Characteristic Functions and State Variables in Fourier Option Pricing

Summary

The document raises a notation issue in Fourier option pricing with the Heston model: some characteristic functions include a term for the current log-price, while others show only terms involving the model’s variance state. The distinction is whether the function represents the distribution of the future log-price itself or a conditional return or log-price change with the current level removed. A characteristic function for the future level carries the current price contribution; one for the increment can omit it.

For an OTC option priced with the Lewis transform, the relevant choice depends on the variable used in the transform and the payoff representation. The question provides no answer or derivation, so it does not establish a preferred convention or demonstrate a pricing formula. Readers should check the definitions of the random variable, initial state, and transform inputs in the chosen source, and ensure that the current price is included exactly once in the pricing setup.

Key ideas

• Characteristic-function formulas can differ because they describe different random variables.
• A future log-price includes the current log-price state contribution.
• A log-price increment can omit the current price term.
• Fourier pricing requires matching the characteristic function to the transform’s variable and payoff representation.
• The document poses the convention question but does not supply a derivation or answer.

Tags

# characteristic function - fourier pricing


# characteristic function - fourier pricing












Some literature states that, for instance for the Heston model, the characteristic function is given by: $$\varphi_{\mathrm{H}}(u, t, T)=\exp \left(A(u, t, T)+B(u, t, T) V(t)+i u X(t)\right)$$

Other literature states that the characteristic function is given by $$\varphi_{\mathrm{H}}(u, t, T)=\exp \left(A(u, t, T)+B(u, t, T) V(t)\right)$$,

without the $i u X(t)$ part. Why is someone including the last term, and other do not? I will price over-the-counter options using Lewis Fundamental Transform. Which of these should I use?

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