Problem $2$: Characteristic Functions and Op tion Pricing In finance, the Heston model describes the evolution of the volatility of an un derlying asset assuming that the volatility of the asset follows a CIR stochastic process. The analytical solution of a vanilla option is solved by using the char acteristic function method. To understand this method, we now apply it to solve the Black$-$Scholes model and derive the pricing formula. $2\mathrm{.}1$ Black$-$Scholes Model The Black$-$Scholes model assumes that asset price St follows: dSt $=$ rStdt$+$StdWt where r is the risk$-$free rate and is constant volatility. Q$2$: Let Xt $=$lnSt$.$ By It$$o$$s lemma, present the SDE of Xt