> the theory of unbiased random walks assumes constant volatility throughout the year

No. I’m pretty sure it doesn’t. If you assume a brownian motion with a constant volatility as your stochastic process for computing the walk then of course vol is constant by definition, but you can use a stochastic vol process (eg Heston[1]), one with jumps or even an SVJJ process to compute the walk[2] if you want to. As long as you don’t have a drift term and the jumps are symmetrical the process will still (I think) be unbiased.

There are technical reasons why it may or may not be important to use stochastic vol, but if I recall correctly, it only really matters if you care about “forward volatility” (eg the volatility of Nvidia one year from some future point in time) which you would if pricing something that uses forward-starting options. Then the term structure of the volatility surface at a future date is important so you need a stochastic vol model. If you care about the price evolution but not the future volatility then you can validly make the simplifying assumption that jumps will cancel each other out over time and that volatility is a locally deterministic function of time and price (if not constant, which it obviously is not) and use something like a Dupire model.[3]

More significantly, implied volatility is just the market price of a particular option expressed in terms of volatility. This is convenient for traders so they can compare option prices on a like for like basis between underlyers without constantly having to adjust for differences in the underlying price, strike and time. Implied volatility is not actually the overall expected volatility of the underlying instrument. For that, you would have to fit one of the models above to market prices and calculate the expectation over all strikes and times. And that still is just the market’s opinion of the volatility, not an actual probability even if you apply the BoE adjustment thing he does in the article.

[1] https://www.homepages.ucl.ac.uk/~ucahgon/Heston.pdf

[2] “SVJ” means stochastic vol with jumps (ie discontinities) in the underlying price evolution. SVJJ means stochastic vol with jumps both in the price of the underlying and in the volatility. An example of this is the Matytsin model, which everyone just calls “SVJJ” but it’s not the only possible svjj model https://www.maplesoft.com/support/help/maple/view.aspx?path=...

[3] https://www.math.kth.se/matstat/gru/5b1575/Projects2016/Vola...