Approximating Surface-Based Derivative Values with Flat Volatility

Summary

The document asks whether a single constant volatility can approximate prices produced from a curve or surface of Black implied volatilities for products such as floating-rate notes with caps or floors and constant-maturity swaps. It also asks whether a useful proxy could be formed as a weighted average of the volatilities, perhaps using at-the-money values or weights linked to option and swap tenors.

No answer or calculation is included, so the document does not specify a method for choosing the constant volatility or demonstrate how close such a proxy would be. The underlying issue is that a derivative's value can depend on several optionlets or swaptions with different expiries, strikes, and sensitivities; a simple average need not preserve the surface-based price. Any practical proxy would therefore depend on the product, model, and pricing target, and would require calibration or valuation comparisons to assess its accuracy.

Key ideas

• The question concerns replacing a Black implied volatility curve or surface with one constant volatility.
• Potential applications mentioned include capped or floored floating-rate notes and constant-maturity swaps.
• A simple average or at-the-money volatility is proposed as a possible heuristic, but no method is supplied.
• A suitable proxy would depend on the product's exposures to optionlets or swaptions across the surface.
• The document gives no pricing comparison or evidence that a flat-volatility approximation is sufficiently accurate.

Tags

# Weighted average implied optionlet/swaptions volatility


# Weighted average implied optionlet/swaptions volatility












Let an implied volatility curve/surface is made up by optionlets or swaptions Black's implied volatility.

If you wanted to price, say, a FRN with cap and/or floor, a CMS et cetera you would input the array/matrix filled by that IV curve/surface in you pricing model: regardless of the model, it's very likely it takes as input that curve/surface.

My questions:

- is it actually possible to find a constant volatility value which returns a fair value close enough to the one obtained via curve/surface? (*)

- If said value existed, how would it be? Could it be a weighted average of all the curve/surface implied volatilities? Or just the ATM ones?

- If said value existed and it could be expressed by a weighted average, how the weights should be? Maybe something related to swaptions options & swap tenors? Something else?

(*) When I say «close enough» I mean that I would be okay with a rough and/or heuristic proxy, too. I do not actually need any accurate value.

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