Derivatives · Group project · March 2026
Pricing an S&P 500 call.
A small tree makes the assumptions visible. Follow how possible index levels become payoffs, then how those payoffs become a price today.
An assumption at a time
Explore the call value.
Inputs dated
Adjust volatility and time to expiry. The spot, strike and interest rate stay fixed, so you can see what each change does to the same European call.
The original one-step scenario is shown. Enable JavaScript to adjust its assumptions.
9.6278% annual volatility · 6 months to expiry.
Original one-step inputs
281.85Index points · model value
Reproduces the report’s 281.85 headline.
- Spot / strike
- 6,466.58 / 6,466.58
- Risk-free rate
- 3.9984% a year
- Risk-neutral up weight
- 63.12%
- Up / down factors
- 1.07045 / 0.93419
The up weight is a pricing input, not a forecast of the market’s next move. This dated model has no live prices.
The price, branch by branch
S = index level · C = call value
On a small screen, swipe across the tree or open the node table.
Inspect every node
| Path | Months | Index (S) | Call (C) |
|---|---|---|---|
| Start | 0 | 6,466.58 | 281.85 |
| Up | 6 | 6,922.15 | 455.57 |
| Down | 6 | 6,040.99 | 0.00 |
Inspect every node
| Path | Months | Index (S) | Call (C) |
|---|---|---|---|
| Start | 0 | 6,466.58 | 224.72 |
| Up | 3 | 6,785.49 | 383.23 |
| Down | 3 | 6,162.66 | 0.00 |
| Up / up | 6 | 7,120.13 | 653.55 |
| Up / down | 6 | 6,466.58 | 0.00 |
| Down / down | 6 | 5,873.02 | 0.00 |
Assumptions, formulas and source reconciliation
Starting point. The March 2026 group report’s one-step example uses spot and strike of 6,466.58, 9.6278% annual volatility, a 3.9984% annual risk-free rate and six months to expiry, dated 13 August 2025. Its full-sample volatility of 18.57% belongs to a different estimation period.
One consistent convention. For each step Δt = T / n: u = exp(σ√Δt), d = 1/u, R = exp(rΔt), and q = (R − d)/(u − d). At expiry C = max(S − K, 0); at earlier nodes C = [qCup + (1 − q)Cdown]/R. Rates are continuously compounded throughout, with constant volatility and no dividends.
Reconciliation. The report mixes an effective-annual probability formula with continuous discounting, and its printed probability labels disagree. Continuous compounding throughout reproduces 281.8541, rounded to the reported 281.85. The explorer recalculates the probabilities and every node instead of copying those inconsistent labels.
Two-step boundary. Switching to two steps keeps the same dated inputs. It does not reproduce the report’s later 252.20 and 256.59 examples, which use different, partly ex-post volatility and rate estimates. One or two steps illustrate the mechanics; they are a coarse approximation, not a calibrated market option quote. Values are index points with no contract multiplier.
Source: original report, PDF pages 5–6, equations 3–6 . Pricing background: CFA Institute’s one-period binomial model .
Decisions and lessons · Team study
Keep the information date in view.
- Decision
- The report compared simple CRR trees with a later time-varying example to examine how volatility and rates affect the result.
- Lesson
- A model fitted with information realised after issuance answers a different question from a price formed on the issuance date. A more elaborate tree does not remove that distinction.
- Next check
- A further study would estimate every input using only information available at issuance, use one rate convention throughout, and test convergence with more time steps.