TL;DR for operators
Discounting is the quiet plumbing of derivatives. Most option-pricing systems assume a risk-free asset sits somewhere in the background, calmly providing the rate at which future payoffs become present prices. This paper asks what happens when that safe haven is unavailable, unreliable, or merely too theoretical to be useful. Its answer is not to abandon discounting, but to manufacture it from the relative dynamics of two risky assets.1
The mechanism is the important part. The paper extends the Lindquist–Rachev option-pricing framework by giving two risky assets a shared Lévy jump driver. From their drifts, volatilities, and jump sensitivities, it derives an endogenous “shadow” short rate, then uses that rate inside a partial integro-differential equation, or LR-PIDE, for option prices. In plainer desk language: if both assets are exposed to the same jump weather, their relative behaviour can imply the rate the market is behaving as if it were using.
The empirical evidence is useful but not broad enough to deserve a victory parade. On S&P 500 options with an approximately 5.25-month maturity, CGMY delivers the lowest relative RMSE at 8.9%, followed by NIG at 9.5%, with Black–Scholes at 11.2%. The direction is unsurprising: jump models are better at pricing crash insurance than a Gaussian model that still thinks tails are a minor inconvenience.
For operators, the business use is twofold. First, Lévy jump models can improve pricing and risk measurement for skewed option surfaces. Second, the shadow rate can be treated as a market-implied stress diagnostic, especially when Treasury yields are too slow, policy-shaped, or institutionally neat to reflect actual funding conditions. That does not make it a tradable rate. Nor does it make jump risk fully hedgeable. The model gives a sharper instrument panel, not a free arbitrage machine with leather seats.
The rate is not assumed; it is extracted
Classical option pricing begins with a convenient object: the riskless asset. It might be a bank account, a Treasury bill, a money-market account, or some theoretical cousin with no default, no liquidity problem, and no unpleasant behaviour after lunch. The asset allows the model to discount future payoffs and define risk-neutral drifts.
The Lindquist–Rachev move is more interesting. Instead of assuming a riskless asset, it asks whether two risky assets can imply the same economic function. One asset can serve as a numéraire, or the two can be used to construct a locally riskless portfolio. The resulting shadow rate $\bar{r}(t)$ is not a policy rate. It is the rate implied by the relative price dynamics of the risky assets themselves.
The paper’s extension is to stop pretending markets move only continuously. It models two assets, $S(t)$ and $Z(t)$, driven by both Brownian noise and a common Lévy jump process. Their jump times are shared, but their jump sensitivities may differ. That distinction matters. If both assets jump in perfect proportion, the common jump cannot really be neutralised; every portfolio inherits the same lurch. If the jump loadings differ, the relative movement contains more information, and, under the model’s assumptions, can help span the risk.
The shadow rate is given in the paper as:
The first term is the diffusion-style Lindquist–Rachev component: it uses the two assets’ drifts and volatilities. The second term is the jump wedge: it appears because the two assets react differently to the shared jump driver. That wedge is the whole point of importing Lévy processes into the framework. A crash, a policy shock, a liquidity squeeze, or a crypto-wide liquidation does not arrive politely as small Gaussian noise. It jumps. The model wants the discounting machinery to know that.
Common jumps turn pricing into a two-asset problem, not a prettier Black–Scholes
The paper’s market has no traditional riskless asset. It has two risky assets and two risk factors: a Brownian component and a common Lévy jump component. Under a selected risk-neutral measure, both assets are assigned the shadow drift $\bar{r}(t)$, while the jump component enters through the compensated jump measure.
This leads to the LR-PIDE for an option $C(t,S,Z)$ with terminal payoff $H(S,Z)$:
This is not just Black–Scholes with a more expensive alphabet. The pricing operator now has three features that change interpretation.
First, the drift term is $\bar{r}(t)(SC_S+ZC_Z)$, not merely $rS C_S$. The option’s value is sensitive to the joint scaling direction of both risky assets. That is natural in a framework where the “riskless” function is produced by their relative behaviour.
Second, the integral term prices discontinuity. Instead of only asking how the option behaves under infinitesimal movement, the equation asks what happens when both assets jump together, with possibly different magnitudes.
Third, the discount rate is endogenous. It is not imported as the harmless green line from the yield curve spreadsheet, although the paper does use Treasury rates as an initial benchmark and comparison point in its calibration workflow. That is not hypocrisy. It is an implementation compromise. Models, like banks, often need a starting balance.
The paper’s methods are mostly machinery, but useful machinery
Once the LR-PIDE is established, the paper turns to solution methods. The conceptual solution is the usual risk-neutral expectation, except discounted by the shadow rate:
That representation is theoretically clean, but traders and risk systems need numbers. The paper therefore relies on characteristic functions and Fourier pricing methods.
The chosen jump specifications are familiar in quantitative finance:
| Lévy model | What it contributes | Practical reading |
|---|---|---|
| NIG | Semi-heavy tails and skew through a Normal Inverse Gaussian law | Good for equity-return asymmetry with relatively stable calibration |
| CGMY | Flexible tempered-stable jumps with parameters for activity, tail decay, and small-jump structure | Better smile flexibility, especially in the wings, but more calibration risk |
| VG | A CGMY limiting case with finite variation and infinite activity | Useful benchmark family; less central in the reported empirical comparison |
The computational section then maps the theory into a workflow: gather market data, estimate historical volatilities, initialise the shadow rate from a Treasury benchmark, calibrate Lévy parameters by minimising RMSE, compute risk-neutral drifts, update the shadow rate, and iterate until convergence. Figure 1 is best read as an implementation detail, not main evidence. It tells a quant developer how the authors imagine the loop, not whether the framework earns money.
The FFT and COS methods play a similar role. They are not the paper’s conceptual novelty; they are how the model becomes computationally usable. FFT is attractive when pricing many strikes for a maturity. COS is efficient for individual or moderate batches of options and can handle payoff expansion neatly. Both rely on the same useful fact: Lévy models often have tractable characteristic functions, which makes transform pricing possible without simulating every unpleasant jump path by hand. There are healthier hobbies.
The evidence says “better smile fit,” not “solved derivatives”
The main empirical pricing evidence is the S&P 500 option calibration. The paper reports results for an approximately 5.25-month maturity:
| Model | Parameters reported | Relative RMSE | Interpretation |
|---|---|---|---|
| Black–Scholes | $\sigma = 0.1579$ | 11.2% | Baseline Gaussian model; weakest fit |
| NIG | $\alpha=8.214$, $\beta=-1.235$, $\delta=0.184$ | 9.5% | Better fit from skew and heavy tails |
| CGMY | $C=1.128$, $G=12.347$, $M=14.562$, $Y=0.312$ | 8.9% | Best fit, helped by extra smile-shape flexibility |
The magnitude is worth reading carefully. CGMY improves relative RMSE by 2.3 percentage points versus Black–Scholes and by 0.6 percentage points versus NIG. That is meaningful in option-pricing calibration, but it is not a universal declaration that CGMY is “the model.” The paper itself is more sober: CGMY can be more flexible, but it can also be less stable. Multiple local minima can produce similar errors, especially because parameters such as $C$ and $Y$ can trade off against each other.
The parameter interpretation is also coherent. In the CGMY fit, $M=14.562$ is greater than $G=12.347$, which the paper interprets as a heavier negative jump tail relative to the positive side. In the NIG fit, $\beta=-1.235$ points in the same direction: negative skew. That matches the economics of index options. Investors pay for downside protection, and the option surface remembers crashes even when the index is pretending to be composed.
The risk-neutral moment discussion reinforces this. The paper reports risk-neutral variance above realised variance, strongly negative skewness of around -1, and excess kurtosis of 6 or more. The economic reading is simple: the risk-neutral distribution prices more downside jump intensity than the realised historical distribution. That gap is not a bug; it is the jump risk premium.
The shadow rate is a diagnostic, not a money printer
The paper also extracts time-varying shadow rates from SPX–NDX and BTC–ETH pairs between 2020 and 2024, then compares them with the three-month U.S. Treasury-bill yield. Figure 2 is best read as exploratory diagnostic evidence. It does not validate the entire pricing system by itself. It shows that the shadow-rate construct produces economically interpretable movements.
The reported behaviour is striking. For equities, the shadow rate falls to deeply negative values during March 2020 stress, around -50% in the paper’s discussion. For crypto, it surges during early 2021 exuberance, with values around +100%. These are not normal “interest rates” in the banking sense. They are relative-market stress readings expressed in rate form. Treat them like a warning gauge, not a deposit product.
The arbitrage language needs discipline. The paper notes that when the shadow rate exceeds the benchmark risk-free rate, borrowing at the benchmark and investing in the $S$–$Z$ portfolio may suggest arbitrage. The word “may” is doing actual work. Transaction costs, shorting constraints, funding limits, margin calls, liquidity gaps, and instrument mismatch can eat theoretical arbitrage for breakfast and ask for seconds.
A better business interpretation is this: the shadow rate can flag when relative asset dynamics are implying funding conditions that official yields do not show. In equity markets, that may help risk teams identify stress in index relationships. In crypto, it may help diagnose regime shifts between major assets where no stable, universally accepted risk-free benchmark exists. It is a market-implied stress signal with modelling assumptions attached.
How to classify the paper’s evidence
The paper contains theory, numerical workflow, empirical calibration, diagnostic plots, robustness discussion, and an appendix. These should not be treated as equal claims.
| Component | Likely purpose | What it supports | What it does not prove |
|---|---|---|---|
| LR shadow-rate derivation | Main mechanism | Shows how discounting can be generated from two risky assets with jump exposure | Does not prove the resulting rate is directly tradable |
| LR-PIDE | Main theoretical contribution | Extends no-riskless-asset pricing to common Lévy jumps | Does not eliminate all practical hedging gaps |
| FFT and COS discussion | Implementation method | Shows the framework can be priced efficiently when characteristic functions are known | Does not itself validate empirical fit |
| Calibration workflow / Figure 1 | Implementation detail | Explains how parameters and shadow rate are iteratively estimated | Does not show out-of-sample robustness |
| SPX RMSE table | Main empirical evidence | Shows NIG and CGMY outperform Black–Scholes on the reported maturity | Does not establish cross-maturity or cross-asset dominance |
| Shadow-rate plot / Figure 2 | Exploratory diagnostic extension | Suggests shadow rates move meaningfully in stress and exuberance regimes | Does not guarantee executable arbitrage |
| Calibration-seed discussion | Robustness / sensitivity test | Shows CGMY can fit well but suffer parameter instability | Does not fully map the parameter-identification problem |
| Appendix verification | Mathematical consistency check | Verifies the continuous LR PDE solution structure in the diffusion-only case | Does not separately validate the Lévy empirical model |
That table is the difference between reading a paper and being impressed by it. The paper is strongest where mechanism and pricing fit reinforce each other. It is weakest where a narrow calibration is asked to carry too much operational ambition.
What this means for derivatives desks
For a derivatives desk, the immediate relevance is smile pricing. Black–Scholes is still useful as a language, but it is a poor description of markets that price crash risk aggressively. Lévy models allow the return distribution to carry skewness, kurtosis, and jump intensity directly. The paper’s CGMY and NIG results are consistent with that practical advantage.
The desk-level question is not “Should we replace Black–Scholes?” That was settled years ago, mostly by the market humiliating anyone who took Gaussian tails too literally. The sharper question is: when does the extra flexibility of CGMY justify its calibration fragility?
The paper points to a reasonable division of labour:
| Use case | More suitable reading | Reason |
|---|---|---|
| Fast benchmark pricing | Black–Scholes | Simple baseline and communication layer |
| Stable skew-aware calibration | NIG | Fewer effective degrees of freedom; more stable estimation |
| Best fit to a single pronounced smile | CGMY | More flexible curvature and wing control |
| Multi-maturity surface production | CGMY alone is not enough | Static Lévy dynamics struggle with term-structure effects |
| Stress and funding diagnostics | Shadow rate | Converts relative asset dynamics into a rate-like signal |
The practical ROI is not magic alpha. It is fewer systematic mispricings in skewed options, better loss distribution modelling, and cleaner separation between observed Treasury rates and market-implied funding stress. That is dull in the right way. Good risk infrastructure usually is.
What this means for risk teams
Risk teams should care less about the calibration leaderboard and more about the distributional shape. A model that prices out-of-the-money puts better is usually also telling a more realistic story about tail loss. That matters for expected shortfall, scenario analysis, and hedging policy.
The framework also nudges risk teams away from treating the risk-free curve as a universal truth. In calm regimes, the paper reports that the calibrated shadow rate sits within a few basis points of the Treasury benchmark. That is comforting, but not the interesting case. The interesting case is stress, when relative asset behaviour can imply funding or liquidity conditions that official rates do not move fast enough to capture.
Cognaptus inference: a shadow-rate series could become part of a risk dashboard alongside implied volatility, skew, funding spreads, basis trades, and liquidity metrics. It should not replace those metrics. It should sit beside them as a relative-pricing stress gauge. The value is triangulation, not prophecy.
The misconception: two risky assets do not magically complete reality
The obvious wrong reading is that two risky assets replace the risk-free asset and therefore solve the hedging problem. The paper does not really say that, although some of the formal language can tempt a hurried reader.
The careful reading is more conditional. The two-asset structure can support a pricing framework under assumptions about shared risk drivers, different exposures, frictionless trading, and a selected equivalent martingale measure. But jump risk is stubborn. Large jumps occur before the hedge can be rebalanced. If the two assets jump too similarly, their combination cannot cancel the common shock. If the second asset is not liquid, available, or economically aligned with the first, the hedge is a diagram, not a trade.
The paper acknowledges this practical incompleteness. It notes that additional securities such as options or jump-risk instruments may be required, and that in empirical use one might need proxies such as related futures, variance swaps, or static hedges with out-of-the-money options.
That matters because the business value is not “riskless replication without a bond.” It is better relative pricing in a world where the clean riskless asset is sometimes a modelling convenience and sometimes a regulatory fairy tale.
Boundaries that matter before implementation
The first boundary is empirical scope. The headline calibration result is based on S&P 500 options at a reported maturity of about 5.25 months. That is enough to show promise, not enough to certify a production model across maturities, assets, and regimes.
The second boundary is parameter identification. CGMY fits better, but the paper reports sensitivity to initial seeds and local minima. When different parameter combinations produce similar RMSE, the desk can get a good price for the wrong structural reason. That is less dangerous for marking a vanilla surface, more dangerous for hedging, scenario analysis, and stress testing.
The third boundary is term structure. Pure Lévy models can capture a static smile but often struggle when the shape of the implied volatility surface changes across maturities. The paper explicitly suggests that combinations with stochastic volatility, stochastic clocks, or time-changed Lévy processes may be needed for consistent performance.
The fourth boundary is execution. A shadow-rate arbitrage signal is not an executable arbitrage. It is a model-implied discrepancy. Execution requires borrow capacity, collateral, liquidity, shorting access, basis stability, and tolerance for the market remaining irrational longer than the model remains elegant. Finance has a rich tradition of confusing those two timelines.
The operator’s implementation checklist
A desk or risk team considering this framework should not begin with the LR-PIDE. It should begin with operational questions:
| Question | Why it matters |
|---|---|
| What are the two assets, and why should they share a jump driver? | The mechanism depends on common shocks with interpretable relative exposure |
| Are their jump sensitivities different enough to contain information? | If exposures are too similar, the shadow-rate inference weakens |
| Which model is being used for which purpose: pricing, hedging, or diagnostics? | A good calibration model is not automatically a good hedge model |
| Is the target a single maturity or a full surface? | Static Lévy models are more defensible for snapshots than term structures |
| How stable are parameters under seed changes and data windows? | Calibration instability can masquerade as market insight |
| What transaction costs and funding constraints block the implied trade? | The paper’s arbitrage interpretation is theoretical unless execution survives reality |
| How is the shadow rate monitored against Treasury rates, funding spreads, and volatility metrics? | A single diagnostic should not become a religion |
The right implementation posture is incremental. Use the framework first as a research layer. Compare its option prices against current production models. Track shadow-rate deviations through known stress periods. Study parameter stability by maturity and regime. Only then decide whether it belongs in pricing, risk, or merely the excellent drawer labelled “interesting but dangerous.”
Conclusion: the safe asset was doing more work than it admitted
The paper’s contribution is not simply that CGMY beats Black–Scholes on a reported SPX calibration. That result is useful, but unsurprising. The more interesting contribution is conceptual: discounting can be reconstructed from risky relative dynamics, then carried into a jump-aware option-pricing equation.
That mechanism matters because modern markets often behave as if the official risk-free rate is only one of several discounting stories. Funding conditions, liquidity stress, collateral quality, and relative asset dynamics can all diverge from the polite benchmark curve. The shadow rate is an attempt to formalise that divergence without abandoning arbitrage logic altogether.
For business users, the framework is best treated as a disciplined diagnostic and pricing extension. It can improve smile-aware pricing, sharpen tail-risk measurement, and reveal stress signals that benchmark rates may hide. It does not abolish hedging error. It does not make jump risk disappear. It does not turn two risky assets into a Treasury bill by force of notation.
Still, the paper does something valuable. It makes the safe asset negotiable. In a market where safety often turns out to be a state-dependent opinion, that is not a small thing.
Cognaptus: Automate the Present, Incubate the Future.
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Ziyao Wang, “Lévy-Driven Option Pricing without a Riskless Asset,” arXiv:2507.20338, 2025. https://arxiv.org/abs/2507.20338 ↩︎