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Quantitative Finance · المسرد

ما معنى Carr–Madan formula؟

يُعرف أيضًا باسم: Carr--Madan formula

Definition 24.3 Derivatives and Volatility · الفصل 24 — Fourier Pricing and Calibration Engineering

The Carr–Madan formula writes the damped call eαkC(k)e^{\alpha k}C(k), α>0\alpha>0, as a Fourier integral in the log-strike: with F=1F=1,

C(k)=e−αkπ∫0∞Re⁡(e−ivkψ(v))dv,ψ(v)=P(0,T) φ(v−(α+1)i)α2+α−v2+i(2α+1)v,C(k)=\frac{e^{-\alpha k}}{\pi}\int_0^\infty\operatorname{Re}\bigl(e^{-ivk}\psi(v)\bigr)dv,\qquad \psi(v)=\frac{P(0,T)\,\varphi\bigl(v-(\alpha+1)i\bigr)}{\alpha^2+\alpha-v^2+i(2\alpha+1)v},

which on a grid in vv and kk is a discrete Fourier transform: the fast Fourier transform returns the calls on a whole grid of log-strikes at once.

Largest error over 17 strikes of a one-year Heston call against the number of terms or grid points: COS converges exponentially to the reference; the FFT converges fast and then stalls on its interpolation and damping errors. Data: the tutorial.
Figure 24.1. Largest error over 17 strikes of a one-year Heston call against the number of terms or grid points: COS converges exponentially to the reference; the FFT converges fast and then stalls on its interpolation and damping errors. Data: the tutorial.

أمثلة

Example 24.4 (Three pricers, one smile)

One-year calls in chapter 10’s base Heston model, at 17 strikes from 60 to 140, against a reference (the Lewis integral by Gauss–Legendre panels to u=1 000u=1\,000, accurate to about 10−910^{-9}; at the money it gives 6.7509). The COS method’s largest error is 0.012 with 64 terms, 8.6×10−58.6\times10^{-5} with 128 and 4.1×10−74.1\times10^{-7} with 256. Carr–Madan with interpolation between the grid’s log-strikes reaches 2.1×10−42.1\times10^{-4} with 1 024 points, 1.8×10−61.8\times10^{-6} with 4 096, and stalls near 2×10−72\times10^{-7} (Figure 24.1). The truncation interval matters as much as the terms. With [a,b][a,b] at twelve standard deviations instead of sixteen, COS stops at 3.0×10−53.0\times10^{-5} however many terms it gets, because Heston’s left tail is fatter than a normal’s.

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