S&P 500 Hedging AnalysisVolatility surface and collar pricing

The surface, day by day

Implied volatility across moneyness (strike over spot) and time to expiry. Dots are the market quotes the surface was fitted to. The white line traces the at-the-money term structure, the pale line the 30-day smile. Drag to rotate; press Play to run the half-year.

How the surface is built

1. Quotes. End-of-day SPY option chains (bid, ask) for 122 trading days, three expiries a day (about 2, 4 and 8 weeks), strikes from roughly −25% to +17% around spot.
2. Underlying at the snapshot. The quotes are not always synchronous with the official close (on 28 March the chain implies SPY near 568 against a 555.66 close), so the spot is inferred from put-call parity: the median across the near-the-money strikes of all expiries.
3. Forwards and rates. Forward = spot × e^(r−q)T, with r from the Treasury curve (1M to 1Y, FRED) and a 1.25% dividend yield.
4. Implied vols. Black-76 on the forward, out-of-the-money options only, from bid, mid and ask. Quotes with a bid below $0.02, a spread above 50% of mid or fewer than 5 days to expiry are dropped.
5. SVI per expiry. Gatheral's raw SVI fitted in vol space, weighted by the inverse bid-ask spread, with a robust loss. Penalties enforce no butterfly arbitrage (g(k) ≥ 0), Lee's wing bound and no crossing with the previous expiry.
6. Across maturities. Linear in total variance at fixed log-forward moneyness between listed expiries. Beyond them, a √T skew-scaling rule with the level anchored to Cboe VIX9D, VIX3M and VIX6M.
7. Out-of-sample check. The 30-day VIX is never used. Rebuilding it from the surface (continuous-strike variance swap) gives a mean error of vol points and a correlation of with the Cboe index.
8. Arbitrage audit. 3.4 million grid checks: no butterfly violations; calendar breaches on 0.015% of points, all in the far call wing (beyond +17%, outside the plotted range) and below 4×10⁻⁵ in total variance.

Limitations

• SPY options are American and SPY is not the S&P 500 index; the Cboe indices are computed on SPX. Early exercise matters little for the out-of-the-money options used here, but the VIX comparison carries a small basis.
• Only three expiries a day are available. Tenors shorter than about 2 weeks or longer than about 8 weeks come from the √T rule and the Cboe anchors, not from quotes.
• The hedge is marked on the model surface. On the 12 occasions the exact strikes were quoted, model and market mids differ by $ on average.
• The book is assumed to move one-for-one with SPY (beta 1, no dividends in the window). Real portfolios carry basis risk to the index.

Data and sources

1. SPY end-of-day option chains and prices: post-no-preference/options and stocks, DoltHub
2. VIX, VIX9D, VIX3M, VIX6M, SKEW and S&P 500 closes: Cboe historical index data
3. Treasury constant-maturity yields (DGS1MO, DGS3MO, DGS6MO, DGS1): FRED, Federal Reserve Bank of St. Louis
4. Gatheral & Jacquier (2014), Arbitrage-free SVI volatility surfaces, Quantitative Finance
5. Lee (2004), The moment formula for implied volatility at extreme strikes, Mathematical Finance
6. Breeden & Litzenberger (1978), Prices of state-contingent claims implicit in option prices, Journal of Business
7. 2025 stock market crash, timeline of the April 2025 tariff shock
8. Dow surges 2,900 points, S&P 500 posts biggest gain since 2008 on Trump tariff reversal, CNBC (9 April 2025)
9. Stocks soar after U.S. temporarily cuts China's tariffs, NPR (12 May 2025)