Short-time behavior of the At-The-Money implied volatility for the jump-diffusion stochastic volatility Bachelier model

TL;DR

The paper analyzes the short-time behavior of at-the-money implied volatility (ATM-IV) in a jump-diffusion stochastic volatility Bachelier model and proves that the ATM-IV level converges to the instantaneous volatility $\sigma_t$ as maturity approaches the current time, independently of the jump law. Using Malliavin calculus and a Hull-White type decomposition, it derives explicit asymptotics for the ATM-IV skew that depend on the Lévy jump structure through the first moment $c_1$ and on the Malliavin regularity of the volatility process via a Hurst-like parameter $H$. The results hold first for compound Poisson jumps and extend to general pure-jump Lévy processes by truncation and approximation arguments, with distinct limits depending on whether $H\ge 1/2$ or $H<1/2$. Numerical experiments across fractional Bergomi, SABR-like, CGMY, and NIG jump settings validate the theory, showing level robustness to jump activity and skew behavior that aligns with the predicted $c_1$-dependent additive effect and $H$-dependent scaling. The findings extend short-time ATM-IV results from lognormal (Black-Scholes/Bates) frameworks to a non-lognormal Bachelier setting, offering practical guidance for calibrating models in environments where asset prices may become negative and jumps are present.

Abstract

In this paper we use Malliavin Calculus techniques in order to obtain expressions for the short-time behavior of the at-the-money implied volatility (ATM-IV) level and skew for a jump-diffusion stock price. The diffusion part is assumed to be the stochastic volatility Bachelier model and the jumps are modeled by a pure-jump Lévy process with drift so that the stock price is a martingale. Regarding the level, we show that the short-time behavior of the ATM-IV level is the same for all pure-jump Lévy processes and, regarding the skew, we give conditions on the law of the jumps for the skew to exist. We also give several numerical examples of stochastic volatilities and Lévy processes that confirm the theoretical results found in the paper.

Figures (13)

• Figure 1: In red we can see the plot of the identity map $(\sigma_0,\sigma_0)$. In blue we can see how the simulated pairs $(\sigma_0, I_0(k^*))$ with $T = 10^{-5}$ for $\sigma_0 \in \{0.1, \dots, 1.4\}$ are overlapped with the theoretical results (plotted in red).
• Figure 2: In blue, we can see the plot of the simulated map $\sigma_0 \mapsto \partial_kI_0(k^*)$ in the case where the jumps follow a $N(-0.01, 0.2)$ law. In red, we can see the plot of the simulated map $\sigma_0 \mapsto \partial_kI_0(k^*)$ in the case were the jumps follow a $N(0,0.2)$ law. In green we can see the plot of the simulated map $\sigma_0 \mapsto \partial_kI_0(k^*)$ in the case were the jumps follow a $L(0.01,1)$ law.
• Figure 3: (a) Theoretical vs experimental ATM implied volatility when the jumps follow a $L(0.1, 0.1)$ distribution. (b) Experimental skew when the jumps follow a $N(\delta, 0.2)$ distribution with $\delta \in \{-0.1, 0, 0.1\}$
• Figure 4: Plot of the maps $T \mapsto \partial_kI_0(k^*)$. In blue, we plot the map in the case where the jumps follow a $N(-0.001, 0.2)$ law ($c_1 = -0.005$). In green we plot the map in the case where the jumps follow a $N(0.001, 0.2)$ law ($c_1 = 0.005$). In red and purple we plot the map in the case where the jumps follow a $N(0,0.2)$ law and we have absence of jumps respectively ($c_1 = 0$).
• Figure 5: Overlapping of simulated pairs $(\sigma_0, I_0(k^*))$ (blue) with the theoretical curve of results $(\sigma_0,\sigma_0)$ (red).

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Remark 3.1
• Theorem 3.2
• Remark 3.3
• Remark 3.4
• Proposition 4.1
• Remark 4.2
• Remark 4.3
• proof