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A functional Itô’s calculus approach to convex risk measures with jump diffusion

https://doi.org/10.1016/j.ejor.2015.10.032Get rights and content

Highlights

  • A functional Ito’s calculus approach is adopted to evaluate convex risk measures.

  • A non-Markovian jump-diffusion model is considered.

  • Functional partial differential-integral equations for convex risk measures are obtained.

  • An entropic risk measure is also considered.

  • Partial differential-integral equations for convex risk measures are obtained in the Markovian case.

Abstract

Convex risk measures for European contingent claims are studied in a non-Markovian jump-diffusion modeling framework using functional Itô’s calculus. Two representations for a convex risk measure are considered, one based on a nonlinear g-expectation and another one based on a representation theorem. Functional Itô’s calculus for càdlàg processes, backward stochastic differential equations (BSDEs) with jumps and stochastic optimal control theory are used to discuss the evaluation of convex risk measures. FPDIEs and PDIEs for convex risk measures are derived in the Markovian and non-Markovian situations, respectively. An entropic risk measure, which is a particular case of a convex risk measure, is discussed.

Introduction

Quantitative risk management is one of the important topics in the interface between modern finance and actuarial science. Different risk metrics have been introduced and their uses in the practices of risk management have been studied in both the finance and actuarial science literature. One of the popular risk metrics in both finance and insurance industries is Value at Risk (VaR). The history of VaR may be traced back to the well-known “4:15 report” in J.P. Morgan and an early use of VaR methodology in practice may be attributed to the Risk-Metric Group. The uses of VaR as a risk metric and for the purpose of risk regulatory capitals have once been recommended by some regulators such as the Bank for International Settlements (BIS). There has been a large amount of research on VaR in both the academic and industrial communities. The classic monograph by Jorion (1996) provides a comprehensive discussion on VaR and its practical implementation.

In a landmark paper by Artzner, Delbaen, Eber, and Heath (1999), the use of VaR for risk measurement was questioned from a theoretical point of view. In particular, it was noted in Artzner et al. (1999) that VaR does not satisfy the sub-additive property which means that diversification may increase risk. This seems counter-intuitive from the perspective of traditional finance. Artzner et al. (1999) employed an axiomatic approach to study risk measures and introduced a class of coherent risk measures satisfying four desirable properties including the sub-additive property. Two typical examples of coherent risk measures are the generalized scenario expectation and the expected shortfall. The generalized scenario expectation is given by a representation of a coherent risk measure by Artzner et al. (1999) while the expected shortfall is given by the average loss when the loss level exceeds a certain threshold given by the Value at Risk. Coherent risk measures have been applied to evaluate (unhedged) risks arising from trading derivative securities. Some works are, for example, Siu and Yang (2000), Yang and Siu (2001), Siu, Tong, and Yang (2001), Boyle, Siu, and Yang (2002), Elliott, Chan, and Siu (2006), Elliott, Siu, and Chan (2008) and Hao and Yang (2011), amongst others.

Liquidity risk has received much attention in the aftermath of the Global Financial Crisis (GFC) of 2008. Its importance has been highlighted in Basel III, where some liquidity requirements were introduced. The incorporation of liquidity in risk measurement was studied in the academic literature before the GFC. For example, Föllmer and Schied (2002) and Frittelli and Rosazza-Gianin (2002) introduced a class of convex risk measures which is a generalization of the class of coherent risk measures of Artzner et al. (1999). The key idea is that the convexity property of a risk measure can incorporate nonlinearity attributed to liquidity risk of a large trading position. A precursor of a convex risk measure is the convex premium principle pioneered by Deprez and Gerber (1985). Rosazza-Gianin (2006) studied the relationship between convex risk measures and nonlinear g-expectations, where the latter can be represented as solutions of backward stochastic differential equations. Convex risk measures have been used to evaluate (unhedged) risks arising from trading derivatives, (see, for example, Elliott, Siu, 2012, Elliott, Siu, 2013a, Elliott, Siu, 2013b, Siu, 2012, and Elliott, Siu, and Cohen, 2015).

In this paper, convex risk measures for European-style contingent claims are studied in a non-Markovian, jump-diffusion modeling framework. A key feature of such model is that the model coefficients such as the drift, volatility and jump amplitude depend on the price path of the underlying security up to the current time. Functional Itô’s calculus is used to discuss the evaluation of convex risk measures. This calculus was introduced by Dupire (2009) and was further developed in the works of Cont, Fournie, 2010, Cont, Fournie, 2013 and Levental, Schroder, and Sinha (2013), amongst others. It is an extension of classical Itô’s calculus with a view to dealing with functionals of stochastic processes. Two representations for convex risk measures are considered here. For the first representation, as in Rosazza-Gianin (2006) and Øksendal and Sulem (2009) for example, a convex risk measure is related to a nonlinear g-expectation in the jump-diffusion modeling set up, where the latter is represented as the solution of a backward stochastic differential equation (BSDE) with jumps. Using functional Itô’s calculus for RCLL, or càdlàg, processes in Levental et al. (2013), martingale representation for functionals of jump-diffusion processes as well as the unique decomposition of a special semimartingale, the solution of the BSDE corresponding to the nonlinear g-expectation, or the convex risk measure, is identified. More specifically, under some “smoothness” assumptions of the functional representing the convex risk measure, a functional partial differential-integral equation (FPDIE) satisfied by the convex risk measure is obtained while the control components of the solution of the BSDE are identified by a functional derivative and a functional difference. The second representation follows from the representation theorem of a convex risk measure in Föllmer and Schied (2002) and Frittelli and Rosazza-Gianin (2002). In this case, a convex risk measure is specified by two major components, namely a family of probability “scenarios” and a penalty function. Here the family of probability “scenarios” is specified by a family of stochastic exponentials while the penalty function is specified by the expectation of a path integral of a convex function as in Øksendal and Sulem (2009) and Delbaen, Peng, and Rosazza-Gianin (2010) for example. In this case, using the functional Itô’s calculus in Levental et al. (2013) in coupled with a BSDE approach to stochastic optimal control, a FPDIE satisfied by the convex risk measure is obtained. A particular case of convex risk measures, namely the entropic risk measure, is considered. A (F)PDIE for the entropic risk measure is derived in the (non)-Markovian situation.

Functional Itô’s calculus was used in Siu (2012) to evaluate convex risk measures for (unhedged) risks arising from trading derivative securities. The present paper is related to Siu (2012). It may perhaps be thought of as an extension of Siu (2012) to a non-Markovian jump-diffusion process, except that unlike Siu (2012), dynamic convex risk measures and forward-backward stochastic differential equations are not considered here. The non-Markovian jump-diffusion process may be quite flexible in incorporating financial and insurance risks modeled by diffusion processes and compound Poisson processes, respectively. In Bülhmann, Delbaen, Embrechts, and Shiryaev (1996), the no-arbitrage pricing under general semimartingales was considered so that a general approach may be provided for pricing financial and insurance risks. Subscribing to the general philosophy of Bülhmann et al. (1996), the evaluation of risk measures under non-Markovian jump-diffusion processes is considered here with the hope that this may provide some insights into developing a general approach for measuring financial and insurance risks. In Siu (2013), a risk-minimizing optimal asset allocation problem was considered in a non-Markovian stochastic volatility models with jumps. Siu and Shen (2014) considered a risk-minimizing pricing of contingent claims in a non-Markovian, regime-switching jump-diffusion model. Convex risk measures and BSDEs were used in Siu (2013) and Siu and Shen (2014). However, functional Itô’s calculus was not adopted in Siu (2013) and Siu and Shen (2014). In Jazaerli and Saporito (2013), functional Itô’s calculus was used to discuss the computation of Greeks of derivative securities in a non-Markovian diffusion situation.

The rest of the paper is organized as follows. In the next section, the modeling framework, a convex risk measure as well as its relationship with a nonlinear g-expectation and a BSDE with jumps are briefly discussed. In Section 3, using functional Itô’s calculus, the FPDIE and the PIDE satisfied by the convex risk measure are derived. The second representation for the convex risk measure is considered in Section 4. The entropic risk measure is discussed and the corresponding (F)PDIE is derived in Section 5.

Section snippets

Convex risk measures in non-Markovian jump-diffusion

A standard continuous-time financial model with two tradable primitive securities, namely a risk-free bond and a risky share, is considered. The time horizon of the financial model is the finite horizon T:=[0,T], where T < ∞. As usual, uncertainty is described by a complete, filtered probability space (Ω,F,F,P), where P is a reference probability measure from which a family of real-world probability measures is generated and F is a filtration {F(t)|tT} satisfying usual conditions, say the

BSDEs with jumps and functional Itô’s calculus

In this section the solution of the BSDE with jumps corresponding to the convex risk measure, or the nonlinear g-expectation, is identified using functional Itô’s calculus with jumps in Levental et al. (2013). More specifically, the first component of the solution, which is the convex risk measure, is identified as the solution of a functional partial differential-integral equation (FPDIE). Whereas, the control components of the solution are identified as a functional derivative and a

Representation of convex risk measures based on stochastic exponentials

Instead of representing the convex risk measure as a nonlinear g-expectation, we consider here an alternative representation of the convex risk measure based on the representation theorem in Föllmer and Schied (2002) and Frittelli and Rosazza-Gianin (2002) which is stated as follows:

Theorem 3

A convex risk measureρ:Hp is represented as: ρ(H)=supQP{EQ[H]η(Q)},HHp,for some familyP of probability measures which are absolutely continuous with respect toP and some convex “penalty” functionη:P.

FPDIE and PDIE for entropic risk measure

In this section a particular case of convex risk measures, namely an entropic risk measure, is considered. By specializing some results in Section 4, a FPDIE for the entropic risk measure is derived. Again in the Markovian case, the corresponding PDIE for the entropic risk measure is obtained.

An entropic risk measure is a convex risk measure with penalty function given by a relative entropy between two measures Pθ and P. It is related to an exponential premium principle in actuarial science,

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