add damping factor to support integration of Π1 and Π2 integrands which are singular at 0

[crowlogic]

3.2 Endpoint singularities and infinite intervals The methods in Pari/GP and mpmath are designed to support typical integrals with infinite intervals or endpoint singularities, which often arise in applications. Arb requires finite a, b and a bounded Rf to return aR finite result, but a manual truncation may be provided (say 0 f (x)dx ≈ ε f (x)dx) to work around this restriction. Tail bounds must then be added based on symbolic knowledge about f . This is not ideal in terms of usability or efficiency, but since the Petras algorithm works well even with an endpoint very close to a singularity (or ∞).

This is required for FFT pricing so needs to be done anyway

[crowlogic]

Numerical integration in arbitrary-precision ball arithmetic.pdf

[crowlogic]

The damping factor method, when applied properly, can yield an exact result for the call price in the Heston model via inverse Fourier transform. The damping factor method involves modifying the integrand with a damping factor and compensating for the damping in the inverse Fourier integral.

Here's an outline of the steps to compute the call price with the damping factor method:

• Introduce a damping factor: Multiply the characteristic function with e^(iωη), where η is a real damping factor, and i is the imaginary unit. This can help in dealing with the singularity at ω=0.

• Modify the Fourier integral: Modify the Fourier integral by dividing it by the damping factor e^(iωη). This ensures that the introduction of the damping factor does not affect the final result. The modified Fourier integral can now be written as:

C(S, K, T) = e^(-rT) (S Π1 - K * Π2),

where Π1 and Π2 are the modified integrals.

• Compute the modified integrals: Compute the modified integrals Π1 and Π2 numerically using a suitable integration method, such as quadrature rules or adaptive integration.

• Compute the call price: Use the modified integrals Π1 and Π2 to compute the call price, as mentioned in step 2.

By introducing the damping factor and modifying the Fourier and inverse Fourier integrals appropriately, you can obtain an exact result for the call price. The damping factor method helps in converging the integrals faster and avoids the singularity issue at ω=0. However, it is still essential to choose a suitable value for the damping factor η and test the accuracy and stability of the implementation.