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 ✓

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] == (1*sigma^2*D[u[x, t], {x, 2}])/2; 
ic  = u[x, 0] == k*Exp[x - 1]*HeavisideTheta[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, t], {x, t}, Assumptions -> k > 0], 60*10]];
 

\[\left \{\left \{u(x,t)\to \fbox {$\frac {1}{2} e^{\frac {t \sigma ^2}{2}+x-1} k \sqrt {\frac {1}{\sigma ^2 t}} \sqrt {\sigma ^2 t} \left (\text {erf}\left (\frac {\sqrt {\frac {1}{\sigma ^2 t}} \left (t \sigma ^2+x\right )}{\sqrt {2}}\right )+1\right )\text { if }\Re \left (\frac {1}{\sigma ^2 t}\right )>0$}\right \}\right \}\]

Maple ✓

restart; 
pde := diff(u(x,t),t) = 1/2*sigma^2*diff(u(x,t),x$2); 
ic  := u(x, 0) = k*exp(x - 1)*Heaviside(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,ic],u(x,t)) assuming k>0),output='realtime'));
 

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

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