# Transforming and minimisation of the BS PDE

I'm trying a novel numerical substitution/fitting method to solve the BS PDE, but the issue is that due to the large range of magnitude of prices $V(s,t)\in[10^{-20},10^1]$, when I try to minimise the error $E = \sum_{i=0}^{N}|L\hat{V}(s_i,t_i)-r\hat{V_i}(s_i,t_i)|$ where $L = \frac{\partial }{\partial t} +0.5\sigma^2s^s\frac{\partial^2 }{\partial s^2} + rs\frac{\partial }{\partial s}$ is the BS differential operator and $\hat{V_i}$ is the trial solution, the error terms are dominated by the errors where $\hat{V}(s_i,t_i)$ are larger prices for the ITM options resulting in poor relative accuracy for OTM options.

As such I want to transform the problem so that I solve it in a space where the magnitude of prices $\hat{V}(s_i,t_i)$ are more similar i.e perhaps taking the log transform $U = log(\hat{V}(s_i,t_i))$ such that $U\in[-20,1]$

so that I can minimise the error $$E = \sum|L_uU-rU|$$ where $L_u$ is the transformed differential operator (and I then use the inverse transform on $U(s_i,t_i)$ to obtain the true price after minimisation)

• Why did you delete your previous question which looked pretty much identical? quant.stackexchange.com/questions/39066 – LocalVolatility Mar 29 '18 at 18:44
• I thought I would make the problem statement clearer, there is a subtle difference that the solution needs to retain the form of $log$, if just by using the substitution of $U=log(V)$, the minimisation still minimises $V$ not the $log$ of the solution – Sam Palmer Mar 29 '18 at 18:52
• Better to make an edit to your original question in such a case. – Bob Jansen Mar 30 '18 at 14:05