Paper: arXiv 2407.03960

Authors: Tahir Choulli, Ella Elazkany, Michèle Vanmaele

Abstract

In this paper, we introduce the second-order Esscher pricing notion for continuous-time models. Depending whether the stock price $S$ or its logarithm is the main driving noise/shock in the Esscher definition, we obtained two classes of second-order Esscher densities called linear class and exponential class respectively. Using the semimartingale characteristics to parametrize $S$, we characterize the second-order Esscher densities (exponential and linear) using pointwise equations. The role of the second order concept is highlighted in many manners and the relationship between the two classes is singled out for the one-dimensional case. Furthermore, when $S$ is a compound Poisson model, we show how both classes are related to the Delbaen-Haenzendonck’s risk-neutral measure. Afterwards, we restrict our model $S$ to follow the jump-diffusion model, for simplicity only, and address the bounds of the stochastic Esscher pricing intervals. In particular, no matter what is the Esscher class, we prove that both bounds (upper and lower) are solutions to the same linear backward stochastic differential equation (BSDE hereafter for short) but with two different constraints. This shows that BSDEs with constraints appear also in a setting beyond the classical cases of constraints on gain-processes or constraints on portfolios. We prove that our resulting constrained BSDEs have solutions in our framework for a large class of claims’ payoffs including any bounded claim, in contrast to the literature, and we single out the monotonic sequence of BSDEs that ``naturally" approximate it as well.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework extending Esscher transforms to second-order continuous-time semimartingale models and linking pricing bounds to novel constrained BSDEs. However, it is entirely theoretical with no empirical evaluation, backtesting, or market data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Develop 2nd-Order Esscher Pricing for Continuous-Time Models] --> B[Model Stock Price S via Semimartingale Characteristics]
    B --> C[Derive Two 2nd-Order Esscher Classes: Exponential & Linear]
    
    C --> D[Analyze Compound Poisson Case]
    C --> E[Apply to Jump-Diffusion Model & Price Bounds]
    
    D --> F[Outcome 1: Link Both Classes to Delbaen-Haenzendonck Measure]
    E --> G[Outcome 2: Characterize Pricing Interval Bounds via Constrained BSDEs]
    G --> H[Outcome 3: Prove Existence of Solutions & Approximation via Monotonic BSDE Sequences]

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0 of 57 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.