Digitals, Barriers, and the Replication Mindset

The final option-pricing skill this course teaches is a mindset: when handed an unfamiliar payoff, don't reach for a model — try to build the payoff out of things you can already price. Digitals and barriers are the classic practice ground, and they close the loop with the implied-distribution lesson.

Digitals: the call-spread limit

A digital (binary) call pays \$1 if$S_T > K$, nothing otherwise. Replication: buy$\frac{1}{h}$call spreads at strikes$K$and$K+h$; as$h \to 0$ the payoff converges to the digital, so

$$\text{Digital}(K) = -\frac{\partial C}{\partial K} = P(S_T > K)$$

— the tail probability from Breeden–Litzenberger, now as a tradeable instrument. Two practical layers on top:

• The smile adjustment: differentiating market prices (with the smile) rather than the flat-vol formula adds a skew term — equity digitals are cheaper than naive Black–Scholes says on the call side, richer on the put side. "Price a digital" at interview wants the call-spread answer plus "and I'd price it off the smile, not a single vol."
• The hedging pathology: near expiry with spot at the strike, the digital's delta explodes (the payoff is a cliff). Desks don't hedge cliffs; they overhedge by actually holding the finite call spread — accepting a conservative payoff bound in exchange for bounded Greeks. General principle, worth stating: discontinuous payoffs are priced by the replication you're willing to hold, not the idealized limit.

Barriers: options that switch on or off

Knock-out options die if spot touches a barrier; knock-ins are born there. In-out parity — knock-in + knock-out = vanilla (same strike/expiry) — halves every pricing problem and is the first tool to state. The elegant special case: for a driftless underlying, an up-and-in at-the-barrier structure can be statically replicated using the reflection principle — the same symmetry that solved first-passage problems in the random-walk lesson: paths that touch the barrier and finish below mirror to paths finishing above, letting a barrier claim be matched by vanillas struck at the reflection of the payoff across the barrier. Real markets (drift, smile) break exact symmetry, but the reflection picture remains the intuition scaffold — and the interview answer.

Why traders respect barriers: like digitals, they concentrate risk at a point. As spot approaches a knock-out barrier, the option's value cliff-drops; hedging near the barrier means enormous, flickering deltas — and historically, barrier-related hedging flows have visibly pushed FX spot around popular barrier levels (a microstructure story: hedge flow becomes price impact). "What's hard about barriers?" — not the math; the neighborhood of the barrier.

The replication hierarchy

The course's synthesis, worth holding as a ladder: static replication with vanillas (parity, spreads, butterflies, the variance-swap strip) — model-free, the strongest kind of pricing; dynamic replication (delta-hedging: the binomial and hedging lessons) — model-dependent, path-dependent costs; model pricing (Monte Carlo from the probability course, trees) — the fallback when neither replication exists, inheriting every model assumption. Interviewers reliably reward candidates who reach for the highest available rung: "before simulating, can I bound or build this from vanillas?"

The interview version

"Price a contract paying \1M if the stock is above \120 in 3 months." — A digital: 1M call spreads at 120/121 (or off $-\partial C/\partial K$ from the chain), skew-adjusted, and quote the hedging-cliff caveat if asked why you'd width the spread. "An up-and-out call is knocked out at 130 — worth more or less than the vanilla, and when is the difference largest?" — Less (in-out parity: you've given away the knock-in), and the gap yawns when spot nears 130 with time remaining. Payoff decomposition first, model second, Greeks-at-the-cliff caveat last — the complete house style of this course.