[Paper Review] Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures

Pauline Barrieu, Nicole El Karoui|London School of Economics and Political Science Research Online (London School of Economics and Political Science)|Aug 7, 2007

Risk and Portfolio Optimization6 references123 citations

TL;DR

This paper proposes a framework for pricing, hedging, and optimally designing derivatives by minimizing convex risk measures, shifting from utility maximization to risk-based optimization. It establishes that optimal risk transfer between agents—especially in incomplete markets with non-tradable risks—can be solved via inf-convolution of convex risk measures, with explicit solutions under dilated or regularized conditions.

ABSTRACT

The question of pricing and hedging a given contingent claim has a unique solution in a complete market framework. When some incompleteness is introduced, the problem becomes however more difficult. Several approaches have been adopted in the literature to provide a satisfactory answer to this problem, for a particular choice criterion. In this paper, in order to price and hedge a non-tradable contingent claim, we first start with a (standard) utility maximization problem and end up with an equivalent risk measure minimization. This hedging problem can be seen as a particular case of a more general situation of risk transfer between different agents, one of them consisting of the financial market. In order to provide constructive answers to this general optimal risk transfer problem, both static and dynamic approaches are considered. When considering a dynamic framework, our main purpose is to find a trade-off between static and very abstract risk measures as we are more interested in tractability issues and interpretations of the dynamic risk measures we obtain rather than the ultimate general results. Therefore, after introducing a general axiomatic approach to dynamic risk measures, we relate the dynamic version of convex risk measures to BSDEs.

Motivation & Objective

• To address the challenge of pricing and hedging contingent claims in incomplete markets where traditional arbitrage-free pricing fails.
• To reformulate indifference pricing based on exponential utility into a convex risk measure framework that preserves key economic properties like cash translation invariance and convexity.
• To develop a general method for optimal risk transfer between agents, including financial markets and insurers, by reducing the problem to inf-convolution of convex risk measures.
• To provide constructive solutions for both static and dynamic hedging problems using BSDEs and regularization techniques.
• To ensure tractability and interpretability by linking dynamic convex risk measures to backward stochastic differential equations (BSDEs).

Proposed method

• Transforms the utility-based indifference pricing problem into a risk measure minimization framework using the certainty equivalent as a convex functional.
• Applies the inf-convolution operation to model optimal risk transfer between two agents, reducing the problem to minimizing the sum of their risk measures.
• Uses the dilatation of convex functions to achieve exact inf-convolution solutions without additional constraints.
• Employs Moreau-Yosida and Lipschitz regularization via inf-convolution with quadratic and linear kernels to ensure differentiability and stability.
• Relates dynamic convex risk measures to solutions of backward stochastic differential equations (BSDEs), enabling time-consistent risk evaluation.
• Characterizes optimal solutions using subdifferential calculus, particularly when the intersection of subdifferentials is non-empty.

Experimental results

Research questions

• RQ1 How can indifference pricing in incomplete markets be re-expressed using convex risk measures instead of utility maximization?
• RQ2 What conditions ensure the existence and uniqueness of optimal risk transfer between two agents with non-tradable risks?
• RQ3 In what cases does the inf-convolution of two convex risk measures yield an exact solution, and how can it be computed explicitly?
• RQ4 How can dynamic risk measures be constructed and linked to BSDEs to ensure time consistency and tractability?
• RQ5 What role do regularization techniques like Moreau-Yosida play in ensuring differentiability and numerical stability in risk minimization problems?

Key findings

• The certainty equivalent derived from exponential utility is a convex risk measure satisfying cash translation invariance, making it suitable for generalization to broader classes of risk measures.
• When both risk measures are dilated from a common base function, their inf-convolution is exact and yields a closed-form solution: $ g^A \square g^B = g_{\gamma_A + \gamma_B} $.
• Optimal solutions to the inf-convolution problem exist when one function is bounded below and the other satisfies a qualification condition involving its recession function.
• The inf-convolution is exact at zero if the subdifferentials of the two functions at zero intersect, and the result is centered if both functions are centered.
• Moreau-Yosida regularization ensures differentiability of the risk measure, with the gradient given by $ \nabla g_{[k]} = k(z - J_k(z)) $, and the resolvent $ J_k(z) $ is 1-Lipschitz.
• Lipschitz regularization via $ b_k(z) = k|z| $ yields a convex, Lipschitz-continuous function that converges pointwise to the original function at interior points of its domain.