Originally published on Risk.net, 24 September 2026. Reproduced with permission.
Key takeaways
No model, no guesswork. The forward volatility curve is extracted directly from listed option prices — no assumptions about how volatility behaves over time.
Zero-cost to implement. The hedge is structured so it costs nothing to put on at inception.
Isolates the risk you actually care about. It separates ordinary spot risk from pure volatility risk, so you can hedge volatility exposure specifically — without touching your spot risk.
Tested against a real market shock. Run against a real multi-expiry SPX portfolio, the hedge held steady under a significant volatility shock, while an unhedged book moved materially.
No stock position required. The entire strategy is built and proven using options alone.
Introduction
Risk managers use forwards to organise risk. An equity or foreign exchange forward is a price fixed today for delivery later, replicated by a static position in spot and cash. No dynamics assumption is required: no-arbitrage alone pins the price down, and every desk agrees on it.
Options carry no such anchor: any answer to ‘what will my book look like in three months?’ requires a model of the dynamics, and desks disagree. This feature constructs the missing reference – a forward volatility curve extracted from current option prices with no modelling assumption – and shows it is achievable by an explicit, zero-cost static hedge.
Implied Probability Density Functions
We work throughout in log-moneyness k = ln(K/F), with the forward at k = 0; this centres the smile at the origin and simplifies every formula that follows. A call is then:
C(S) = e−rT · F · c(k)
where c(k) is the price per unit of forward. The discount factor and the forward enter every expression below as the same two multipliers, so we suppress them and work with c(k) throughout.
An option’s smile and its implied risk-neutral density p(k) carry the same information; p(k) is the second strike-derivative of the call price. The density is the more useful of the two: it is the price, per unit notional, of an infinitesimal butterfly struck at k – a payoff converging to a Dirac delta at expiry. We write [p(k)dk] for an actual butterfly, of value p(k)dk – used throughout. Any payoff f(k) is therefore priced and hedged as ∫f(k)·[p(k)dk].
The forward density p12(k) – prevailing over [T1,T2] – is pinned down by the convolution: the T2 outcome is the T1 outcome convolved with the move over the forward period.
p2(k) = ∫ p12(k − k′) · p1(k′) dk′
Solved by Fourier transform, φ12 = φ2/φ1, then inverse transform, with light smoothing to remove deconvolution artefacts. Reconvolving p12 with p1 reproduces p2 to within 0.1% of its mass on real S&P 500 (SPX) data. Figure 1 plots all three:
p₁ and p₂ – the fitted risk-neutral densities at T₁ and T₂ – and the forward density p₁₂ recovered between them by Fourier deconvolution. T₁ = 25, T₂ = 81 days.
Implied vol follows in two steps: integrating each density against the call payoff gives c1(k), c2(k), and c12(k); inverting each through Black-Scholes gives σ1, σ2, and σ12, the forward volatility curve over [T1,T2]. Figure 2 plots all three.
The T₁ and T₂ implied vol smiles σ₁(k), σ₂(k), and the forward volatility curve σ₁₂(k) between them, recovered from the real forward density p₁₂ via c₁₂(k) and Black-Scholes inversion. T₁ = 25, T₂ = 81 days.
Fundamental Equation
The same construction, on call prices rather than densities, delivers the paper’s central result.
Write the T2 call as an integral over forward-weighted T1 butterflies, each weighted by the forward call price c12:
c2(k) = ∫ c12(k − k′) · ek′ p1(k′) dk′
The ek′ is a unit conversion: the butterfly [ek′p1(k′)dk′] pays the realised T1 level, the unit c12 is quoted in. Its weights integrate to one – with p = d²C/dK², ∫ekp(k)dk integrates by parts to the zero-strike call, which must equal the forward. p1 and p2 inherit that from the listed calls behind them; the convolution passes it to p12.
Mark each side as a traded position – in brackets – and take the difference, that is the fundamental equation, zero for every strike today, now written as an options position:
Δ(k) = [c2(k)] − ∫ c12(k − k′) · [ek′p1(k′)dk′] = 0
The right is a strip of forward-weighted T1 butterflies [ek′p1(k′) dk′], each held in the quantity c12(k−k′) – a price, not a traded quantity. Both terms have identical value today, so the position – long the T2 call, short the strip, the forward-vol contract – costs nothing at inception.
Project forward to T1, over a range of realised stock prices. At the forward, the difference is no longer zero: it is the gap between the forward call price under the vol curve that actually realises and the curve locked in today:
Δ(k) at T1 = c12(k; σ̂12) − c12(k; σ12)
This is a clean, one-dimensional bet on the level of the forward vol curve. Figure 3 plots Δ(k) at T1 for the T1 = 25, T2 = 81-day pair (SPX July 27, 2026), under an illustrative +1.0 vol-point shock.
Δ(k) at T₁ – the forward-vol contract’s payoff – under a +1.0 vol-point shock to the fitted forward smile σ₁₂(k) (realised versus locked-in), for three contract strikes, k = −5%, 0 and +5%, against realised spot at T₁. T₁ = 25, T₂ = 81 days.
Because each contract is individually zero-cost, any linear combination across strikes is zero-cost too – a trader can assemble, at no premium, an arbitrary target vega profile by choosing quantities at a handful of strikes.
Hedging an Options Portfolio
This is where the framework becomes operational. The unrealised profit and loss (P&L) of an options portfolio P over [0,T1] decomposes as:
P(S1; Σ1) − P(S0; Σ0)
= [P(S1; Σf) − P(S0; Σ0)] + [P(S1; Σ1) − P(S1; Σf)]
where Σf is the forward surface locked in today and Σ1 is whatever prevails at T1. The first bracket is the P&L if volatility evolved onto the model-free forward – realised-spot risk only. The second is pure volatility risk: the realised surface’s deviation from the locked-in forward, conditional on spot. The fundamental equation replicates the first term at zero cost, supplying an overlay that isolates – and can neutralise – the second.
A single ‘portfolio vega’ is misleading: a three-month and a three-year option both carry vega, but summing them is a category error, like netting the dollar value of a basis point across maturities. Figure 4 takes a realistic multi-expiry SPX book – 72 option legs and four forward hedges, September to December 2026 – and decomposes its forward vega by expiry and in aggregate: each maturity contributes a differently shaped curve in spot and vega.
The book’s forward vega at T₁, by expiry and in aggregate, against spot, from the fitted forward smile at each expiry. Market date: July 27, 2026, T₁ = 25 days. Vega in $ per vol point, at the standard 100× index multiplier.
The fundamental equation modifies this profile at zero cost; here we zero out vega. Figure 5 fits an independent seven-contract strip to that target at each expiry and tests the combined hedge under a +5 vol-point shock: the hedged curve stays close to the unshocked baseline while the unhedged book drifts on the downside.
The same book’s unrealised P&L at T₁, hedged and unhedged, under a +5 vol-point shock at every expiry, against the unshocked forward-vol baseline. Market date: July 27, 2026; today’s cost basis ($11,930 net debit) from each leg’s own listed vol.
Forward-Starting Options: Completing the Circle
A forward-starting option – struck at ek·F1 when spot first arrives at T1, expiring at T2 – is worth exactly c12(k)·F1, the forward call price this article has built throughout. One could posit that a c2(k) call is the expected value of a strip of forward-starting calls, which would look exactly like the fundamental equation. But that is pedagogical, not a proof.
The actual proof is by dynamic hedging – and, notably, in options alone, with no position in the underlying stock. That omission matters: the result assumes nothing about how the stock moves. We know of no other dynamic-hedging argument that trades only options.
The result is established using the hedging portfolio [c12(k)·F1] − Δ(k): long the forward-starting option, short the zero-cost contract Δ(k) built above. Examining its value one period later – not just at expiry – shows this to be a self-financing hedging strategy with negligible error I(k).
We omit the error bound.2 Figure 6 shows I(k) versus the forward-starting option price, measured daily over the T1 horizon, earlier in the same SPX sample.
![The daily hedge error I(k)]](https://tsimagine.com/wp-content/uploads/2026/09/Fig-6.jpg.webp)
The daily hedge error I(k) at k = 0, against the forward call price c₁₂(k) it perturbs, and its cumulative sum ΣI(k) – live-engine values, not simulated. A 90-day forward-starting option, T₁ = 46, T₂ = 136 days, SPX, Feb 2 – Mar 16, 2026 (30 rebalancing dates, through the T₁ horizon). Mean I(k) = −3.9×10⁻⁵, std = 9.4×10⁻⁵, cumulative ΣI(k) = −1.2×10⁻³, against c₁₂(k) ∈ [0.0285, 0.0386].
From forward-vol theory to real-time risk
The vega decomposition and stress scenarios in this article are exactly what RiskSmart runs live, across your full book. See your exposure, margin, and volatility risk update in real time, across every asset class, buy-side or sell-side.
1. DT Breeden and Litzenberger, RH (1978), Prices of state-contingent claims implicit in option prices, The Journal of Business, 51 (4), pp. 621–651.
2. Forward volatility III: Forward-starting options, TS Imagine – Volatility Research, working paper.
This article was originally published by Risk.net as sponsored content produced with TS Imagine – Volatility Research.
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