One of the most sacred assumptions in financial modeling is the existence of a traded risk-free asset. It anchors discounting, defines arbitrage boundaries, and supports the edifice of Black–Scholes. But what happens when you remove this pillar? Can we still price options, hedge risk, or extract information about funding conditions? In a striking extension of the Lindquist–Rachev (LR) framework, Ziyao Wang shows that not only is it possible — it may reveal financial dynamics that conventional models obscure.
From Bonds to Shadow Rates
The original LR framework showed how two risky assets could span a complete market in the absence of a risk-free bond. One asset serves as the numéraire, and an endogenous shadow short rate emerges to fill the role of risk-neutral drift. Wang extends this idea to a setting where asset prices jump, skew, and fatten their tails — by introducing common Lévy jump dynamics shared across both risky assets.
This leap matters. Financial returns are riddled with discontinuities: crashes, announcements, liquidity shocks. Standard diffusions can’t capture these behaviors, but Lévy processes like NIG and CGMY can. Embedding them in a bondless pricing model challenges both our pricing methods and economic intuitions.
A Tale of Two Assets and a Jump
In Wang’s framework, two risky assets $S(t)$ and $Z(t)$ share a Brownian motion and a common Lévy jump process $L(t)$, but respond with different sensitivities $\kappa_S$ and $\kappa_Z$. This induces co-jumps — simultaneous, correlated jumps that mimic systemic shocks.
The absence of a riskless bond means no externally defined short rate. Instead, Wang derives the shadow short rate $\bar{r}(t)$ using consistency conditions between the two asset drifts under the physical and risk-neutral measures. Crucially:
$\bar{r}(t) = \frac{\mu_S \sigma_Z - \mu_Z \sigma_S}{\sigma_Z - \sigma_S} + \frac{\lambda(\kappa_Z - \kappa_S)}{\sigma_Z - \sigma_S}$
This formula not only governs discounting in pricing equations but becomes a signal of market sentiment. When calibrated to real asset pairs, $\bar{r}(t)$ exhibits sharp dives during stress and exuberant spikes during bubbles — sometimes diverging drastically from observed Treasury rates.
LR-PIDE: A New Equation for a Bondless World
Using Itô–Lévy calculus, Wang derives a partial integro-differential equation (PIDE) — the LR-PIDE — governing option prices under this jump-augmented LR framework. It has the familiar drift and diffusion terms, but with key twists:
- Discounting and drifts are driven by $\bar{r}(t)$, not an exogenous $r$.
- Jump integrals appear via the Lévy measure $\nu(dx)$, reflecting infinite activity in the case of CGMY.
- Hedging requires using both $S$ and $Z$, since they co-span the market’s jump and diffusion risks.
Numerical solutions use Carr–Madan’s FFT and COS methods, enabled by the tractable characteristic functions of NIG and CGMY.
Empirical Calibration: A Smile That Talks
Wang calibrates the model to S&P 500 index options and finds clear improvements over Black–Scholes:
| Model | RMSE | Key Parameters |
|---|---|---|
| Black–Scholes | 11.2% | $\sigma = 0.1579$ |
| NIG | 9.5% | $\alpha = 8.21, \beta = -1.24$ |
| CGMY | 8.9% | $C=1.13, G=12.35, M=14.56, Y=0.31$ |
CGMY delivers the most flexible fit thanks to its tunable jump activity and asymmetry. The negative skew and high kurtosis in parameters align with the empirical demand for crash protection — an important realism that Black–Scholes famously lacks.
Moreover, the time series of shadow rates $\bar{r}(t)$ extracted from SPX–NDX and BTC–ETH pairs show strong reactions to macro stress (e.g. March 2020 crash, 2021 crypto mania), revealing funding signals absent in conventional interest rates.
Why This Matters
This work reframes the very meaning of “risk-free” in markets where trust is scarce, or where safe assets are illiquid, manipulated, or absent — think crypto exchanges, emerging markets, or systemic stress scenarios. Rather than assuming away such realities, the LR jump model endogenizes risk-free pricing through relative arbitrage between risky assets.
It also hints at new ways of extracting market-implied funding stress. Shadow short rates, when inferred from traded asset pairs, offer a lens into liquidity sentiment and contagion dynamics — possibly more responsive than Treasury yields distorted by QE or regulatory arbitrage.
Finally, the computational tractability via characteristic functions makes this model not just theoretically rich but practically usable — for market-makers, risk managers, and researchers seeking smarter hedging in incomplete markets.
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