#### 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.

Mathematica

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

Maple

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

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