Brownian Bridge™

1. A down-and-out call (DOC) has strike $K = 100$ , barrier $H = 80$ , and the same maturity as a vanilla call. Without using any model, what is the price of the corresponding down-and-in call (DIC) if the vanilla call is worth $\$ 10.50 $and the DOC is worth$ \ $8.30$ ?

DIC = $2.20, by knock-in/knock-out parity: KI + KO = Vanilla.DIC = $10.50, because the down-and-in call always equals the vanilla call when H < K.DIC = $18.80, because knock-in options are more expensive than vanilla options.DIC cannot be determined without knowing the model — parity only holds under Black-Scholes.

2. Under Black-Scholes with $r = q = 0$ and constant $\sigma$ , the down-and-out call formula simplifies to: $C_{DOC} = c_{BS}(S_0, K) - \frac{H}{S_0}\, c_{BS}\!\left(\frac{H^2}{S_0}, K\right)$ What is the probabilistic interpretation of the image term $\frac{H}{S_0}\, c_{BS}(H^2/S_0,\, K)$?

It prices the contribution of reflected Brownian paths that cross the barrier — paths that are subtracted to enforce the knock-out condition, using the reflection principle.It is the rebate paid to the holder when the barrier is breached.It is the price of a vanilla call with spot H²/S₀, which must be subtracted to avoid double-counting terminal payoffs.It is the risk-neutral probability of the spot ending above K, conditional on having crossed H.

3. A down-and-out call (DOC) has $S_0 = 100$ , $K = 100$ , $H = 90$ , $T = 1$ week. As spot moves from 95 to 91 (approaching the barrier from above), what happens to the option's delta?

Delta becomes large and negative: the option is about to knock out to zero value, so the value collapses rapidly as spot approaches H, creating an enormous negative delta.Delta becomes large and positive: the option is deep in-the-money as spot falls.Delta is approximately zero near the barrier because the option has negligible time value.Delta is approximately +1 near the barrier because the option will either survive to pay the full vanilla payoff or knock out.

4. A desk holds a long down-and-out call with $H = 80$ and current spot $S = 85$ , maturity $T = 3$ months. Is the option's vega positive or negative at this spot level, and why?

Negative: with spot close to H, higher vol increases the probability of breaching the barrier during the remaining 3 months, reducing the DOC value.Positive: higher vol always increases option values by inflating the tails of the distribution.Zero: barrier options have no vega because the barrier condition removes the vol sensitivity.Positive for S > K and negative for S < K, regardless of the barrier level.

5. An equity barrier option is specified as a down-and-out put monitored daily at market close. The continuous-barrier BS formula gives a price of $\$ 3.20 $. The Broadie-Glasserman-Kou (1997) continuity correction for daily monitoring ($ \Delta t = 1/252 $) uses$ \beta \approx 0.5826 $. With$ \sigma = 25\%$, in which direction does the correction shift the effective barrier, and does it increase or decrease the DOC price?

The correction shifts the down-barrier upward (H_eff = H · exp(+β·σ·√Δt) > H), making breach less likely → DOP is worth more than the continuous-barrier price.The correction shifts the down-barrier downward (H_eff < H), making breach more likely → DOP is worth less.The correction does not shift the barrier but adjusts the vol upward to compensate for infrequent monitoring.The correction shifts the barrier upward for calls and downward for puts.

6. A down-and-out call is priced under Black-Scholes with ATM vol $\sigma_{ATM} = 20\%$ at $\$ 5.00 $. The equity vol surface has a strong negative skew: implied vol at the barrier level$ H = 80 $is$ \sigma(H) = 28\%$. Under a smile-consistent model (local vol calibrated to the full surface), in which direction does the DOC price move relative to the flat-vol BS price, and why?

The DOC price decreases below $5.00: higher local vol near H increases the probability of breaching the barrier, reducing the knock-out option value.The DOC price increases above $5.00: higher vol at H inflates the option's intrinsic value near the barrier.The DOC price is unchanged: smile adjustments affect only vanilla options, not barriers.The DOC price could go either way, depending on whether the barrier is OTM or ITM.

7. Which of the following barrier option structures is most commonly embedded in autocallable notes sold to retail investors, and what is the key risk for the issuing desk?

Down-and-in put: if the underlying falls below a barrier (e.g., 60% of initial spot), the investor receives the (depreciated) underlying instead of par — the desk is long this DIP and must manage its barrier Greeks.Up-and-out call: the investor benefits if the underlying rises but loses the upside above a cap — the desk hedges with barrier-free calls.Down-and-out call: used to reduce premium cost; if the underlying falls below the barrier, the call is cancelled and the investor loses all premium.Up-and-in put: the investor is protected against large upward moves in the underlying.

8. A quant analyst is asked to price a barrier option where the underlying can jump. Under a pure Black-Scholes framework (continuous paths), the barrier is always triggered by a continuous crossing. What is the key pricing error introduced when jumps are present?

The underlying can jump over the barrier without triggering it — the continuous-path formula underestimates the barrier survival probability and overprices knock-out options.Jumps always increase the probability of barrier breach, so the BS formula underprices knock-out options.Jumps are irrelevant for barrier pricing because only the terminal distribution of S_T matters.Jumps make the Black-Scholes formula invalid only for up-barriers, not down-barriers.

Quiz: Barrier Options — Pricing and Risk

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