Brownian Bridge™

1. In the raw SVI formula $w(k) = a + b[\rho(k-m) + \sqrt{(k-m)^2 + \sigma^2}]$ , what are the asymptotic slopes of $w(k)$ as $k \to +\infty$ and $k \to -\infty$ respectively?

b(1+\rho)

for

k \to +\infty

and

b(\rho-1)

for

k \to -\infty

(in absolute value:

b(1-\rho)

)

b\rho

for both wings, since

\rho

controls the asymmetry

b

for both wings, since

\sqrt{(k-m)^2 + \sigma^2} \approx |k-m|

in both limits

0

for both wings, since

w(k)

converges to a constant

2. The no-butterfly-arbitrage condition for an SVI slice requires $g(k) \geq 0$ for all $k$ , where $g(k)$ involves $w$ , $w'$ , and $w''$ . Why is this connected to the Dupire local variance?

Negative

g(k)

implies negative call prices at some strikesThe Dupire local variance is

\sigma_{\mathrm{loc}}^2 = \partial_T w / g(k)

; negative

g

makes the local variance negative, which is an arbitrageNegative

g(k)

causes the Newton-Raphson implied vol inverter to divergeThe condition

g(k) \geq 0

is equivalent to the vol smile being convex

3. In the SSVI parametrisation, the sufficient condition $\theta_t \phi(\theta_t)(1 + |\rho_\infty|) < 4$ must hold for global no-butterfly-arbitrage. This condition is related to:

The Feller condition for the CIR variance process underlying SSVILee's moment formula, which bounds the wing slope of total variance as

|k| \to \infty

The positive semidefiniteness of the Jacobian used in SSVI calibrationThe no-calendar-spread condition across consecutive maturities

4. A vol surface calibrated by fitting each maturity slice independently with raw SVI is automatically free of calendar spread arbitrage.

truefalse

5. The jump-wings (JW) parametrisation of SVI replaces $(a, b, \rho, m, \sigma)$ with financially meaningful quantities. Why is JW preferred over raw SVI for numerical calibration?

JW reduces the number of free parameters from 5 to 3JW parameters correspond directly to observable market quantities (ATM vol, skew, wing slopes), giving better-conditioned initial guesses and a more orthogonal parameter spaceJW guarantees the calibrated smile is free of butterfly arbitrageJW uses a different functional form that is easier to differentiate analytically

6. An SVI calibration produces parameters $(a=0.04, b=0.2, \rho=-0.7, m=0, \sigma=0.1)$ for the 1-year slice. What is the approximate ATM total variance $w(0)$ and the ATM implied vol?

w(0) = 0.04 + 0.2 \times 0.1 = 0.06

;

\sigma_{\mathrm{ATM}} \approx 24.5\%

w(0) = 0.04 + 0.2 \times (-0.7 \times 0 + 0.1) = 0.06

;

\sigma_{\mathrm{ATM}} = \sqrt{0.06/1} \approx 24.5\%

w(0) = 0.04 + 0.2 \times (0 + 0.1) = 0.06

;

\sigma_{\mathrm{ATM}} \approx 6\%

w(0) = a = 0.04

;

\sigma_{\mathrm{ATM}} \approx 20\%

Quiz: SVI Parametrisation

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