European baskets in discrete-time continuous-binomial market models

Paper: arXiv 2301.04996

Authors: Jarek Kędra, Assaf Libman, Victoria Steblovskaya

Abstract

We consider a discrete-time incomplete multi-asset market model with continuous price jumps. For a wide class of contingent claims, including European basket call options, we compute the bounds of the interval containing the no-arbitrage prices. We prove that the lower bound coincides, in fact, with Jensen’s bound. The upper bound can be computed by restricting to a binomial model for which an explicit expression for the bound is known by an earlier work of the authors. We describe explicitly a maximal hedging strategy which is the best possible in the sense that its value is equal to the upper bound of the price interval of the claim. Our results show that for any $c$ in the interval of the non-arbitrage contingent claim price at time $0$, one can change the boundaries of the price jumps to obtain a model in which $c$ is the upper bound at time $0$ of this interval. The lower bound of this interval remains unaffected.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides rigorous theoretical proofs regarding no-arbitrage bounds and hedging strategies for basket options in incomplete markets using advanced probability theory. However, it is purely theoretical with no empirical backtesting, market data validation, or code implementation provided.

Research Flowchart

  flowchart TD
    A["Research Goal: No-arbitrage price bounds & hedging for European basket options in continuous-jump models"] --> B["Inputs: Discrete-time multi-asset market model & European basket call option payload"]
    B --> C["Compute Lower Price Bound: Apply Jensen's Inequality"]
    B --> D["Compute Upper Price Bound: Restrict to extremal binomial model"]
    C --> E["Outcome: Lower bound equals Jensen's bound"]
    D --> F["Outcome: Explicit upper bound & optimal maximal hedging strategy"]
    F & E --> G["Key Finding: Calibrating jump boundaries shifts upper bound to any no-arbitrage price 'c' while preserving lower bound"]