Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 48 / 3 / b / Solution

Write $h=\Delta x$, $k=\Delta t$ and march backward from the terminal payoff. At the interior grid point $(x_i,t_j)$ useThe central finite difference spatial approximation and backward-in-calendar-time explicit Euler method giveHere $a_i^j=\sigma (e^{x_i},t_j{)}^2/2$, $b_i^j=r-a_i^j$, and $u_i^m=(e^{x_i}-K{)}^{+}$. At a far-left boundary use $u\simeq 0$; at a far-right boundary the call asymptotics are $u\simeq e^{r(T-t)}{e}^x-K$. These boundary values are inserted into the adjacent interior updates. Smooth-solution Taylor expansion gives consistency order $O(k+h^2)$ in the evolution equation.