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How does one calculate the implied liquidity of a specific option contract given a set of vanilla puts and calls with various strikes and maturities on a single underlying?

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From Implied Liquidity : Towards stochastic liquidity modeling and liquidity trading

We will call the parameter, fitting the bid-ask spread (under a symmetric distortion) around the mid price, the implied liquidity parameter. Hence for the European Call option (strike K and maturity T ) with given market bid (b) and ask (a) prices, the implied liquidity parameter is the specific λ > 0, such that:

a = − exp(−rT )Eλ [−(ST − K )+ ] and b = exp(−rT )Eλ [(ST − K )+ ]

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