving some linear system of equations. We illustrate our theory on two examples of different nature. The first is a general exponential Lévy model with stochastic volatility, and the second is a generalization of an exponential Lévy model with regime switching.

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.

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.

On the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Levy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical examples for two typical exponential Levy models: Merton models and variance gamma models.

Hedging With Credibility When Assets Can Jump

Hedging options in non-Gaussian models is a well-known and difficult task, yet remaining important for practitioners. Hence, solutions through the quadratic hedging methods have been suggested, see the recent contributions of Riesner (2006), Vandaele and Vanmaele (2008) and that in Cont and Tankov (2004). Although their suggested ratios to invest in the underlying asset for an optimal replication are different from each to other, they, however, share a common structure which makes their implementation non obvious. This structure originates in the integral part of the partial integro-differential equation and stems from the expectation of option prices taken over the Lévy measure. Although non-straightforward numerical integrations can be used to implement this quantity, they have to be modified and adapted to suit the choice of the random jump size distributions, resulting in a cumbersome task. Hence, implementation efficiency has still to be addressed. Using a locally risk-minimizing hedging strategy together with an elegant result of Hille and Phillips (1957), this paper shows how to efficiently compute the expectation of the option prices taken over the Lévy measure of any Lévy processes, be they of the finite or of the infinite activity. Hence, all the optimal ratios suggested by the aforementioned authors can be evaluated by adding a minor factor to a fast Fourier transform pricing formula and thereby gaining its computation efficiency.

Risk-minimizing pricing and hedging foreign currency options under regime-switching jump-diffusion models

ABSTRACTThis article mainly investigates risk-minimizing European currency option pricing and hedging strategy when the spot foreign exchange rate is driven by a Markov-modulated jump-diffusion model. We suppose the domestic and foreign money market floating interest rates, the drift, and the volatility of the exchange rate dynamics all depend on the state of the economy, which is modeled by a continuous-time hidden Markov chain. The model considered in this article will provide market practitioners with flexibility in characterizing the dynamics of the spot foreign exchange rate. Using the minimal martingale measure, we obtain a system of coupled partial-differential-integral equations satisfied by the currency option price and find the corresponding hedging strategies and the residual risk. According to simulation of currency option prices in the special case of double exponential jump-diffusion regime-switching model, we further discuss and show the effects of the parameters on the prices.

Hedging with Credibility When Assets Can Jump

Hedging options in non-Gaussian models is a well-known and difficult task, yet remaining important for risk practitioners from banks to insurance companies. Hence, solutions through the quadratic hedging methods have been recently suggested, see Cont and Tankov (2004), Riesner (2006) and Vandaele and Vanmaele (2008). Although their suggested ratios to invest in the underlying asset for an optimal replication are different from each to other, they, however, share a common structure which makes their implementation non obvious. This structure originates in the integral part of the partial integro-differential equation and stems from the expectation of option prices taken over the random jump sizes. Although non-straightforward numerical integrations can be used to implement this quantity, they have to be modified and adapted to suit the choice of the random jump size distributions, resulting in a cumbersome task. Hence, implementation efficiency has still to be addressed. Using a locally risk-minimizing hedging strategy together with an elegant result of Hille and Phillips (1957), this paper shows how to efficiently compute the expectation of the option prices taken over the random jump sizes of any Levy processes, be they of the finite or of the infinite activity. Hence, all the optimal ratios suggested by the aforementioned authors can be evaluated by adding a minor factor to a fast Fourier transform pricing formula and thereby gaining its computation efficiency.

Risk-minimizing hedging strategy for an equity-indexed annuity under a regime switching model

The equity-indexed annuity (EIA) contract offers a proportional participation in the performance of a specified equity index, in addition to a guaranteed return on the single premium. How to manage the risk of the EIA is an important issue. This paper considers the hedging of the EIA. We assume that the parameters of the financial model depend on a continuous-time finite-state Markov chain and the Markov chain is observed, that is the Markov regime switching model. The state of the Markov chain can be interpreted as the state of an economy. Under the regime switching model, we obtain the risk-minimizing hedging strategy for the EIA.

GKW representation theorem under restricted information: An application to risk-minimization

We provide Galtchouk–Kunita–Watanabe representation results in the case where there are restrictions on the available information. This allows one to prove the existence and uniqueness of solution for special equations driven by a general square integrable càdlàg martingale under partial information. Furthermore, we discuss an application to risk-minimization where we extend the results of Föllmer and Sondermann, Hedging of non-redundant contingent claims, to the partial information framework and we show how our result fits in the approach of Schweizer, Risk-minimizing hedging strategies under restricted information.

Measuring the Model Risk of Quadratic Risk Minimizing Hedging Strategies with an Application to Energy Markets

Measures of model risk based on the residual error from hedging in a misspecified model were recently proposed in (Detering and Packham, 2013). These measures rely on the assumption that the model used for hedging represents a complete financial market. We show that under certain conditions, in a diffusion setup, markets can be completed to derive measures of model risk for the original market. If the market can not be completed, as it is the case in most market models that allow for jumps, we derive measures that are applicable in a more general setup. In a case study we measure the model risk that is present when hedging options on energy futures with a simplified model compared to a model that better fits the empirical returns observed in the market.

A benchmark approach to risk-minimization under partial information

The goal of this paper is to investigate (locally) risk-minimizing hedging strategies under the benchmark approach in a financial semimartingale market model where there are restrictions on the available information. More precisely, we characterize the optimal strategy as the integrand appearing in the Galtchouk–Kunita–Watanabe decomposition of the benchmarked contingent claim under partial information and provide its description in terms of the integrand in the classical Galtchouk–Kunita–Watanabe decomposition under full information via dual predictable projections. Finally we show how these results can be applied to unit-linked life insurance contracts.

Risk-minimizing portfolio selection for insurance payment processes under a Markov-modulated model

This paper extends the model in Riesner (2007) to a Markov modulated Lévy process. The parameters of the Lévy process switch over time according to the different states of an economy, which is described by a finite-state continuous time Markov chain. Employing the local risk minimization method, we find an optimal hedging strategy for a general payment process. Finally, we give anexample for single unit-linked insurance contracts with guarantee todisplay the specific locally risk-minimizing hedging strategy.

Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces. Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim. Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.

Risk Minimizing Hedging of Crude-Oil Options: Theory and Empirical Performance

We study the hedging problem for European-style options written on crude-oil futures. Locally risk-minimizing hedging strategies are derived under the assumption that the dynamics of crude-oil futures are described by a Merton-type jump-diffusion. These are then tested empirically using historical data from the NYMEX West Texas Intermediate (WTI) from the 2002-2007 period. We use Black-Scholes-Merton delta hedges as a benchmark for comparison and find that in crude-oil option markets locally risk-minimizing hedges systematically outperform the benchmark by a margin of 13%-18%. In addition to this real world performance test, we also investigate hedging strategies for robustness against model uncertainty and mis-specification, using simulated data. Our results show that locally risk minimizing strategies are much more robust than its classical alternatives.

Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering

In this paper we propose a new, information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup market prices of traded credit derivatives are given by the solution of a nonlinear filtering problem. The innovations approach to nonlinear filtering is used to solve this problem and to derive the dynamics of market prices. More- over, the practical application of the model is discussed: we analyse calibration, the pricing of exotic credit derivatives and the computation of risk-minimizing hedging strategies. The paper closes with a few numerical case studies.

Locally risk-minimizing hedging strategies for unit-linked life insurance contracts under a regime switching Lévy model

This paper extends the model and analysis in that of Vandaele and Vanmaele [Insurance: Mathematics and Economics, 2008, 42: 1128–1137]. We assume that parameters of the Levy process which models the dynamic of risky asset in the financial market depend on a finite state Markov chain. The state of the Markov chain can be interpreted as the state of the economy. Under the regime switching Levy model, we obtain the locally risk-minimizing hedging strategies for some unit-linked life insurance products, including both the pure endowment policy and the term insurance contract.