Implied Skew Reconstruction — Decrement Baskets

Consider a custom basket of $N$ stocks with initial weights $w_i$, subject to a fixed decrement $d$ (typically 50 bps/year or 5% annualised). This basket is the underlying of an autocall structure.

No options trade on this basket — so there is no observable implied vol surface. However, listed options exist on each individual component $S_i$, giving us full implied vol surfaces $\sigma_i^{\text{imp}}(K, T)$.

The goal: reconstruct the implied vol surface of the basket from the single-name surfaces and a correlation structure, in a way that is arbitrage-free and consistent with market data.

The basket performance index at time $t$, before decrement, is defined as:

For a proportional decrement at rate $d$ per annum:

The decrement acts as a forward shift — it lowers the forward of the basket without changing its volatility directly. The adjusted forward is:

Instantaneous variance of the basket. Under a multi-asset diffusion where each stock follows its local vol dynamics, the instantaneous variance of the basket is:

where $\tilde{w}_i(t) = w_i \, S_{i,t} / B_t$ are the effective weights at time $t$, which drift dynamically as stocks diverge from each other.

Step 1 — Calibrate local vol (or SLV) on each component. For each stock $i$, invert its listed implied surface to obtain $\sigma_i^{\text{loc}}(S, t)$ via Dupire. For better forward-skew dynamics (critical for autocalls with distant barriers), calibrate a stochastic-local vol model (e.g. Heston + local vol lever function).

Step 2 — Specify the correlation structure. Build $\rho_{ij}$ either from historical realised correlations or — better — by extracting implied correlation from a benchmark index. If the benchmark index has vol $\sigma_{\text{idx}}^{\text{imp}}$, known weights $\omega_i$, and the component vols are known, invert:

for a flat implied correlation $\hat{\rho}$, or fit a parametric structure. Extract a term structure by doing this per maturity.

Step 3 — Multi-asset Monte Carlo simulation. Generate $M$ paths (typically 200k–500k for smooth skew) for all $N$ stocks simultaneously. At each time step:

• Draw correlated Gaussians via Cholesky decomposition of $\rho$.
• Evolve each $S_i$ under its own local vol $\sigma_i^{\text{loc}}(S_i, t)$.
• Compute $B_t$ and apply the decrement to get $I_t$.
• If the basket rebalances, reset weights at rebalancing dates.

Step 4 — Price vanillas on the basket and invert. From the simulated distribution of $I_T$, compute call and put prices on the basket for a grid of strikes $K$ and maturities $T$:

Then invert Black-Scholes for each $(K, T)$ to recover:

Step 5 — Smooth and parametrise. Fit the raw implied vol points with a parametric model — SVI (Gatheral) or SABR — to obtain a smooth, arbitrage-free, interpolable surface. This is what gets passed to the autocall pricer.

The reconstructed skew of the basket is not assumed — it emerges mechanically from the simulation. It has three distinct drivers:

Single-name skew propagation

Each stock's local vol surface encodes its own skew: $\sigma_i^{\text{loc}}(S, t)$ is an increasing function of moneyness on the downside. In crash scenarios, all stocks move into regions of higher local vol, which inflates the basket variance. This propagates individual skew into the basket.

Correlation regime shift

In practice, correlations rise in down markets (the "correlation smile"). Even with a fixed $\rho_{ij}$ matrix, the effective correlation of the basket increases in down scenarios because individual skews push all vols higher simultaneously. If using stochastic or regime-dependent correlation, this effect is amplified further. This fattens the left tail of the basket distribution.

Weight concentration (dispersion effect)

In a fixed basket (no rebal), a stock that drops 40% sees its effective weight $\tilde{w}_i$ shrink, naturally de-risking the basket. But in a rebalanced basket, the weights are reset — you re-concentrate exposure on the loser, which now has the highest vol. This creates a pro-cyclical vol injection mechanism. Quantitatively, the post-rebal basket variance is higher than the pre-rebal variance because:

because reverting to equal weights (or initial weights) from post-crash drifted weights increases the contribution of high-vol stocks.

An autocall investor is long coupons + early redemption and short a down-and-in put (KI barrier, typically 50–60% of initial). This makes the investor structurally short vol and short skew.

Vega exposure. The Vega of the autocall is net negative, dominated by the short put embedded in the KI. Higher basket vol → higher KI put value → lower autocall price to the investor. The rebalanced basket, being structurally more volatile, therefore commands a higher coupon (compensation for taking more risk) than the same structure on a fixed basket.

Skew exposure. The autocall's short KI put is deep OTM — its price is directly driven by the left tail of the basket distribution. A steeper skew means higher implied vol at the KI strike, which increases the embedded put value disproportionately. The pricing difference between a correctly reconstructed skew and a flat vol-ratio proxy can be significant — on the order of 50–150 bps of coupon for a typical 3Y–5Y autocall, depending on basket composition and barrier level.

Decrement amplification. The decrement lowers the forward of the basket, which has two effects:

• The ATM vol reference shifts (the "moneyness" of the KI barrier changes relative to the forward).
• The basket starts "closer" to the KI barrier in forward space, making the put more in-the-money in forward terms and increasing the sensitivity to the downside vol.

As $d$ increases, the effective moneyness of the KI barrier increases, making the embedded put more expensive and pushing the coupon higher.

Computational cost. Multi-asset MC with local vol on 20–50 underlyings requires 200k–500k paths for a smooth skew surface. This is GPU-amenable (embarrassingly parallel). Variance reduction via antithetic variates and control variates (using the benchmark index as control) can reduce the path count by 3–5x for the same accuracy.

Correlation calibration. The correlation matrix is the least observable and most sensitive input. Best practice is to calibrate on index dispersion trades (long index vol / short single-stock vol) or on correlation swap levels when available. The key sensitivity to test is the correlation skew — how much correlation increases in down scenarios — since this is the primary driver of the basket's left tail.

Rebalancing frequency. Discrete rebalancing (monthly, quarterly) must be modelled explicitly in the simulation. More frequent rebalancing → more vol injection → steeper skew. The rebalancing schedule is contractually fixed in the index rules, so it's not a modelling choice but a constraint to respect.

Stability. The reconstructed surface should be checked for calendar spread and butterfly arbitrage. The SVI/SABR fit enforces no-arbitrage constraints. Day-to-day stability of the surface depends on the stability of the single-name inputs and the correlation matrix — bumping each by 1 standard deviation and re-running the pipeline gives a practical measure of model risk.