Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 29K / c

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- Solution c

Solution

The digital call option and digital put option indicators partition the possible terminal stock prices:

1_{{S_{T} \geq K}} + 1_{{S_{T} < K}} = 1.

Thus the digital put-call parity is

D_{call} (t, S_{t}) + D_{put} (t, S_{t}) = e^{- r (T - t)} .

At time zero, part b gives

D_{call} (0, S_{0}) = e^{- r T} Φ (d_{-})

, so

D_{put} (0, S_{0}) = e^{- r T} (1 - Φ (d_{-})) = e^{- r T} Φ (- d_{-}) .

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