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Black-Scholes Characteristic Function(CF)

CF is defined as:

\[ \phi_X(u) = \mathbb{E}\left[e^{iuX}\right] \]

It’s the expected value of a complex exponential, and it uniquely determines the distribution of the random variable X — like a frequency signature of the distribution.

Deriving the Black-Scholes Characteristic Function

We assume that:

\[ s_T = \log S_T \sim \mathcal{N}\left(s_0 + \left(\mu - \frac{1}{2} \sigma^2\right)T, \sigma^2 T \right) \]

That’s just the log of the stock price under GBM, which is normally distributed.

Then using the Definition of the CF, substitute in the normal PDF and solve:

\[ \phi_{s_T}(u) = \int_{-\infty}^\infty e^{ius_T} f(s_T)\, ds_T \]

Then combine the exponentials, we get:

\[ \phi_{s_T}(u) = \int_{-\infty}^\infty e^{ius_T - \frac{(s_T - \hat{\mu})^2}{2 \hat{\sigma}^2}} \, ds_T \]

Where:

  • \(\hat{\mu} = s_0 + \left(\mu - \frac{1}{2}\sigma^2\right)T\)
  • \(\hat{\sigma} = \sigma \sqrt{T}\)

This is just a normal PDF wrapped in a complex exponential.

To solve the integral, we rewrite the exponent using a completing-the-square trick:

\[ ius_T - \frac{(s_T - \hat{\mu})^2}{2 \hat{\sigma}^2} = -\frac{(s_T - y)^2}{2\hat{\sigma}^2} + \text{constants} \]

We end up with:

\[ \phi_{s_T}(u) = \exp \left( iu \hat{\mu} - \frac{1}{2} u^2 \hat{\sigma}^2 \right) \]

Substituiting back:

\[ \phi_{s_T}(u) = \exp \left( iu \left(s_0 + \left(\mu - \frac{1}{2} \sigma^2\right)T \right) - \frac{1}{2} u^2 \sigma^2 T \right) \]

This is the closed-form CF of \(\log S_T\) in the Black-Scholes model.

This is actually the same as the CF of a normal distribution:

\[ \phi_X(u) = \exp \left(iu \mu - \frac{1}{2} u^2 \sigma^2 \right) \]

So this confirms that:

  • \(\log S_T\) is normal
  • The CF matches the theory

We derived the characteristic function of \(\log S_T\) under Black-Scholes by plugging the normal distribution into the definition of CF and simplifying the integral using a completing-the-square trick. This gives us a clean formula that’s perfect for plugging into Fourier pricing methods.