Log-Price Transformations for Black–Scholes PDE Fitting

Summary

The document raises a numerical fitting problem for the Black–Scholes partial differential equation: option prices span a wide range, so an absolute residual objective can be dominated by in-the-money observations. The author is concerned that this leaves out-of-the-money prices with poor relative accuracy.

The proposed idea is to take the logarithm of the trial price function before fitting, which would compress the range of magnitudes, then transform the fitted values back to price space. This is posed as a question, not a demonstrated solution. The document does not derive the transformed differential operator or show that minimizing the suggested residual is mathematically equivalent to solving the original PDE; the operator, boundary conditions, and numerical evidence remain unspecified.

Key ideas

• A broad range of option prices can make absolute PDE residuals favor larger prices.
• The author proposes fitting log prices to reduce differences in scale across observations.
• The intended benefit is improved relative accuracy for out-of-the-money options.
• The transformed operator and validity of the proposed objective are not established in the document.

Tags

# Transforming and minimisation of the BS PDE


# 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)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.