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I want to ask a perhaps simple question: Why do we use delta on the x-axis instead of the strike price when discussing volatility smile or volatility surface? In the book I'm currently reading, it is written, 'Different deltas correspond to different strike prices,' but I don't understand why.

Book: Derivatives: Theory and Practice of Trading, Valuation, and Risk Management (Jiří Witzany)

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    $\begingroup$ The Delta($\Delta$) of the option serves as a good measure of the moneyness of the option and different moneyness levels trade (are quoted) at different IV levels in the market, which is the smile/skew you observe. $\endgroup$
    – Quasar
    Commented Jan 13 at 16:56

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It's important to note that this is only done for certain markets, predominantly foreign exchange, and almost always OTC, where there is no set number of available strikes and no direct price quotes.

FX is vol quoted. Delta is a neat choice because it makes IV comparable across tenors. Depending on time to expiry, the same strike will be very different in terms of how "far" from ATM it is. Just as delta is an increasing function in vol, it also grows in time.

Comparing a 1y 10D call vs a 10y 10D call, with same $\sigma = 10\%$, $r_{CCY1}=-1\%$ and $r_{CCY2}=1\%$ gives use a strike that is much farther from the forward for the long maturity option.

function GKMSpot(S, K,t,ccy1,ccy2,σ)
    d1 = ( log(S/K) +  ( ccy2 -ccy1 + 0.5*σ^2)*t ) / (σ*sqrt(t))
    d2 = d1 - σ*sqrt(t)
    c  = S*exp(-ccy1*t)*N(d1)-K*exp(-ccy2*t)*N(d2)
    delta = exp(-ccy1*t)*N(d1)
  return c, round(delta*100,digits=2)
end

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Likewise, it's easy to incorporate premium into delta for hedging purposes which is something unique to FX, where the choice of premium currency has an impact (e.g. in stock options you wouldn't pay in shares typically, but it's perfectly fine to pay in EUR or USD).

A lot of details are explained in a paper by Uwe Wystup and Dmitri Reiswich, which is a mist read for anyone I treated in FX options.

Some useful resources here (there are plenty of you search):

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It's about giving the best "relative" measure. Using Moneyness rather than strike quotes gives more context around the option, thus giving more information. An option being $2 OTM could be deep-OTM, or it could be basically ATM. Delta is just another way of quoting a relative strike. Generally you want to quote with as much information as possible, but context matters. OTC derivatives will be quoted using butterflies, strangles etc because the investors trading those products will know what they mean and have the required tools to analyse them, i.e. Bloomberg. But someone wanting to buy 300, \$0.05 0DTE SPY puts on Robinhood probably wouldn't understand what they were buying if SPY options were being quoted in moneyness levels.

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