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Pricing a digital from a call spread

MediumFinance & Derivatives~3m

A cash-or-nothing binary call pays $1 if ST>100S_T > 100 (strike $100). You have no pricing model, but you can see two vanilla call quotes on the same stock and expiry: the $99-strike call trades at $6.00 and the $101-strike call at $5.20. Using only these, estimate the fair value of the $100 digital, in dollars to two places.

Show hints (2)+
  1. A digital call is the slope −∂C/∂K-\partial C/\partial K - replicate it with a narrow, width-scaled call spread.
  2. Estimate: (C(99)−C(101))/(101−99)(C(99)-C(101))/(101-99), dividing by the $2 strike gap.

Answer

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0.4 (± 0.001)

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