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Individual-Rate Vega of a Normal-Model Interest Rate Spread Option

Article Quant Q&A · Author: babaji
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Summary

The document asks how to translate a spread option’s sensitivity to spread volatility into sensitivities to the volatilities of its two component rates. It gives the normal-model spread variance as the sum of the component variances less twice their correlation-weighted product, then differentiates that variance with respect to the first rate’s volatility. This suggests the spread variance can decline as that component volatility rises when the correlation-weighted volatility of the second rate is large enough.

The question is whether this variance derivative determines the option’s individual-rate vega, or whether the rate levels also matter. The post provides no answer, derivation of the full option sensitivity, or empirical evidence. Its formula concerns spread variance; option value sensitivity additionally depends on how option value responds to spread volatility and on the model’s other assumptions. The discussion is framed for a long spread option in a normal-volatility setting, so it does not establish a general result for other models or contracts.

Key ideas

  • Spread variance depends on both component volatilities and their correlation.
  • The derivative of spread variance with respect to one component volatility can be negative.
  • The post asks whether that derivative alone determines individual-rate option vega.
  • It does not resolve the role of rate levels or provide a complete option-pricing sensitivity.

Tags

OptionsFixed incomeVolatilityDerivatives pricing
Full text
# Spread vol for interest rate spread options in normal environment


# Spread vol for interest rate spread options in normal environment












Suppose I am long spread option with underlying : rate A - rate B. The vega on the option would be positive. But if I want to compute the option vega with respect to individual rates, can I use the below formula?

sprdVar = (v1)^2 – 2 * rho * v1 * v2 + (v2)^2

where sprdVar = spread variance, i.e. sprdVol^2 v1, v2 = vol of the assetA/B (in normal vol)

d(sprdVar)/d(v1) = 2*(v1) – 2 * rho * v2

So for the spread vols to be negative wrt asset A, v1- rho*v2 < 0 . Will this be correct or the actual rate levels will also play a role in determining the option vega sensitivity from individual rates? Thanks

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