Fourier-Based Option Pricing¶

In simple models like Black-Scholes, we have closed-form solutions (thanks to Ito calculus). But for more complex models (like Heston, Jump-Diffusion, etc.), there’s no closed-form formula for the price.

So instead, we use the characteristics function (CF), and price the option using Fourier transforms

Generic workflow for Fourier-based option pricing¶

Here’s a step-by-step workflow for Fourier-based option pricing in stochastic modeling:

Model Specification: Define the model for the underlying asset prices, including the stochastic process that governs their movements.
1. Defuine the Stochastic process for the underlying asset. 1. Black-Scholes: GBM 2. Heston: Stochastic volatility 3. Merton: Jump-diffusion
2. The key is that we must know the distribution of $\log(S_T) $ or at least its characteristic function.
Characteristic Function: Derive the characteristic function of the log of the underlying asset’s price. This function is crucial for the Fourier transform method.
1. Find the characteristic function $\phi(u) = \mathbb{E}[e^{iu \log S_T}]$ under the risk-neutral measure $\mathbb{Q}$.
2. This is the bridge into frequency space. Even if the PDF is unknown, CF is often tractable.
Fourier Transform of Payoff Function: Apply the Fourier transform to the option’s payoff function. This converts the payoff into a form that can be used in the Fourier domain.
1. Fourier transform the option payoff, such as $(S_T - K)^+$.
2. This step ensures you’re working in the same frequency space as the CF.
3. In Carr-Madan: this involves modifying the payoff to make it square-integrable (e.g. adding a damping factor $e^{-\alpha k}$).
Integrate Product: Integrate the product of the characteristic function and the transformed payoff function. This step involves complex mathematical integration.
1. Compute the integral: $\text{Option Price} = \int_{-\infty}^{\infty} \phi(u) \cdot \hat{g}(u) \, du$ Where: $\phi(u)$ = CF of log price , $\hat{g}(u)$ = transformed payoff function
2. This step is often where FFT or quadrature is used.
Inverse Fourier Transform: Perform the inverse Fourier transform to convert the result back to the time domain. This gives the option price.
1. Bring the result back to the real domain to obtain the option price: $f(x) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{-iux} \hat{f}(u) \, du$
2. Often the inverse is embedded inside the pricing formula (e.g. Carr-Madan or Lewis), so this step is implicit.
Adjust for Model Characteristics: Make any necessary adjustments to account for specific characteristics of the model or market conditions. (like discounting)
1. Finally, multiply by $e^{-rT}$ to account for the present value of the expected payoff under $\mathbb{Q}$.
2. Some models require further tweaks (e.g. jump compensators, FFT damping, etc.)

flowchart TD
    A["1.Model Specification: Define the stochastic process (e.g. GBM, Heston, etc.)"]
    B["2.Derive Characteristic Function: Compute φ(u) = E[e^{iu log S_T}] under Q"]
    C["3.Fourier Transform Payoff: Convert payoff function (e.g. (S - K)^+) to frequency space"]
    D["4.Integrate Product: Compute φ(u) × Fourier-transformed payoff"]
    E["5.Inverse Fourier Transform: Convert result back to real domain (e.g. via inverse FFT)"]
    F["6.Adjust/Discount: Apply discounting (e^{-rT}) and model-specific corrections"]

    A --> B
    B --> C
    C --> D
    D --> E
    E --> F

Fourier Transform¶

The Fourier Transform converts a function (like a PDF) into frequency space.

\[ \hat{f}(u) = \int_{-\infty}^{\infty} e^{iux} f(x) dx \]

We can also recover the function using the Inverse Fourier Transform:

\[ f(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iux} \hat{f}(u) du \]

Paeseval’s relation connects the dot product in original space and frequency space.

The PDF(probability densitiy function) is a curve, that tells you how likely it is to see different outcomes of a random variable. The area under the curve between two points tells you the probability of the variable falling in that range.

Dartboard

Suppose you’re guessing where a dart will land on a board. You can’t predict exactly where it will land… But you can describe how likely it is to land in a certain area. That’s what a PDF dose.

Let’s say $X=log(S_T)$ (the log of future stock price)

The PDF $f(x)$ of $X$ answers:

“What’s the probability that the log of the stock price ends up around x?”

A normal distribution PDF looks like a bell curve.
The peak is the most likely value.
The tails show how much chance there is of extreme events.

To price an option, we want to average all possible future outcomes, weighted by how likely they are.

Call option price:

\[ C_0 = e^{-rT} \mathbb{E}^{\mathbb{Q}}[(S_T - K)^+] \]

We can compute this as:

\[ C_0 = e^{-rT} \int_0^\infty (S - K)^+ \cdot f(S)\, dS \]

Where:

$(S - K)^+$ is the payoff
$f(S)$ is the PDF of the stock price under risk-neutral measure

So: The PDF tells us how likely each payoff is, and we multiply that by the payoff and add it all up to get the option price.

For complex models (e.g. Heston), we often don’t know the PDF of S_T… but we can compute its characteristic function — and that’s where Fourier methods come in!

Characteristic Function (CF)¶

\[ \hat{q}(u) = \mathbb{E}^{\mathbb{Q}}[e^{iuX}] \]

This is the Fourier transform of the PDF of a random variable $X$
It fully characterizes the distribution, even if the PDF is not known

This is the key: even if we can’t write down the PDF of $S_T$, we can often get its CF

Imagine you have a mysterious black box (a random variable). We can’t always see its PDF, but we still want to understand its behavior - like:

Where it’s centered
How wide it is
How it behaves over time

The Characteristic Function is a tool that lets you describe any random variable in a way that’s super flexible - even when you don’t have the PDF.

The characteristic function of a random variable $X$ is defined as: $\phi_X(u) = \mathbb{E}[e^{iuX}]$, where:

$u$ is a real number (like a frequency),
$e^{iuX}$ is a complex exponential (don’t panic — it’s like encoding the shape of a wave),
$\mathbb{E}[\,]$ means you’re taking an expected value — i.e., an average.

The term $e^{iuX}$ comes from Euler’s formula: $e^{iuX} = \cos(uX) + i\sin(uX)$

So what are we really doing? -> We’re taking the cosine and sine of the random variable, then averaging them! This tells us: How “aligned” the distribution is with the wave of frequency u

A characteristic function is like a “frequency signature” of a random variable. It encodes all its behavior — even if we don’t have the PDF. For option pricing, the CF is often much easier to work with than the PDF, and we can use Fourier inversion to get prices directly.

Lewis (2001) Approach¶

Uses the Fourier inersion to get the call price:

\[ C_0 = S_0 - \frac{\sqrt{S_0 K} e^{-rT}}{\pi} \int_0^{\infty} \text{Re}\left[ e^{izk} \phi(z - i/2) \right] \frac{dz}{z^2 + 1/4} \]

Where:

$\phi(u)$: characteristic function of $\log S_T$
$k = \log(S_0/K)$

Carr-Madan (1999) with FFT¶

They modify the call price to make it square-integrable and apply FFT:

\[ \Psi(\nu) = \frac{e^{-rT} \phi(\nu - i(\alpha+1))}{\alpha^2 + \alpha - \nu^2 + i(2\alpha + 1)\nu} \]

Then recover the price via inverse Fourier transform:

\[ C_0 = \frac{e^{-\alpha k}}{\pi} \int_0^{\infty} e^{-i \nu k} \Psi(\nu) d\nu \]

This is used in practice for fast numerical pricing using FFT