2.6.1 classic Black Scholes model from finance, European call version

problem number 96

From Mathematica symbolic PDE document.

Solve for \(V(S,t)\) where \(V\) is the price of the option as a function of stock price \(S\) and time \(t\). \(r\) is the risk-free interest rate, and \(\sigma \) is the volatility of the stock. \[ \frac {\partial V}{\partial t} + \frac {1}{2} \sigma ^2 S^2 \frac {\partial ^2 V}{\partial S^2} = r V - r S \frac {\partial V}{\partial S} \] With boundary condition \( V(S,T) = \max \{ S-k,0 \}\)

Reference https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation See the European call version at bottom of the page.


\[\left \{\left \{u(x,t)\to \frac {1}{2} k e^{\frac {\sigma ^2 t}{2}+x-1} \left (\text {erf}\left (\frac {\sigma ^2 t+x}{\sqrt {2} \sqrt {t} | \sigma | }\right )+1\right )\right \}\right \}\]


\[u \left (x , t\right ) = {\mathrm e}^{-1} \left (-i \mathcal {F}^{-1}\left (\frac {{\mathrm e}^{-\frac {s^{2} \sigma ^{2} t}{2}}}{s +i}, s , x\right )+\mathcal {F}^{-1}\left ({\mathrm e}^{-\frac {s^{2} \sigma ^{2} t}{2}} \mathcal {F}\left ({\mathrm e}^{x}, x , s\right ), s , x\right )\right ) k\]