Paper: arXiv 2305.10849

Authors: Alexander Gairat, Vadim Shcherbakov

Abstract

This paper concerns a local volatility model in which volatility takes two possible values, and the specific value depends on whether the underlying price is above or below a given threshold value. The model is known, and a number of results have been obtained for it. In particular, option pricing formulas and a power law behaviour of the implied volatility skew have been established in the case when the threshold is taken at the money. In this paper we derive an alternative representation of option pricing formulas. In addition, we obtain an approximation of option prices by the corresponding Black-Scholes prices. Using this approximation streamlines obtaining the aforementioned behaviour of the skew. Our approach is based on the natural relationship of the model with Skew Brownian motion and consists of the systematic use of the joint distribution of this stochastic process and some of its functionals.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced stochastic calculus and functional properties of Skew Brownian motion to derive alternative option pricing formulas and asymptotic skew behavior. However, it is purely theoretical with no real-world market data backtesting or empirical validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive option pricing & ATM skew<br/>for discontinuous local volatility model] --> B[Input Model: 2-state local volatility<br/>with threshold at-the-money]
    B --> C[Methodology: Apply Skew Brownian Motion<br/>& joint density of its functionals]
    C --> D[Derive alternative exact<br/>option pricing formulas]
    C --> E[Develop Black-Scholes<br/>price approximation]
    D & E --> F[Outcomes: Streamlined proof of power law<br/>implied volatility skew & efficient pricing]