Brownian Bridge™

1. A portfolio has zero net vega ( $\sum_k n_k \nu_k = 0$ ) but consists of a long ATM straddle and a short OTM strangle. What risk does this portfolio carry that zero net vega does not reveal?

Delta risk: the straddle and strangle have different deltasVolga risk: OTM options have positive volga, so the portfolio is long volga despite zero net vegaTheta risk: the positions have different time decaysRho risk: the two positions have different rate sensitivities

2. Under a sticky-delta smile model, the portfolio delta is $\Delta_{\text{model}} = N(d_1) + \nu \cdot \partial\hat{\sigma}/\partial S$ . For a long call with positive vega and a downward-sloping smile (implied vol rises as spot falls, $\partial\hat{\sigma}/\partial S < 0$ ), how does the model delta compare to the Black-Scholes delta?

Model delta > BS delta: the smile adjustment adds to the hedgeModel delta < BS delta: rising vol as spot falls reduces the effective deltaModel delta = BS delta: the smile adjustment cancels under put-call parityThe sign depends on whether the option is a call or a put

3. The vanna of an OTM put ( $d_2 < 0$ ) is positive: $\text{Vanna} = -n(d_1)d_2/\sigma > 0$ . A risk-reversal is long OTM call and short OTM put. What is the sign of the risk-reversal's net vanna?

Zero: the call vanna and put vanna cancelPositive: long OTM call contributes positive vanna; short OTM put contributes negative vanna (short position flips sign)Negative: the short put position dominatesIndeterminate without knowing the specific strikes and notionals

4. DV01 (dollar value of a basis point) is defined as the P&L from a 1bp increase in all rates. Its sign is negative for long bond positions.

truefalse

5. Vanna-Volga pricing of FX exotics uses three vanilla options (ATM, 25-delta call, 25-delta put) to construct a hedge that replicates the vanna and volga of the exotic. What additional risk does this hedge leave unhedged?

Delta riskThird-order vol sensitivities (e.g.,

\partial^3 C/\partial\sigma^3

) and path-dependent smile risk not captured by the static vanna-volga replicationGamma riskTheta risk

6. A portfolio aggregation shows zero net delta and zero net gamma, but significant positive volga across all positions. Which scenario would most harm this portfolio?

A large spot move with no vol changeA moderate spot move combined with a large vol move in one direction (vol either much higher or much lower)A quiet market day with no price changesA vol move of exactly the same magnitude as the daily average vol move

Quiz: Portfolio Greeks Aggregation

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