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The Mean Field Market Model Revisited

Manuel Hasenbichler, Wolfgang Müller, Stefan Thonhauser

Abstract

In this paper, we present an alternative perspective on the mean-field LIBOR market model introduced by Desmettre et al. in arXiv:2109.10779. Our novel approach embeds the mean-field model in a classical setup, but retains the crucial feature of controlling the term rate's variances over large time horizons. This maintains the market model's practicability, since calibrations and simulations can be carried out efficiently without nested simulations. In addition, we show that our framework can be directly applied to model term rates derived from SOFR, ESTR or other nearly risk-free overnight short-term rates -- a crucial feature since many IBOR rates are gradually being replaced. These results are complemented by a calibration study and some theoretical arguments which allow to estimate the probability of unrealistically high rates in the presented market models.

The Mean Field Market Model Revisited

Abstract

In this paper, we present an alternative perspective on the mean-field LIBOR market model introduced by Desmettre et al. in arXiv:2109.10779. Our novel approach embeds the mean-field model in a classical setup, but retains the crucial feature of controlling the term rate's variances over large time horizons. This maintains the market model's practicability, since calibrations and simulations can be carried out efficiently without nested simulations. In addition, we show that our framework can be directly applied to model term rates derived from SOFR, ESTR or other nearly risk-free overnight short-term rates -- a crucial feature since many IBOR rates are gradually being replaced. These results are complemented by a calibration study and some theoretical arguments which allow to estimate the probability of unrealistically high rates in the presented market models.
Paper Structure (12 sections, 3 theorems, 52 equations, 6 figures, 4 tables)

This paper contains 12 sections, 3 theorems, 52 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction and Motivation
  2. Theoretical Considerations
  3. Existence and Uniqueness
  4. Total Implied Variance Structures
  5. Damping Thresholds and Market Consistency
  6. Calibration Results
  7. Calibration
  8. Calibration Results
  9. Dampening Effects
  10. Simulation Results
  11. Summary
  12. Blow Up Probabilities

Key Result

Proposition 1.1

MercurioLyashenko Let $0 < S < T$. Then, $F^B(t,S,T) = F(t,S,T) \quad \forall \, t \in [0,S]$.

Figures (6)

  • Figure 1: Total implied volatility structure, exemplary plot
  • Figure 2: Damping factor, exemplary plot
  • Figure 3: Model ATM caplet vol. versus quoted ATM caplet vol. with various expiries.
  • Figure 4: Model ATM swaption vol. versus quoted ATM swaption vol. with various expiries (x-axis) and periods to maturity (y-axis).
  • Figure 5: Empirical distribution's tails of $F_{60}(59)$ with different damping approaches.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Proposition 1.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Corollary 2.1.1
  • proof
  • Remark 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 7 more