In option pricing, Fourier Transform methods are typically used together with characteristic functions to handle complex integrals of exponential functions. The main advantage of using Fourier Transform in pricing options is that it allows to compute the price of options more efficiently, and it can be easily applied to several option pricing models, like the Black-Scholes Model, Heston Model, etc.
I will provide a brief explanation of the Fourier Transform within finance and show step-by-step how to use the Fourier Transform in pricing European call options as an example.
### 1. Fourier Transform and characteristic functions
The Fourier Transform is a mathematical technique that transforms signals between time and frequency domains. It’s particularly useful in finance for pricing derivative products, as the transform can help represent complex functions, such as characteristic functions, in a more manageable manner.
In pricing options, the characteristic function ϕ(u) of a random variable X is the expected value of eiuX:
ϕ(u) = 𝔼[eiuX]
The characteristic function can be used to derive the distribution of returns in the option pricing models. In particular, it can be used to derive the density function p(X) from the characteristic function using the inverse Fourier Transform:
$$p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-i ux} \phi(u) \, du$$
### 2. Option pricing using Fourier Transform
Let’s consider European-style call options. Using the risk-neutral pricing method, the call option price can be expressed as the discounted expected payoff under the risk-neutral measure:
C = e − rT𝔼Q[max(ST−K,0)]
where C is the option price, ST is the underlying stock price at expiry date T, K is the strike price, and r is the risk-free interest rate.
### 3. Pricing European call options using Fourier Transform
To price European call options using the Fourier Transform, we need to compute the integrals involving the characteristic function and the discounted expected payoff. First, let’s rewrite the expected payoff by slightly modifying it:
$$\begin{aligned}
C &= e^{-rT} \mathbb{E}_Q\left[\max(S_T - K, 0)\right] \\
&= e^{-rT} \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-i ux} \phi(u) \, du \\
&= \frac{e^{-rT}}{2\pi} \int_{-\infty}^{\infty} \frac{e^{-i ux} \phi(u)}{-iue^{iuk}} e^{iku} \, du \\
\end{aligned}$$
Let’s denote
$$C(u) = e^{-iuk}\frac{\phi(u)}{iu}$$
Now, we can rewrite the call option price:
$$C = e^{-rT} \, \text{Re} \left(\frac{1}{\pi} \int_{0}^{\infty} e^{-i\ln{(K)}u} C(u) \, du\right)$$
Here, Re denotes the real part.
### 4. Specializing to Black-Scholes Model
For a Black-Scholes Model, the characteristic function is derived from the standard Black-Scholes-Merton assumptions, which states that the log return follows a normal distribution:
$$p(x) = \frac{1}{\sqrt{2 \pi \sigma^2 T}} e^{-\frac{(x-\left(\mu - \frac{\sigma^2}{2}\right)T)^2}{2 \sigma^2 T}}$$
The characteristic function for this distribution is:
$$\phi(u) = e^{i u \left(\mu - \frac{\sigma^2}{2}\right)T - \frac{1}{2}\sigma^2 u^2 T}$$
To compute the price of a European call option using the Fourier Transform method in the Black-Scholes-Merton framework, you can simply substitute this characteristic function above into the call option price formula:
$$C = e^{-rT} \, \text{Re} \left(\frac{1}{\pi} \int_{0}^{\infty} e^{-i\ln{(K)}u} C(u) \, du\right)$$
with
$$C(u) = e^{-iuk}\frac{e^{i u \left(\mu - \frac{\sigma^2}{2}\right)T - \frac{1}{2}\sigma^2 u^2 T}}{iu}$$
This will allow you to efficiently calculate the price of European call options using the Fourier Transform method. Similar approaches can be applied for put options and other models, like the Heston model, by finding their respective characteristic functions.
In summary, the Fourier Transform method in pricing options has the advantage of computational efficiency, which makes it a powerful tool when dealing with complex option pricing models. It essentially allows us to simplify complex integrals and work with characteristic functions to efficiently compute the option price.