Digital Options and Digital Cliquets — Model Sensitivity and Hedging

A digital call pays $\text{\char36}1$ if $S_T > K$ and nothing otherwise — the simplest binary payoff. As a benchmark instrument, digitals reveal the direct connection between option pricing and the implied probability density: the digital price is the discounted risk-neutral probability $e^{-rT}\mathbb{Q}(S_T > K)$. For a digital cliquet (a sequence of forward-start digitals), this probability becomes conditional on the vol path, making the price highly sensitive to the forward smile — a fingerprint of whether the model has realistic forward dynamics.

• price a digital call from the call-price skew (Breeden-Litzenberger);
• define the digital cliquet and write its pricing formula;
• compute the model sensitivity of a digital cliquet to the forward skew (local vol vs stochastic vol comparison);
• construct the "super-replication" hedge for a digital call.

1. 1.1 Digital options and the implied density
2. 1.2 The digital cliquet
3. 1.3 Forward smile sensitivity