Should Theta Be Included in a Minimum-Variance Interest Rate Hedge?

Summary

The document poses a practical hedging question for an interest rate derivative. It assumes a hedge is built from linear positions in instruments such as interest rate swaps and forward rate agreements, and seeks positions that make the hedge’s change in value track the derivative’s change over a chosen interval. The objective is expressed as minimizing expected squared hedging error under the physical probability measure. Because the hedge is not self-financing, the question is whether the passage-of-time effect, or theta, should be included when selecting its positions.

The document does not provide an answer or a comparison of hedging methods. It gives no market data, empirical results, or guidance on how to estimate the relevant changes in value. Its useful contribution is to frame the modeling choice clearly: whether to include time decay depends on the intended hedging objective and how changes in the derivative and hedge are measured. Readers need further analysis to determine the appropriate treatment for a particular instrument and hedging horizon.

Key ideas

• The proposed hedge combines linear positions in swaps and forward rate agreements.
• The stated objective is to minimize expected squared changes in hedging error under the physical measure.
• The hedge is assumed not to be self-financing.
• The document asks whether time passage should enter hedge construction but leaves the question unanswered.

Tags

# Should you hedge theta?


# Should you hedge theta?












Consider an arbitraty interest rate derivative $C$. Assume that in order to hedge it, you are allowed to construct portfolio $H$ of linear combination of simple instruments such as Interest Rate Swaps (IRS) and Forward Rate Agreement (FRA). By construction such portfolio is not self-financing. During some time interval, for example 1 week, you want to have change in value of your hedging portfolio $H$ equal to the change in value of $C$, i.e.: $$E^P[(\Delta H(w)-\Delta C)^2]=0$$ or $$argmin_wE^P[(\Delta H(w)-\Delta C)^2]$$ where $w$ is vector of units of each trade (IRSes and FRAs) in the hedging portfolio $H$, the superscript $P$ indicates physical measure.

Should you consider time passage (theta) in order to construct hedging portfolio $H$?

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