Separate an expected premium from a realized spread
Bali and Zhou define the variance risk premium as risk-neutral expected variance minus physical, or real-world, expected variance over the same future interval. Both expectations are conditional on information at time t. This page uses that implied-minus-physical sign convention; papers using the reverse convention report the opposite sign. A single option IV squared is only a proxy for model-free implied variance, since its strike and pricing model matter.
- The last bracket is a forecast error. One quiet month can produce a large realized spread even when the expected premium was modest.
- Square root of expected variance is not generally expected volatility. The volatility spread above is a reporting transformation, not the variance premium in another unit.
- IV minus trailing historical volatility can be an observable proxy if trailing volatility is your declared forecast. It is not a measurement of what subsequently happened, and that forecast needs its own evaluation.
q(t,T) = option-implied expected variance
m(t,T) = physical expected variance estimated at t
R(t,T) = variance realized later, between t and T
Expected variance premium = q(t,T) - m(t,T)
Realized variance spread = q(t,T) - R(t,T)
= [q(t,T) - m(t,T)] + [m(t,T) - R(t,T)]
Realized volatility spread = sqrt(q(t,T)) - sqrt(R(t,T))