Implied Volatility and P&L Attribution for Single-Stock Options

Summary

The document raises a question about P&L attribution for a single stock option. It defines daily P&L as the change between desk market values and proposes explaining that change through underlying-price delta and gamma effects, implied-volatility vega, and theta. The setup assumes that each market value is associated with an implied volatility for that date through a pricing function, potentially Black–Scholes.

The author questions whether using implied volatility inferred from the market-value change to explain that same P&L is circular reasoning. No answer or resolution is included, so the document does not establish whether this is a logical error or how a robust attribution should be constructed. It is useful as a prompt to examine the distinction between observed marks, model inputs, and factor-based explanations, while recognizing that the proposed expansion is approximate and depends on the chosen pricing model and inputs.

Key ideas

• The setup defines option P&L as the difference between successive desk market values.
• A Taylor-style attribution decomposes the change into delta, gamma, vega, and theta contributions.
• Implied volatility is inferred from each market value using a selected pricing function.
• The author asks whether attributing P&L to an implied-volatility change derived from those same marks is circular.
• The document supplies no resolution or validation of an attribution method.

Tags

# Logical mistake in PL attribution


# Logical mistake in PL attribution












We are attributing the PnL of a single stock option to risk factors, solving the PLA problem. We have desk quotes $MV(T-1), MV(T)$, and $PnL_{T}=MV(T)-MV(T-1)$. We associate $MV$'s to $\sigma_{iv}^{T-1}, \sigma_{iv}^{T}$, thus $MV(t)=f(\sigma_{iv}^t;t)$ for the appropriate pricing function $f$ (suppose even that is the Black-Scholes formula).

Then we attribute $PnL_{T}$ to $\sigma_{iv}$ change (particularly): $$ PnL_{T}\approx\Delta_{T-1}(S_T-S_{T-1})+\frac{1}{2}\Gamma_{T-1}(S_T-S_{T-1})^2+Vega_{T-1}(\sigma_{iv}^{T}-\sigma_{iv}^{T-1})+\theta_{T-1} $$

We explain PnL via the change in implied vol, computed wrt to MV change (that is explained via PnL again). That is a logical mistake, isn't it?

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