Autocall Hedging: LSV vs LV · STELLA

01

Skew and Forward Skew under LV and LSV

Under $\text{LV}$ , forward skew flattens significantly. Under $\text{LSV}$ , forward skew is steeper than LV — but still flattens with tenor.

FigLSV vs LV forward skew

Forward starting smile comparison. LSV preserves more skew in the forward smile than LV.

FigForward skew in LSV (term structure)

Even under LSV, forward skew flattens with tenor: 6m1y is flatter than 3m1y.

◎Explanation

Under LSV, forward skew is steeper than LV. But even under LSV, it still flattens with tenor: $6\text{m}1\text{y}$ is flatter than $3\text{m}1\text{y}$ .

02

Why Is an Autocall Short Skew but Long Forward Skew?

Setup: effective life $= 1Y$ , maturity $T = 3Y$ . At $t = 1M$ , the trader is already exposed to instantaneous forward skew. The forward skew $3M \to 1Y$ is steeper under LSV — forward-starting vanillas in the hedge cost more.

Vega, Vanna and re-hedging

FigVega & Vanna — Trading View

Vega profile of an autocall with barrier and autocall trigger levels marked. Vanna (dVega/dS) drives re-hedging costs.

Vanna < 0 right of vega peak: when $S \downarrow$ , vega $\uparrow$ — trading gets longer vol on the way down. Must re-hedge by buying vol at higher levels. Under LSV this costs more (steeper forward skew). Exposure concentrated around the barrier.

⚠Coupon effect — partial offset

KI barrier → short forward skew (trading). Snowball coupons → long forward skew (opposite sensitivity) (trading). Large coupons partially compensate → net LSV/LV gap shrinks.

03

Another Explanation: Bergomi's SSR

SSR Definition(3.1)

\text{SSR}_\tau = \frac{1}{skew} \cdot \frac{d\sigma_F}{d\ln S_t}

$skew$ : slope of vol surface at forward level. $d\sigma_F$ : change of ATM forward vol. $d\ln S_t$ : log return of spot.

SSR = how implied vol moves with spot, in units of skew. $\text{SSR} = 2$ : 1% spot move shifts ATM vol by $2 \times skew$ .

Regression: $\Delta\sigma_{\text{ATM}} = m \cdot (\text{log ret} \times skew) + b$

⚠LV overprices SSR

Under LV: $\text{SSR} \to 2$ as $T \to 0$ , $\text{SSR} \to 3$ as $T \to \infty$ . So $\text{SSR}_{\text{LV}} \in [2, 3]$ — systematically too high vs realized.

SSR and forward skew pricing

$\text{SSR} \uparrow$ ⟹ fwd skew flattens ⟹ autocall cheaper
$\text{SSR} \downarrow$ ⟹ fwd skew steepens ⟹ autocall more expensive

Under LSV, $\rho$ controls spot-vol. $\rho$ lower → more SV → vol rises more when $S \downarrow$ → steeper fwd skew → autocall more expensive.

FigSX5E 3m Realized SSR — Rolling 20-Day Windows

Realized SSR sits mostly below the LV band [2,3], confirming LV systematically overstates spot-vol coupling.

04

LSV Calibration

The term structure of $|\rho(S, \sigma)|$ is concave increasing. We observe it at discrete tenors and compute a weighted average to calibrate $\bar\rho$ .

FigTerm structure of |ρ(S, σ)| — weighted average

Observed correlation at each tenor (dots), fitted curve, and the vega-weighted average ρ̄ used for calibration.

Weighted average and limitations

Weighted Average(4.1)

\bar{\rho} = \sum_i w_i \cdot \rho_i

Weights $w_i$ = autocall's vega exposure at each tenor bucket.

⚠Limitation

A single $\bar{\rho}$ does not capture the term structure of vol-of-vol. Realized $\rho$ varies across tenors (concave curve) — flat $\bar{\rho}$ overestimates at short dates, underestimates at long dates → mispricing at both ends.