The Little Heston Trap in Duffie–Pan–Singleton Transforms

Summary

The document asks whether a discounted characteristic function written in the Duffie, Pan, and Singleton representation already avoids the little Heston trap. It presents formulas for the transform’s coefficients, including a square-root term and a logarithm, and asks whether the integral component also needs modification to handle the issue.

The question concerns numerical and algebraic stability when expressing Heston-type characteristic functions, where equivalent-looking formulas can behave differently in computation. However, the document gives no answer, derivation, or numerical test, so it does not establish whether the stated representation is trap-free or what changes would be required. It also refers to a transform definition in the original paper without reproducing it. Readers should treat the formulas as the subject of an unresolved technical question rather than as a validated implementation guide.

Key ideas

• The question examines whether the Duffie–Pan–Singleton discounted transform handles the little Heston trap.
• The displayed coefficients include square-root and logarithmic terms that invite comparison with other Heston formulations.
• The author specifically asks whether the integral term also requires a change.
• No derivation or computational evidence is provided to resolve the issue.

Tags

# The little Heston Trap in DPS representation


# The little Heston Trap in DPS representation












I was wondering if the representation by Duffie, Pan, and Singleton (2000) is already accounting for the little Heston trap. DPS represent their 'general' discounted characteristic function as: $$ \begin{align} \psi(u,(y,v),t,T) = exp(\alpha(u,T-t) + yu + \beta(u,T-t)v), \end{align} $$ where

$$ \begin{align} \beta(\tau,u) &= -a\frac{1-\exp{(-\gamma\tau)}}{2\gamma-(b+\gamma)(1-\exp{(-\gamma\tau)})}\\ \alpha_{0}(\tau,u) &= -r\tau +(r-\xi)\tau u - \kappa\sigma\left(\frac{b+\gamma}{\sigma^{2}}\tau + \frac{2}{\sigma^{2}}\log\left(1 - \frac{b+\gamma}{2\gamma}\left(1-\exp{(-\gamma\tau)}\right)\right)\right)\\ \alpha(\tau,u) &= \alpha_{0}(\tau,u) - \bar{\lambda}\tau(1+\bar{\mu}u) + \bar{\lambda}\int^{\tau}_{0} \theta(u,\beta(s,u))ds,\\ a &= u(1-u),\\ b &= \sigma\rho u - \kappa,\\ \gamma &= \sqrt{b^{2} + a\sigma^{2}},\\ \bar{\lambda} &= \lambda_{y} + \lambda_{v} + \lambda_{c}. \end{align} $$ The transform $\theta(c_{1},c_{2})$ can be found in their paper. When comparing this discounted characteristic function to other variations of the original Heston characteristic function, they look quite different from each other. It is giving me a hard time to figure out if this representation already takes care of 'the little Heston trap'. In the case what would I need to change in here to handle the little Heston trap. Do I also need to change something in the integral?

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