Fourier transform pricing

Definition Let $X = \ln S_T$ be the log of the terminal price under a pricing measure $\mathbb{Q}$. Its characteristic function is

In this lesson $\varphi$ denotes this function, not the normal density, which does not appear. It exists for every real $u$ because $\lvert e^{iuX} \rvert = 1$, it is uniformly continuous, and

It determines the distribution of $X$. If $\int_{\mathbb{R}} \lvert \varphi(u) \rvert\, du < \infty$, then $X$ has a bounded continuous density, given by

This condition is sufficient, not necessary, and without it $X$ need not have a density at all. Pricing with complex arguments of $\varphi$ needs more, namely finite moments of $S_T$, which the next card describes.

Definition For the terminal price $S_T > 0$ under $\mathbb{Q}$ let

By Hölder’s inequality $I$ is an interval. It contains $0$, and it contains $1$ when $\mathbb{E}^{\mathbb{Q}}[S_T] < \infty$, which holds when the discounted price is a $\mathbb{Q}$-martingale. For complex $u$ one has $\lvert e^{iuX} \rvert = e^{-(\operatorname{Im} u)X} = S_T^{-\operatorname{Im} u}$, so the expectation $\varphi(u) = \mathbb{E}^{\mathbb{Q}}[e^{iuX}]$ converges absolutely exactly on the horizontal strip of complex $u$ with $-\operatorname{Im} u \in I$, and it is analytic in the interior of that strip. The interval $I$, or this strip, is called the moment strip. The Fourier transform of the damped call in the next card evaluates $\varphi$ at $u = v - i(\alpha + 1)$, where $-\operatorname{Im} u = \alpha + 1$, so it needs $\alpha + 1 \in I$. In GBM the interval is all of $\mathbb{R}$, because

for every real $p$. With jumps or stochastic volatility $I$ can be bounded, and in models such as Heston its upper end depends on the parameters and on the maturity, an analysis that is model specific and not covered here. A closed-form expression for $\varphi$ often continues analytically beyond the strip, where it no longer equals the expectation.

Definition Let $k = \ln K$ be the log strike and $C(K) = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[(S_T - K)^{+}]$ the time 0 price of a European call, with a constant rate $r$ and any dividend yield already contained in the law of $S_T$. The undamped price is not integrable in $k$, because $C(K) \to e^{-rT}\mathbb{E}^{\mathbb{Q}}[S_T] > 0$ as $K \to 0$. A damping parameter $\alpha > 0$ defines the damped call $c(k) = e^{\alpha k} C(e^k)$ and its Fourier transform, for real $v$,

If $\mathbb{E}^{\mathbb{Q}}[S_T^{\alpha + 1}] < \infty$, then

and the denominator equals $\alpha^2 + \alpha - v^2 + i(2\alpha + 1)v$. To derive it, fix a value $x$ of $X$. The payoff $(e^x - e^k)^{+}$ vanishes for $k \ge x$, so

where the terms at $-\infty$ vanish because $\alpha > 0$. Taking $\mathbb{E}^{\mathbb{Q}}$ and using $\mathbb{E}^{\mathbb{Q}}[e^{(\alpha + 1 + iv)X}] = \varphi(v - i(\alpha + 1))$ gives the formula. Exchanging the expectation and the integral is justified by Fubini’s theorem, because the absolute value of the integrand has integral

which at $x = X$ is $S_T^{\alpha + 1}/(\alpha(\alpha + 1))$, with finite expectation by assumption. The same computation shows that $c$ is integrable, with $\int c(k)\,dk = \widehat{c}(0) = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[S_T^{\alpha + 1}]/(\alpha(\alpha + 1))$.

Definition With the convention $\widehat{f}(v) = \int_{\mathbb{R}} e^{ivk} f(k)\,dk$, the Fourier inversion theorem states that if $f$ and $\widehat{f}$ are both integrable, then

at every $k$ where $f$ is continuous. It applies to the damped call when $\alpha > 0$ and $\mathbb{E}^{\mathbb{Q}}[S_T^{\alpha + 1}] < \infty$. The function $c$ is integrable by the previous card and continuous because $\lvert C(K_1) - C(K_2) \rvert \le e^{-rT} \lvert K_1 - K_2 \rvert$. Its Fourier transform is integrable because $\lvert \varphi(v - i(\alpha + 1)) \rvert \le \mathbb{E}^{\mathbb{Q}}[S_T^{\alpha + 1}]$ and the denominator has modulus at least $\max\bigl(\alpha(\alpha + 1), v^2\bigr)$. Since $c$ is real, $\widehat{c}(-v) = \overline{\widehat{c}(v)}$, the two half-lines combine, and for every strike $K > 0$ with $k = \ln K$

Every admissible $\alpha$ gives the same price, and the choice changes only how well the integral behaves numerically.

Definition The discrete Fourier transform of complex numbers $a_0, \dots, a_{N-1}$ is

Evaluated term by term, the $N$ sums take about $N^2$ operations. A fast Fourier transform, or FFT, is an algorithm that computes all of them, exactly up to rounding, in a number of operations proportional to $N \log N$, for example when $N$ is a power of 2. To use it for the inversion integral, sample frequencies $v_j = j\,\Delta v$ and log strikes $k_m = k_0 + m\,\Delta k$ with

Then $e^{-iv_j k_m} = e^{-ij\Delta v k_0}\, e^{-2\pi i jm/N}$, so with quadrature weights $w_j$ the approximations

are $e^{-\alpha k_m}/\pi$ times the real parts of the discrete Fourier transform of $a_j = e^{-ij\Delta v k_0}\, \widehat{c}(v_j)\, w_j\, \Delta v$, and one FFT returns all $N$ prices. It is a fast way to evaluate the sum and nothing more. It removes neither the error of cutting the integral off at $N\Delta v$ nor the error of sampling it with step $\Delta v$. For a fixed $N$, a finer frequency step forces a coarser strike step, and the strike grid spans a log strike window of width $N\Delta k = 2\pi/\Delta v$. Implementation choices such as Simpson weights and the placement of the strike grid are beyond this lesson.

Definition A computed price replaces the integral over $[0, \infty)$ by a finite sum with step $\Delta v$ up to a cutoff $V$, and each replacement has its own error. The truncation error is the omitted tail

and the bounds of the inversion card give the simple estimate

which ignores any decay of $\varphi$ and is often far from sharp. Aliasing is the error of sampling the frequencies. Let $L = 2\pi/\Delta v$. When the shifted copies $c(k + nL)$ for $n \in \mathbb{Z}$ have a uniformly convergent sum, the Poisson summation formula, together with the absolute summability of the samples of $\widehat{c}$ that follows from the bounds of the inversion card, shows that the trapezoid approximation of the price, summed over all of $[0, \infty)$ with half weight at $0$, equals $e^{-\alpha k} \sum_{n \in \mathbb{Z}} c(k + nL)$. So the sampled price is $C(K)$ plus the images

which are prices at strikes a factor $e^{nL}$ away, wrapped back onto $K$. On an FFT grid the period $L$ equals the width of the strike window. The images with $n < 0$ are deep in-the-money calls, each about $e^{-\alpha \lvert n \rvert L} \bigl(S_0 e^{-qT} - K e^{-\lvert n \rvert L - rT}\bigr)$, so a small $\alpha$ or a coarse step makes them large. The images with $n > 0$ carry the factor $e^{\alpha n L}$ and depend on how fast call prices fall at high strikes. Rounding adds a third error far from the money, where a tiny price comes out of an integrand of much larger size. Agreement at one strike is a check at that strike only, and varying the cutoff, the step and the strike window separately shows which error dominates.