Checking the Expected Profit Formula for Implied-Volatility Delta Hedging

Summary

The document presents a question about a derivation in a derivatives text for expected profit when an option is delta-hedged using implied volatility. It gives an initial time-integral expression proportional to the difference between actual and implied variance, then a more explicit integral involving the underlying price distribution, discounting, and option gamma. The author reports that a numerical implementation gives a much larger magnitude than expected when using the printed coefficient.

The question compares that coefficient with a different denominator used in accompanying VBA material and says the latter produces values closer to figures in the book. However, the document contains no answer or independent derivation resolving the discrepancy. It therefore serves as a prompt to check algebra, notation, or implementation details rather than as evidence that either coefficient is correct. Any conclusion requires verifying the original derivation and assumptions, including the distribution and integration conventions used.

Key ideas

• The expected hedging profit expression depends on the difference between actual and implied variance.
• The questioner reports a mismatch between the printed coefficient and an implementation in accompanying material.
• Numerical integration is used to compare the formula with plotted results from the referenced text.
• The document offers no resolution, so it does not establish which coefficient is correct.
• The derivation and implementation assumptions need independent verification before using the expression.

Tags

# Confused by Solution to the Expected Profit when Hedging an option using Implied Volatility (from Wilmott 2006)


# Confused by Solution to the Expected Profit when Hedging an option using Implied Volatility (from Wilmott 2006)












Paul Wilmott on Quantitative Finance 2nd Ed (section 12.5.1) gives a solution to the initial expected profit when hedging using delta based on implied volatility as

$$\frac{1}{2}(σ^2 - σ̃^2) \,\int_{t_0}^{T}e^{-r(s-t_0)}S^2\Gamma\,ds$$

from which he then derives the single integral $$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{2\sqrt{2\pi}} \, \int_{t_0}^{T} \frac{1}{\sqrt{σ^2(s-t_0) + σ̃^2(T-s)}} \\ \times \exp\left( - \frac{(log(S/E) + (\mu - 0.5σ^2)(s-t_0) + (r - D - 0.5σ̃^2)(T-s))^2}{2(σ^2(s-t_0) + σ̃^2(T-s))} \right)\\$$

However, and very confusingly, in order to get similar results to those shown by him when comparing expected profit versus various growth rates (as per figures 12.4, 12.5, 12.6) I have to set the initial term to

$$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{\frac{2}{\sqrt{2\pi}}} $$ instead of $$\frac{Ee^{-r(T-t_0)}(σ^2 - σ̃^2)}{2\sqrt{2\pi}} $$ as set out in the text.

My implementation uses numerical integration from the Python SciPy package. When implementing it as shown, the magnitude of the expected profit is clearly excessive but with the change in denominator it's nearly spot on.

In the VBA code accompanying Paul Wilmott Introduces Quantitative Finance 2nd Ed he has an approximate solution which also uses ${\frac{2}{\sqrt{2\pi}}}$ instead of ${2\sqrt{2\pi}}$

Have I misread his equation? I can't find errata for either book and am really confused as to why his implementation differs from the derivation shown. Am I missing something (obvious)?

Thanks

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