Case study

# Market-Implied Probability Explorer — SPY Oct 16, 2026

**Dataset:** SPY — broad market ETF
**As of:** Sep 25, 2026 · **Spot:** $578.40 · **Expiration:** Oct 16, 2026 (21 days)

## Research question
What terminal price distribution does the SPY option chain imply for Oct 16, 2026, and is that distribution skewed toward downside or upside outcomes?

## Data and cleaning
- 74 raw quotes for this expiration; 73 survived cleaning, 1 were filtered.
- Mid prices computed from bid/ask; calls and puts separated, with calls used for the density estimate.
- Filter reasons:
  - Implied vol could not be solved: 1
- Implied volatilities solved by bisection where not supplied.

## Method
1. Fit the call-side implied volatility smile with Gaussian kernel regression on log-moneyness IV (bandwidth 0.055 in log-moneyness).
2. Re-price calls with Black-Scholes on a 401-point strike grid using r = 4.30% and q = 1.30%.
3. Apply Breeden-Litzenberger: q(K) = e^{rT} · ∂²C/∂K², computed as a central second difference of the smooth call curve.
4. Clip negative density values (0 grid points) and normalize total probability to 1.

## Findings
| Measure | Value |
| --- | --- |
| Implied expected price | $579.35 |
| Implied median | $580.16 |
| Implied mode | $580.80 |
| 5th percentile | $543.29 |
| 95th percentile | $612.92 |
| 1σ implied move | ±3.7% |
| P(finish below spot) | 46.5% |
| P(finish 10% lower or worse) | 0.9% |
| P(finish 10% higher or better) | 0.1% |

## What the distribution implies
Options imply a broadly symmetric distribution. Left and right tails are close (6.1% vs 6.0%), with an implied drift of 0.2% to the risk-neutral mean.

## Limitations
- Risk-neutral probabilities embed a risk premium; they are not real-world forecasts.
- Second derivatives amplify quote noise, so results depend on the smoothing bandwidth.
- The density is only defined across the quoted strike range; tails beyond it are truncated.
- Black-Scholes re-pricing treats these listed options as European and ignores early exercise.
- Demo chains are deterministic synthetic data for reproducibility, not live market quotes.

## Next steps
- Add a no-arbitrage constrained smile fit (SVI or spline with convexity constraints).
- Extend to a full term structure and compare densities across expirations.
- Fit parametric tails (generalized Pareto) beyond the quoted strike range.
- Backtest realized outcomes against implied probabilities to measure the variance risk premium.

_Educational research tool. Not investment advice._