Author: Alessandro Radice · M.Sc. Economics and Business Law (Finance), Università Cattolica del Sacro Cuore, Milan
Live page: alessandroradice.github.io/SP500-hedging-analysis
1 April 2025, after the close. A $500M US equity book, beta 1. Reciprocal tariffs are announced the next day at 4pm. Do we hedge, with which structure, and at what cost?
An equity derivatives case study built on real market data. It rebuilds the S&P 500 implied volatility surface every trading day from January to June 2025 from real SPY option quotes (bid and ask), fits an arbitrage-free SVI smile to each expiry, and checks the result against the Cboe VIX, which the fit never sees. The surface then prices four hedges on 1 April at the real quotes and marks them every day through the crash, the 90-day tariff pause and expiry on 30 May. The main output is an interactive web page with an animated 3D volatility surface. It comes with a Colab notebook, an Excel hedge pricer with live formulas, an investment memo and a presentation deck.
A volatility surface is the object every options desk prices from. This project builds one from raw quotes and then uses it to answer a portfolio question:
- Build the surface properly. Clean quotes, infer the underlying and the forwards, compute implied vols, fit SVI per expiry and join the expiries without static arbitrage.
- Prove it is right. Rebuild the 30-day VIX from the surface out of sample, and audit butterfly and calendar arbitrage on the whole interpolated surface.
- Use it for a decision. Say whether protection was cheap or expensive on 1 April, price four structures, and follow them day by day under two management rules.
The surface
| Check | Result |
|---|---|
| Days and quotes | 122 trading days (2 Jan to 30 Jun 2025), about 51 out-of-the-money quotes a day on 3 expiries |
| Fit error | 0.47 vol points RMSE on average (median 0.41, worst day 1.16) |
| VIX rebuilt out of sample | Mean error −0.15 vol points, correlation 0.996 with the Cboe VIX; worst day 8 April (49.4 vs 52.3) |
| Butterfly arbitrage | 0 violations on 3.4 million grid points |
| Calendar arbitrage | 0.015% of grid points, all in the far call wing (beyond +17%, outside the plotted range), largest breach 4×10⁻⁵ in total variance |
The decision on 1 April (SPY 561.36, 8,907 contracts to cover the book, 30 May expiry)
- Protection was not expensive. 30-day ATM implied vol was 17.7%, below the 20.1% realised over the previous month, and the 25-delta risk reversal (−4.96) sat near its half-year median (−4.51).
- The event was already priced in the short end. VIX9D 24.5 against a VIX of 21.8, an inverted front of the curve.
| Structure | Cost (% NAV) | Max drawdown, held | NAV 30 May, held | NAV 30 May, sold at VIX ≥ 40 (4 Apr) |
|---|---|---|---|---|
| Unhedged | – | −11.8% | $524.4M | $524.4M |
| Long 539 put | 1.49% | −4.6% | $516.9M | $553.0M |
| 539/475 put spread | 1.17% | −7.7% | $518.5M | $541.1M |
| 539/583 collar | 0.25% | −3.4% | $518.0M | $557.6M |
| 539/475/583 put-spread collar | −0.07% (credit) | −6.5% | $519.6M | $545.7M |
- Recommendation: the 539/583 collar, with a monetisation rule decided in advance. It cut the drawdown from −11.8% to −3.4% for 0.25% of NAV, a sixth of the outright put's cost.
- Held to expiry, every hedge lost money. The pause and the US–China truce took SPY back above 580, so the puts expired worthless and the collar gave away the rally above 583 ($6.3M behind the unhedged book).
- Sold on 4 April, when the VIX closed at 45.3, the same collar finished $33.2M ahead of the unhedged book. The monetisation rule mattered more than the choice of strikes.
| Step | Module | What it produces |
|---|---|---|
| 1 | Data | SPY end-of-day option chains (DoltHub), SPY prices, Cboe VIX9D / VIX / VIX3M / VIX6M / SKEW, Treasury yields (FRED) |
| 2 | Cleaning and implied vols | Underlying inferred from put-call parity, forwards, Black-76 implied vols from bid, mid and ask |
| 3 | SVI per expiry | Raw SVI fitted in vol space with no-arbitrage penalties |
| 4 | Surface across maturities | Total-variance interpolation, √T skew rule anchored to VIX9D, VIX3M and VIX6M |
| 5 | Calibration | 122 days with warm starts, cached |
| 6 | Validation | Fit error, out-of-sample VIX, arbitrage audit |
| 7 | Risk metrics | 30-day ATM vol, 25-delta risk reversal, 90% put skew, realised vol, risk-neutral density |
| 8 | Hedging case | Four structures priced at the 1 April quotes, marked daily, hold vs monetise |
| 9 | Charts | Surface before and during the crash, VIX check, portfolio paths |
| 10 | Export | The interactive page SP500_Hedging_Analysis.html |
| 11 | Excel | The hedge pricer SP500_Hedge_Pricer.xlsx, with live formulas |
SP500_Hedging_Analysis.html opens in any browser:
- Animated 3D surface (implied vol by moneyness and days to expiry) with a day slider, a Play button that runs the half-year, and shortcuts to the key events (DeepSeek sell-off, the March correction, 1 April, the 3–8 April crash, the 9 April pause, the Geneva truce). Market quotes are drawn as dots; the ATM term structure and the 30-day smile are traced on the surface.
- For the selected day: smiles on the three listed expiries with bid/ask ranges and the SVI fit, the term structure against the Cboe indices, and the 30-day implied distribution against 1 April.
- History of the VIX against the VIX rebuilt from the surface, ATM and realised vol, risk reversal and put skew.
- The hedging case: the decision table at the real quotes, portfolio paths with a switch between the two rules, and the results.
Cover · Inputs · Pricer · Scenarios · Paths · Checks
- Inputs: the 1 April market (parity-implied SPY, Treasury rate, dividend yield), the SVI smile of the 30 May expiry fitted on 1 April, the portfolio, and the four structures with their real bid/ask quotes.
- Pricer: Black-76 on the SVI implied vol for every leg (model price, delta, vega, execution at the quotes), then net premium, cost, % of NAV and Greeks of each structure.
- Scenarios: the selected structure at expiry across SPY levels, and before expiry under a spot shock, a horizon and a parallel vol shift.
- Paths: daily marks from the fitted surface, 1 April to 30 May, with the book unhedged, hedged and held, or hedged and sold at the VIX trigger.
- Three switches on
Inputs: structure (C5, 1 to 4), management rule (C6, 1 = hold, 2 = sell at the trigger) and VIX trigger (C7). - Banker colour code: blue = hard-coded input, black = formula, green = link to another sheet. 842 formulas, reconciled with the Python engine to the cent for all four structures under both rules.
The volatility surface itself is calibrated in Python: fitting 122 days of SVI smiles with arbitrage penalties is not a spreadsheet job. The workbook takes the 1 April slice and the daily marks as inputs, and everything the hedging decision needs is a live formula.
- 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). The spot is the median parity-implied spot across the near-the-money strikes of all expiries.
- Forwards and implied vols.
F = S e^((r−q)T), r from the Treasury curve (1M to 1Y), q = 1.25%. Black-76 on out-of-the-money options only. Quotes with a bid below $0.02, a spread above 50% of mid or fewer than 5 days to expiry are dropped. The implied vols match the data vendor's to a median of 0.02 vol points. - SVI per expiry. Raw SVI (Gatheral) fitted in vol space, weighted by the inverse bid-ask spread in vol, with a soft-L1 loss. Penalties enforce Gatheral's g(k) ≥ 0 (no butterfly arbitrage), no crossing with the previous expiry (no calendar arbitrage), positive variance and Lee's bound b(1+|ρ|) ≤ 2. The vertex m is kept inside the quoted strikes and ρ ≤ 0.5, which removes "phantom" fits with an exploding unobserved wing.
- Across maturities. Linear in total variance at fixed log-forward moneyness between listed expiries. Outside them, a √T skew-scaling rule
w_T(k) = λ·w_ref(k/√λ), with λ solved so that the slice's variance-swap vol equals VIX9D (9 days), VIX3M (93 days) or VIX6M (182 days). - Out-of-sample check. The 30-day VIX is never used. It is rebuilt from the surface with the continuous-strike variance-swap formula.
- Hedging case. Contracts = NAV / (SPY × 100). Entry at the real quotes (ask for longs, bid for shorts). Daily marks on that day's fitted surface. Monetisation at the first close with VIX ≥ 40, at model mid minus half the quoted spread on each leg.
- SPY options are American and SPY is not the S&P 500 index, while the Cboe indices are computed on SPX options. Early exercise matters little for out-of-the-money options, but the VIX comparison carries a small basis.
- The dataset has three expiries a day (about 2, 4 and 8 weeks). 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 hedges are marked on the model surface. On the 12 occasions the exact strikes were quoted, model and market mids differ by $0.13 on average.
- The portfolio is assumed to move one-for-one with SPY (beta 1, no dividends inside the window). A real book carries basis risk to the index.
- The VIX ≥ 40 rule is one pre-committed rule among many; the result depends on the path the market took in April 2025.
This project is for educational purposes and is not investment advice.
| Requirement | Details |
|---|---|
| Environment | A Google account to run the notebook in Google Colab, free tier is enough. It also runs in any local Jupyter with Python 3.10+. |
| Python libraries | pandas, numpy, scipy, matplotlib, requests, openpyxl. The first cell installs what is missing. |
| Data | Bundled in data/. If the folder is missing, the notebook downloads everything from DoltHub, Cboe and FRED (a few minutes). |
| To open the outputs | Any modern browser for the page (it loads Plotly and the fonts from public CDNs); Microsoft Excel or Google Sheets; any PDF reader. |
| Background knowledge | Black-Scholes and implied volatility, option structures (spreads, collars), put-call parity. |
- Open
SP500_Hedging_Analysis.ipynbin Google Colab. To use the bundled data, upload thedata/folder next to the notebook; otherwise it is downloaded automatically. - Change the case in the Configuration cell if you want: decision date, expiry, NAV, strikes, monetisation trigger. Strikes and expiry must be listed in the chain on the decision date.
Runtime → Run all. The calibration takes about 10 minutes the first time and is cached incalibration.pkl.- The last two cells write
SP500_Hedging_Analysis.htmlandSP500_Hedge_Pricer.xlsxand, in Colab, download them. - In Excel, change
Inputs!C5(structure),C6(hold or sell) orC7(VIX trigger), or edit any blue cell.
├── SP500_Hedging_Analysis.ipynb # the notebook (run this)
├── sp500_hedging_analysis.py # same code as a plain Python script
├── SP500_Hedging_Analysis.html # interactive page with the animated 3D surface
├── index.html # same page, served by GitHub Pages as the live link
├── SP500_Hedge_Pricer.xlsx # Excel hedge pricer with live formulas and switches
├── SP500_Hedging_Memo.pdf # investment memo
├── SP500_Hedging_Deck.pdf # seven-slide presentation
├── data/
│ ├── spy_options_raw.csv # SPY option chains, 16 Dec 2024 to 30 Jun 2025 (bid, ask, vendor IV, Greeks)
│ ├── spy_ohlcv.csv # SPY daily prices
│ ├── cboe_indices.csv # S&P 500, VIX9D, VIX, VIX3M, VIX6M, VIX1Y, SKEW, VVIX
│ └── fred_treasury.csv # 1M, 3M, 6M and 1Y Treasury yields
├── surface_2025-04-08.png # images used in this README
├── hedge_paths.png
├── smiles_term_density.png
└── README.md
- SPY option chains and prices: post-no-preference/options and post-no-preference/stocks, DoltHub
- Cboe historical index data: VIX, VIX9D, VIX3M, VIX6M, SKEW, S&P 500
- FRED, Federal Reserve Bank of St. Louis: DGS1MO, DGS3MO, DGS6MO, DGS1
- Gatheral, J. and Jacquier, A. (2014), Arbitrage-free SVI volatility surfaces, Quantitative Finance 14(1)
- Lee, R. (2004), The moment formula for implied volatility at extreme strikes, Mathematical Finance 14(3)
- Breeden, D. and Litzenberger, R. (1978), Prices of state-contingent claims implicit in option prices, Journal of Business 51(4)
- Cboe, VIX White Paper (variance-swap methodology)
- 2025 stock market crash, Wikipedia (timeline of the April 2025 tariff shock)
- Dow surges 2,900 points, S&P 500 posts biggest gain since 2008 on Trump tariff reversal, CNBC (April 2025)
- Stocks soar after U.S. temporarily cuts China's tariffs, NPR (May 2025)
Python · pandas · numpy · scipy · matplotlib · openpyxl · Plotly.js · Google Colab · Excel