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arxiv: 2605.25450 · v1 · pith:ZPZJHXH5new · submitted 2026-05-25 · 💱 q-fin.MF · q-fin.RM

Valuation of Variable Annuities with Equity Protection Swaps under Jumps and Default Risks

Marek Rutkowski , Huansang Xu This is my paper

Pith reviewed 2026-06-29 19:43 UTC · model grok-4.3

classification 💱 q-fin.MF q-fin.RM
keywords equity protection swapsvariable annuitiesMerton's jump-diffusioncounterparty default riskclosed-form valuationput-call parityhedging under jumpsdefault-adjusted premiums
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The pith

Closed-form valuation formulas for equity protection swaps are obtained under Merton's jump-diffusion and an independent default time model, with default risk producing unhedgeable residual losses that adjust initial premiums.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds pricing frameworks for equity protection swap products that incorporate both market jumps and counterparty default. It derives explicit formulas and put-call parity relations for the embedded European options when the equity follows Merton's jump-diffusion dynamics and default occurs at an independent random time. Hedging analysis shows that static positions fully replicate the product when default is absent, yet default leaves residual losses that cannot be hedged away. These losses are measured explicitly and used to define default-adjusted starting premiums in both the Black-Scholes and jump-diffusion settings. Numerical examples quantify how jump intensity and default probability change hedging costs and required premiums.

Core claim

Under Merton's jump-diffusion model for the equity price and Szimayer's independent random time model for default, closed-form valuation formulas and put-call parity relations for European options are derived. Hedging strategies for equity protection swaps remain effective in the absence of default, but counterparty default risk produces residual losses that cannot be fully hedged; these losses are quantified and employed to define default-adjusted initial premiums under both Black-Scholes and jump-diffusion dynamics.

What carries the argument

Closed-form valuation formulas obtained by combining Merton's jump-diffusion equity dynamics with Szimayer's independent random time default model.

If this is right

  • Static hedging replicates equity protection swaps exactly when default risk is absent.
  • Default risk requires explicit adjustment of the initial premium to cover expected residual losses.
  • Higher jump intensity or default probability increases both hedging costs and the size of the default adjustment.
  • Put-call parity continues to hold for the European options embedded in the swaps under the combined jump and default dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same residual-loss adjustment could be applied to other variable-annuity guarantees that embed similar protection features.
  • Dynamic hedging strategies might reduce but not eliminate the default-induced losses, suggesting a need to quantify the residual after dynamic rebalancing.
  • Calibration of jump and default parameters to observed option and credit spreads would be required before the formulas can be used for live pricing.

Load-bearing premise

Merton's jump-diffusion for equity prices together with an independent random time for default accurately represent the market and credit risks relevant to the equity protection swaps.

What would settle it

Market prices of equity protection swaps that deviate systematically from the closed-form values predicted by the jump-diffusion plus independent default model, after calibration to the same jump and default parameters.

Figures

Figures reproduced from arXiv: 2605.25450 by Huansang Xu, Marek Rutkowski.

Figure 1
Figure 1. Figure 1: Examples of Swap provider’s cash flow after hedge when only third party default [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: EPS provider’s general cash flow after hedge when only third party default [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
read the original abstract

This paper examines the valuation and hedging of standard equity protection swap (EPS) products proposed by Xu et al.. To account for financial crises and counterparty default risk, we develop pricing frameworks based on Merton's jump-diffusion model and Szimayer's independent random time default model, under which closed-form valuation formulas and put-call parity relations for European options are derived. Hedging strategies for EPS products are analysed under jump and default risks. While static hedging remains effective in the absence of default, counterparty default risk leads to residual losses that cannot be fully hedged. These losses are quantified and used to define default-adjusted initial premiums under both Black-Scholes and jump-diffusion settings. Numerical results illustrate the effects of jump characteristics and default intensity on hedging costs and premiums, highlighting the importance of incorporating crisis and credit risks in EPS pricing and risk management.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript examines valuation and hedging of equity protection swap (EPS) products for variable annuities. It incorporates Merton's jump-diffusion dynamics for the equity price and Szimayer's independent random-time default model to derive closed-form valuation formulas and put-call parity relations for European options. Hedging analysis shows that static hedging is effective absent default but that counterparty default produces unhedgeable residual losses; these losses are quantified to define default-adjusted initial premiums under both Black-Scholes and jump-diffusion settings. Numerical illustrations examine the sensitivity of hedging costs and premiums to jump intensity/size and default intensity.

Significance. If the closed-form derivations hold, the work supplies a tractable extension of standard EPS pricing that explicitly quantifies the impact of jumps and unhedgeable default risk on hedging performance and premium adjustments. The reliance on established, analytically convenient models (Merton, Szimayer) is a practical strength, enabling explicit formulas rather than purely numerical methods. The demonstration that default risk leaves residual losses even under static hedging provides a clear, falsifiable insight for risk management of variable annuities.

minor comments (2)
  1. The abstract states that closed-form formulas are derived, but the introduction should explicitly reference the sections (e.g., §3 or §4) containing the derivations and state the precise assumptions under which the put-call parity holds.
  2. Numerical results are mentioned but the manuscript should include a brief description of the parameter values chosen for the jump intensity, jump-size distribution, and default intensity, together with a short robustness check varying one parameter at a time.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive overall assessment. The recommendation for minor revision is appreciated. Since no specific major comments were raised, we will focus on addressing any minor issues identified during the revision process to strengthen the presentation of the closed-form results and numerical illustrations.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained on external models

full rationale

The paper derives closed-form valuation formulas and put-call parity relations for EPS products by applying Merton's jump-diffusion equity model and Szimayer's independent random-time default model. These are standard external frameworks with no indication that any result reduces by construction to a fitted input, self-definition, or self-citation chain. The reference to EPS products proposed by Xu et al. provides context for the instrument but does not serve as a load-bearing premise that forces the pricing formulas. Hedging analysis and default-adjusted premiums follow directly from the stated dynamics without circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on two standard but non-trivial modeling assumptions drawn from prior literature; free parameters such as jump intensity, jump-size distribution, and default intensity are implicit and typically calibrated to market data.

free parameters (2)
  • jump intensity and size parameters
    Merton's model parameters that control frequency and magnitude of jumps; required for closed-form expressions and numerical results.
  • default intensity
    Parameter governing the random default time in Szimayer's model; directly affects residual loss calculations and adjusted premiums.
axioms (2)
  • domain assumption Equity price follows Merton's jump-diffusion process
    Invoked to obtain closed-form option prices under jumps.
  • domain assumption Counterparty default time is independent of the equity process and follows Szimayer's random-time construction
    Used to separate default risk from market risk and derive adjusted premiums.

pith-pipeline@v0.9.1-grok · 5672 in / 1320 out tokens · 33081 ms · 2026-06-29T19:43:08.360849+00:00 · methodology

discussion (0)

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Reference graph

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