Differentiating a Carr–Madan Option Pricing Integral by a Model Parameter

Summary

The note asks how to differentiate a Carr–Madan Fourier integral for option prices with respect to a model parameter. Since the integration limits do not depend on that parameter, the proposed application of differentiation under the integral sign leaves the Fourier kernel unchanged and differentiates the characteristic-function term. Applying the chain rule gives the exponential characteristic-function term multiplied by the parameter derivative of its exponent.

The author reports that evaluating the resulting expression with an FFT gives implausible derivative values, despite checking the exponent derivative separately with another program. The document poses the calculation as a question and supplies no resolution, numerical example, or diagnosis. In particular, differentiating under the integral requires suitable regularity and integrability conditions, and an FFT implementation also has discretization and transform conventions that can affect results. The proposed symbolic step alone therefore does not establish that the numerical derivative is correct.

Key ideas

• The proposed derivative uses differentiation under the integral sign because the bounds are parameter-independent.
• The chain rule multiplies the exponential characteristic-function term by the derivative of its exponent.
• The author reports implausible FFT results despite separately checking the exponent derivative.
• The document does not resolve the issue or verify the required analytic and numerical conditions.

Tags

# How to derive Parameter Derivative within an FFT integral


# How to derive Parameter Derivative within an FFT integral












I have the following function (Carr-Madan) of which I am trying to take the derivative wrt $\theta$:

$c(k)=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} e^{\phi_T(u-(\alpha+1)i,\theta)} du$

Since the integration bounds are not dependent on $\theta$, according to the Leipniz integration rule, I derive the following:

$\frac{dc(k)}{d\theta}=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} \frac{d(e^{\phi_T(u-(\alpha+1)i,\theta)})}{d\theta} du$

$=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} e^{\phi_T(u-(\alpha+1)i,\theta)} \frac{d\phi_T(u-(\alpha+1)i,\theta)}{d\theta} du$

Essentially we can evaluate the term inside the integral of $c(k)$ and multiply such by $\frac{d\phi_T(u-(\alpha+1)i,\theta)}{d\theta}$. Then I use FFT to evaluate that the entire integral expression.

Now I evaluate this term numerically and get complete nonsense for the derivative term. I have already checked numerically if $\frac{d\phi_T(x,\theta)}{d\theta}$ is correct and used other programs (Mathematica) to check its correctness. Have I come to the derivative term correctly?

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