# Characteristic function of a random variable by Fourier transform

this is character function in probability theory

$$\phi(u)=\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}ux}f(x)\mathrm{d}x$$

Let an asset price $$S_t$$ (e.g. a stock) be modeled with a Geometric Brownian motion:

$$\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t$$

where $$W_t$$ is a Wiener process, $$r$$ the risk-free rate and $$\sigma$$ the volatility. Consider a European call option written on $$S_t$$ with strike $$K$$ and maturity $$T$$. We apply the following transformation:

$$x=\log{(\frac{S_0}{K})}$$ and $$y=\log{(\frac{S_T}{K})}$$

Show that the characteristic function of y is given by

$$\phi_y(u)=\mathrm{e}^{\mathrm{i}u(x+(r-\frac{1}{2}\sigma)T)-\frac{1}{2}T\sigma^2u^2}$$

Hint: You may use the fact that the characteristic function of a standard normal distri- bution $$Z$$ is given by $$\phi_Z(u)=e^{-\frac{1}{2}u^2}$$

my problem is i don't even know where to start. Anyone give some tips till the point i can drive this?

• I am hoping you will tell us what you mean by the function f(x), which is currently used but left undefined. Mar 26 at 21:51
• Hint: Use Ito’s lemma to find a solution for S and then look at what you get. Mar 27 at 0:08

I'll admit that I'm not familiar with most if not all stochastic finance terminology.

So, I'll assume that $$K$$ and $$S_{0}$$ are constants.

See that for $$f(x)=\log(x)$$, by Ito's Lemma, you get,

\begin{align}\log(S_{t})-\log(S_{0})&=\int_{0}^{t}f'(S_{s})\,dS_{s}+\frac{1}{2}\int_{0}^{t}\sigma^{2}S_{s}^{2}\cdot f''(S_{s})\,ds\\\\ &=\int_{0}^{t}\frac{1}{S_{s}}(rS_{s}\,ds+\sigma S_{s}dW_{s})+\int_{0}^{t}\frac{1}{2}\sigma^{2}S_{s}^{2}\cdot\frac{-1}{S_{s}^{2}}\,ds\\\\ &=\int_{0}^{t}(r\,ds+\sigma dW_{s})-\frac{\sigma^{2}}{2}\int_{0}^{t}\,ds\\\\ &=\sigma W_{t}+t(r-\sigma^{2}/2) \end{align}

Hence you have $$\log(S_{t}/K)=\log(S_{0}/K)+\sigma W_{t}+t(r-\sigma^{2}/2)$$

Thus, you have

\begin{align}\mathbb{E}(\exp(iu y))&=\exp\bigg(iux+iuT(r-\frac{\sigma^{2}}{2})\bigg)\mathbb{E}(\exp(iu\sigma W_{T}))\\\\ &=\exp\bigg(iux+iuT(r-\frac{\sigma^{2}}{2})-\frac{T\sigma^{2}u^{2}}{2}\bigg)\end{align}

by using the Fourier transform for for Normal distribution.

• This is beautiful. Yes i reckon it's nothing to do with finance but just in probability theory. Thank you very much for this Mar 27 at 10:10