Interpreting Jump Terms in Stochastic Volatility Models

Summary

The document raises a notation and modeling question about the stochastic volatility with correlated jumps framework associated with Duffie, Pan, and Singleton. The author compares a log-price and variance specification that multiplies jump increments by jump sizes with a paper equation that appears to add the jump counting processes directly, and asks why the jump-size terms seem absent. The drift in the log-price equation includes a compensator involving the expected exponential jump size.

No answer or derivation is included, so the apparent discrepancy remains unresolved. The question highlights an important interpretation issue: a jump process can be written using a counting process with an implicit jump mark, or with an explicit random jump size multiplying each event. Comparing equations therefore requires checking how jump sizes and their distributions are defined. The text provides no evidence that the formulations are inconsistent, nor does it derive the log transformation from a price process; it is best read as a prompt to examine the paper’s notation and jump specification.

Key ideas

• The question compares explicit jump-size increments with jump counting processes in a stochastic volatility model.
• A compensator term in the log-price drift accounts for the expected effect of price jumps.
• Jump notation must be checked to see whether event sizes are explicit or implicit in the process definition.
• The document provides no resolution or derivation, so it does not establish that the cited formulations conflict.

Tags

# Stochastic volatility with jumps


# Stochastic volatility with jumps












I'm reading the Duffie, Pan, and Singleton (2000) paper now and I've stumbled upon something that seems to me as an inconsistency. Whenever I look up the SVJJ model, I see that its log-transform is formulated in the following way $$ \begin{align} dY_{t} &= \left(r - \frac{1}{2}V_{t} - \lambda\mathbb{E}[\mathrm{e}^{J^{Y}} - 1]\right)dt + \sqrt{V_{t}}dB^{Y}_{t} + J^{Y}dN^{Y}_{t}\\ dV_{t} &= \kappa(\theta - V_{t})dt + \sigma\rho\sqrt{V_{t}}B^{Y}_{t} + \sigma\sqrt{(1-\rho^{2})V_{t}}B^{V}_{t} + J^{V}dN^{V}_{t}. \end{align} $$ However, Duffie, Pan, and Singleton (2000) seems to just drop the $J$, i.e. they formulate the SVJJ as $$ \begin{align} dY_{t} &= \left(r - \frac{1}{2}V_{t} - \lambda\mathbb{E}[\mathrm{e}^{J^{Y}} - 1]\right)dt + \sqrt{V_{t}}dB^{Y}_{t} + dN^{Y}_{t}\\ dV_{t} &= \kappa(\theta - V_{t})dt + \sigma\rho\sqrt{V_{t}}B^{Y}_{t} + \sigma\sqrt{(1-\rho^{2})V_{t}}B^{V}_{t} + dN^{V}_{t}. \end{align} $$ (I refer to equation 4.1 in their paper). Can someone tell me why this is done or how I should understand this. Papers that perform some type of derivation from the price process to the log-transform price process would also be appreciated.

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