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I have to solve the following exercise: compute and plot the value $V = V(S, t),\ t<T$, ($T=$ maturity) of an European CALL option (with arbitrary $t$, $T$, $K$ (strike price), $r$ (risk-free interest rate), $S>0$, i.e. the asset price at $t>0$), no dividends, but with an (arbitrary) time-varying volatility $\sigma=\sigma(t)$ (I choose it linear, $\sigma(t)=1+t$), by using the implicit finite difference method (implemented in MATLAB). Moreover the exercise says: $"$an approximation $\tilde V=\tilde V(S, t)$ of the value $V$ of this type of option is given by the closed form solution of the B&S equation with constant parameter where the constant volatility $\tilde\sigma$ is approximated by $$ \tilde\sigma = \sqrt{\frac{1}{T-t}\int_t^T\sigma^2(t)dt}." $$ So, my question is: in order to implement the implicit finite difference method, do I have to compute the approximated value $\tilde\sigma$ (with $\sigma(t)=1+t$, choosed by me) and develop the algorithm with this value, or shall I proceed in an other way? Because in the way I think, the exercise seems very simple..

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Ok, problem solved. The Implicit finite difference method must be implemented with $\sigma(t)=1+t$, not with his approximated value.

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