Brownian Bridge™

1. Why can't the Fourier transform of the call price $C_T(k)$ be taken directly (without dampening), even though the characteristic function $\varphi_T(u)$ is well-defined?

The call price is not measurableAs

k \to -\infty

(deep ITM),

C_T(k) \to S_0

(a positive constant), so

C_T \notin L^1(\mathbb{R})

and its Fourier transform does not existThe characteristic function decays too slowly for the inversion to convergeThe call price is discontinuous at

k = \ln(S_0)

2. In the Carr-Madan formula, $\Psi_T(v) = e^{-rT}\varphi_T(v - (\alpha+1)i) / (\alpha^2 + \alpha - v^2 + i(2\alpha+1)v)$ . What happens at $v = \sqrt{\alpha^2 + \alpha}$ ?

The characteristic function

\varphi_T

has a singularityThe real part of the denominator vanishes, potentially creating a near-pole if

\varphi_T

does not cancel itThe imaginary part of

\Psi_T

changes signThe dampening condition fails and the formula is invalid

3. The FFT condition links the frequency spacing $\eta$ and log-strike spacing $\lambda$ by $\eta\lambda = 2\pi/N$ . If you need a finer log-strike grid (smaller $\lambda$ ) without reducing $\eta$ (to maintain integration accuracy), what technique is required?

Reduce

N

to satisfy the constraintUse the Fractional FFT (FRFT), which breaks the

\eta\lambda = 2\pi/N

constraint at the cost of doubling the FFT lengthUse Richardson extrapolation on a coarser gridIncrease

\alpha

to widen the integration domain

4. The Carr-Madan FFT method prices options at all $N$ strikes simultaneously in $O(N\log N)$ operations, compared to $O(N \cdot M_{\text{quad}})$ for $N$ separate numerical integrations, where $M_{\text{quad}}$ is the number of quadrature points.

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5. Aliasing in the Carr-Madan FFT occurs when:

The dampening parameter

\alpha

is chosen too largeThe characteristic function has significant magnitude at frequencies above the truncation point

v_{\max} = \eta N

, causing the periodic wrap of the FFT to contaminate the option pricesThe log-strike grid does not include the at-the-money strikeThe number of paths

N

is not a power of 2

6. For a deep out-of-the-money call ( $K \gg S_0$ ), the Carr-Madan FFT typically gives inaccurate results. What is the preferred approach?

Increase

\alpha

to improve accuracy at high strikesPrice the corresponding OTM put and use put-call parity to recover the call priceUse a finer frequency grid near

v = 0

Switch to Monte Carlo for deep OTM options

Quiz: FFT and Carr-Madan Pricing

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