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References for Pricing Vanilla Options in Affine SV Jump-Diffusion Models

Article Quant Q&A · Author: user11881
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Summary

The document asks where to find vanilla call pricing formulas for affine stochastic volatility jump-diffusion models, including SVJ and SVJJ, and gives Heston’s characteristic-function approach as a comparison. It points readers to three academic sources covering related pricing formulas, along with two books on quantitative finance and quadrature-based option pricing.

The material is primarily a reference guide rather than a derivation: it does not present the requested SVJ or SVJJ formulas or compare model specifications. The cited sources may help readers locate formulas for European vanilla options under different stochastic processes, but the document does not explain which reference treats which jump model or whether the formulas match a particular parameterization. Readers will need to consult the sources and verify those details themselves.

Key ideas

  • The question concerns vanilla call pricing in affine stochastic volatility jump-diffusion models such as SVJ and SVJJ.
  • Heston pricing through a characteristic function is offered as a baseline for comparison.
  • The responses recommend academic papers and two books as places to look for pricing formulas.
  • The document does not derive formulas or specify which references cover each model variant.

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OptionsVolatilityDerivatives pricing
Full text
# Where to find pricing formulas for affine stochastic volatility jump-diffusion models?


# Where to find pricing formulas for affine stochastic volatility jump-diffusion models?












Does anyone know a reference where I can find the pricing formulas for vanilla calls in the affine stochastic volatility jump diffusion class of models such as SVJ and SVJJ?

I am looking for something analogous to the following formulas which apply to the Heston (square root) affine stochastic volatility model:

\begin{align} c(t) & = \frac{e^{-\alpha\log K}}{\pi}\int_0^\infty dv e^{-i v \log K}\rho(v) \\ \rho(v) & = \frac{e^{-r(T-t)}\phi(v-i(\alpha+1);T)}{\alpha^2+\alpha-v^2 + i(2\alpha+1)v} \\ \phi(u;T) & = \mathbb{E}^{Q_B}_t[e^{i u \log S(T)}], \\ \phi(u;T) & = e^{i u[\log S(t)+(r-\delta)(T-t)]-\frac{1}{\sigma_v^2}\left[\bar{v}\kappa\left(a(T-t) + 2\log\beta\right)+v_0 \gamma \right]} \\ \beta & = \frac{1-ge^{-d (T-t)}}{1-g} \\ \gamma & = \frac{a(1-e^{-d (T-t)})}{1-g e^{-d (T-t)}} \\ d & = \sqrt{(i\rho \sigma_v u - \kappa)^2 + \sigma_v^2(iu + u^2)} \\ g & = a/b \\ a & = i\rho\sigma_v u-\kappa + d \\ b & = i\rho \sigma_v u-\kappa - d \end{align}

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/22155

Do these work for you?

P34 of http://web.mit.edu/junpan/www/SVJ.pdf

P1360 of http://www.darrellduffie.com/uploads/pubs/DuffiePanSingleton2000.pdf

P2045 of http://www.math.ku.dk/~rolf/bakshi.pdf

## Answer by roym00 (score 1)

https://quant.stackexchange.com/a/21107

One of these two books may help you:

- A Guide to Quantitative Finance

- Option Pricing via Quadrature

They are both from the same author. Both price European vanilla options under various stochastic processes.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.