Pricing Calls with Time-Varying Volatility Using Finite Differences

Summary

The document clarifies how to approach an implicit finite-difference exercise for a European call when volatility varies over time. The stated example uses a linear volatility function, while the exercise also provides a Black–Scholes approximation that substitutes a constant volatility calculated from the average variance over the remaining life of the option.

The key distinction is that the finite-difference method should use the time-dependent volatility function itself at the relevant time steps. The averaged volatility belongs to the separate closed-form approximation, which can be compared with the numerical solution. The response is brief and supplies no derivation, grid design, boundary conditions, code, or numerical results. It therefore resolves the conceptual choice but leaves implementation details and validation of the approximation to the student.

Key ideas

• Use the specified time-varying volatility directly in the implicit finite-difference calculation.
• The constant volatility derived from average variance is for the separate Black–Scholes approximation.
• The numerical method and closed-form approximation represent distinct calculations that can be compared.
• The answer does not specify discretization choices, boundary conditions, or implementation details.

Tags

# MATLAB exercise on an European call option with time-varying volatility


# MATLAB exercise on an European call option with time-varying volatility












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..

## Answer by Jeji (score 1, accepted)

https://quant.stackexchange.com/a/34651

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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