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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?

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

1 Answer 1

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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.

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

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