Extending the Bates Characteristic Function with Variance Jumps

Summary

The document seeks a characteristic function for a Heston stochastic volatility model that includes jumps in both price and variance, describing this as an extension of the Bates model. It provides a reference form for the Bates characteristic function, decomposed into terms for the initial asset price and carry, Heston variance dynamics, and price jumps. It also defines auxiliary quantities used in that expression, including the complex root and ratio associated with the Heston component.

The material gives a target structure and notation, but it does not supply the requested extension with variance jumps, derive the formula, or specify jump dynamics for variance. It cites a textbook as the source of the Bates example, but offers no numerical results, validation, or discussion of parameter conventions. The useful takeaway is the decomposition of a price-jump stochastic volatility characteristic function and the open modeling step required to add variance jumps.

Key ideas

• The request concerns a Heston model with jumps in both asset price and variance.
• The document gives a Bates characteristic function as a reference for incorporating price jumps.
• The displayed expression separates the Heston dynamics from the price-jump contribution.
• A variance-jump specification and its corresponding characteristic function are not provided.

Tags

# Characteristic function for heston model with jumps in price and variance


# Characteristic function for heston model with jumps in price and variance












I need the characteristic function of the Heston model with jumps in price and variance, or in other words, the characteristic function of the Bates model (1996) adding jumps in the variance dynamics. An example of the Bates characteristic function is given in the book of Gilli, Maringer, Schumann:

φBates = e^(A+B+C+D),

where

A =iωs0+iω(r−q)τ ;

B =θκ/σ^2((κ − ρσiω − d)τ − 2 log((1 − ge^(−dτ))/(1 − g)) ;

C =v0/σ^2((κ − ρσiω − d)(1 − e^(−dτ))/(1 − ge^(−dτ)) ;

D =−λμJiωτ+λτ((1 + μJ)^(iω)e^((1/2)σj^2iω(iω−1))−1 ;

d =squareroot((ρσiω − κ)^2+σ^2(iω + ω2)) ;

g =(κ−ρσiω−d)/(κ−ρσiω+d)

I need something like this. A characteristic function in this terms.

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