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

2025/03/28 by Alòs, Elisa, Burés, Òscar, Vives, Josep
#60H07 60J76 91G20 91G60 (Primary) #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.22282

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.

Related

• The implied volatility of Forward-Start options: ATM short-time level,\n skew and curvature
• On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model
• Short-time asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps
• A new look at short-term implied volatility in asset price models with jumps
• Short-time at-the-money skew and rough fractional volatility