Recap and the Fundamental Equation

Paper I of this series established that the forward volatility surface — the implied volatility curve that would prevail over the forward period \([T_1, T_2]\) — can be constructed from current option prices without invoking any volatility model. Given two implied volatility smiles at expiries \(T_1\) and \(T_2\), the forward probability density \(p_{12}(k)\) is pinned down by the convolution identity

solved by Fourier deconvolution: \(\varphi_{12} = \varphi_2 / \varphi_1\) followed by inverse FFT. Because the raw inversion can carry artificial high-frequency oscillations, a light numerical smoothing is applied; the recovered density is then checked against the identity above, which it satisfies to a minuscule residual. The forward call prices \(c_{12}(k)\) follow from \(p_{12}\) by integration, and the forward volatility curve \(\sigma_{12}(k)\) is recovered by Black–Scholes inversion. None of these constructions invokes any stochastic model.

The central result of Paper I — what we will call throughout this paper the Fundamental Equation — expresses every \(T_2\) call as a continuum of forward-weighted \(T_1\) butterfly positions, weighted by forward call prices:

The bracket notation is from Paper I: brackets denote traded instruments, unbracketed quantities are prices or weights. Both sides of the equation are tradable positions with identical value today. The equation is a static hedge: long a \(T_2\) call, short the strip of forward-weighted \(T_1\) butterflies, at zero net cost, with no rebalancing required until \(T_1\).

We sometimes refer to this static position as a forward volatility contract, by analogy with a forward contract on a stock. A stock forward locks in the delivery price; a forward volatility contract locks in the forward volatility curve over \([T_1, T_2]\). In both cases the lock-in is achieved by no-arbitrage against currently traded instruments, with no dependence on any model for the underlying's dynamics.

What This Paper Does

Paper I closed by stating the Fundamental Equation and demonstrating that the locked-in forward vol becomes the central anchor for risk analysis at horizon \(T_1\). This paper turns that anchor into operational practice.

The vega of a portfolio is a misleading number, in that — like DV01 — it cannot simply be added across all of your options positions. Adding across expiries assumes a one-month vol and a twelve-month vol move together, which they do not. It can be summed by expiry, but even then the sum holds only at today's spot: an option's vega peaks at its own strike, so as spot moves through the book the individual vegas rise and fall, and where the long and short strikes sit on opposite sides of spot the total can change sign. A single number is one point on a curve. In this paper we separate the expiries, mark each one against its own locked-in forward volatility, and hedge the exposure across the whole spot range, at zero cost.

We will go beyond hedging, and show how to create your own custom volatility exposure independent of your current position. Put another way: you can now produce an unrealised P&L risk slide — stepping the stock price — at a horizon, with a known locked-in volatility curve but an arbitrary vega profile.

The remainder of the paper develops this on the SPX market of February 2, 2026 as a running example. All numerical work is performed in the Volatility Laboratory, whose outputs we present as interactive figures throughout. Drag the controls. The math recomputes in real time.

Locking in the volatility of an option: the building blocks of zero-cost vega hedging.

The Fundamental Equation holds today, at \(t = 0\). Both sides are positions of equal value. To track what the hedge does, we follow each side forward in time to \(T_1\).

The left side is a long \(T_2\) call. At \(T_1\) it remains a live \(\Delta T\)-day call, repriced at whatever volatility the market quotes at \(T_1\) for a \(\Delta T\)-day option at the new log-moneyness \(k - k^*\), where \(k^* = \ln(S_{T_1}/F_1)\) is the realized log-moneyness at the observation date. Call this market vol \(\sigma_2(k - k^*;\, T_1)\). The position's value at \(T_1\) is therefore the Black–Scholes call price at that vol.

The right side is the strip of forward-weighted \(T_1\) butterflies \([e^{k'}p_1(k')\,dk']\), with weights given by \(c_{12}(k - k')\) at every \(k'\). Each butterfly \([p_1(k')\, dk']\) expires at \(T_1\). The strip's value collapses to a single number — the weight evaluated at the realized \(k^*\):

The forward-weighted butterfly pays \(e^{k^*}\), and \(e^{k'}c_{12}(k-k') = c_2\bigl(k-k';\sigma_{12}(k-k')\bigr)\) — the same forward call, re-quoted from the forward prevailing at \(T_1\), namely \(F_2e^{k^*}\), onto \(F_2\). So the strip settles into exactly the \(T_2\) call at the locked-in forward volatility \(\sigma_{12}(k - k^*)\). The P&L of the static hedge at \(T_1\) is the difference between the same instrument at two volatilities:

Everything else — parallel shifts in the volatility surface, changes in skew, term-structure moves, even the realization of \(k^*\) itself — is absorbed by the strip. What remains is realized forward vol versus locked-in forward vol, evaluated at the moneyness of a \(\Delta T\)-day option struck \(k - k^*\) away from spot.

The forward volatility contract is a vega hedge

That analogue has an immediate operational consequence. A stock forward, once struck, has no further exposure to the spot level — it has already locked the delivery price, and what remains is a pure bet on where spot actually lands. The forward volatility contract does the same thing one dimension across: once struck, it has no exposure to the level of the surface, only to the gap between the forward volatility that realizes and the one locked in today. Everything else has cancelled. That is what makes it a vega instrument rather than a directional one, and it is why an arbitrary vega profile can be assembled from a handful of them at no premium.

The panel below is that statement, live. Fix a contract at strike \(k\) with the slider. At the horizon the strip collapses to the single butterfly that fires, at whatever log-moneyness \(k^*\) spot has actually reached, and what the contract is worth there is the difference between the market \(T_2\) call and the one locked in today: \[ \text{P\&L}(k^*) \;=\; c_2\bigl(k-k^*;\,\sigma_{12}(k-k^*) + \Delta\sigma\bigr) \;-\; c_2\bigl(k-k^*;\,\sigma_{12}(k-k^*)\bigr). \] The chart runs \(k^*\) across the horizon, for the one contract you have selected, and redraws as you slide it. Set \(\Delta\sigma = 0\) and the curve is identically zero at every \(k^*\): the contract costs zero today, and it settles at zero wherever spot lands, provided the forward volatility realizes as locked in. Any value it does take is forward vega and nothing else. That is the whole claim, and it is worth confirming by moving the slider rather than taking on trust.

For an options book of any meaningful size, this is the right anchor for horizon-date risk. Using anything else — rolling current vols forward, assuming a sticky-strike or sticky-moneyness convention — introduces gratuitous model risk that two desks marking the same book will disagree about. Two institutions can finally agree on what the forward Greeks are, because they agree on \(\sigma_{12}\).

3.3 · The key decomposition

The case for treating the forward vol surface as the horizon-date anchor can be made operational with a single identity. Let \(\Sigma_0\) denote the current (known) volatility surface, \(\Sigma_f\) the forward volatility surface (also known today, locked in by the Fundamental Equation), and \(\Sigma_1\) the volatility surface that will actually prevail at \(T_1\) — currently unknown. Let \(S_0\) be today's underlying price and \(S_1\) the underlying price at \(T_1\), again unknown.

The unrealized P&L of an options portfolio \(P\) over the interval \([0, T_1]\) decomposes as

The decomposition is the key to vega hedging. The first bracket measures the P&L if the surface at \(T_1\) turned out to be exactly the forward surface \(\Sigma_f\) locked in today. It depends only on the realized spot \(S_1\) and on quantities known at \(t = 0\); it has no exposure to the actual realized surface. The second bracket measures the surface's deviation from the locked-in forward, conditioned on the realized spot — pure volatility risk, isolated from the spot path.

The first bracket — the spot-only term at the locked-in surface — is the strategy. The second bracket is the surface-deviation exposure that the strategy unavoidably carries alongside it. The overlay lets us neutralize that exposure, invert its sign, or sculpt it to any conditional shape we want — long vega on the downside, short vega on the upside, peaked at a particular strike, flat in the middle. Every choice is zero-cost. The underlying spot view is preserved bit-for-bit. The rest of this paper is the operational consequence of that separation: how the overlay is constructed, how it is discretized, and what it does to the conditional-vol exposures that aggregate vega calculations conceal.

Example 1: a portfolio of single-expiry options.

We illustrate the framework on a simple portfolio of five options with the same expiry. The Portfolio_T2 file holds five SPX calls, short the two lower strikes and long the three higher ones, together with 10 forwards at that same expiry — a position with mixed vega across the strike range. The panel below marks it on the loaded surface; on the default selection that is 2026-02-02, with the book expiring at T₂ = 74 days and marked at a T₁ = 46-day horizon.

4.1 · Unrealized P&L at the 46-day horizon

We now ask: what is the value of this portfolio in 46 days, under various realizations of the spot price? The portfolio is still alive at \(T_1 = 46\) days, with 28 days of remaining life. Pricing it requires a volatility surface at that future date — and the surface dictated by no-arbitrage is the forward volatility surface \(\sigma_{12}\) from Paper I.

The Volatility Laboratory's Vol Hedger computes the portfolio's value across a grid of spot scenarios at \(T_1\), revaluing each option at the locked-in forward volatility \(\sigma_{12}(k)\) determined by the Fundamental Equation. Subtracting today's marked cost gives the Unrealized P&L at horizon.

4.2 · Vega is spot-dependent

A single number — "the portfolio has $80 of vega" — papers over enormous structural variation. To make this concrete, plot the portfolio's vega (per 1 vol point) at \(T_1\) across the same spot grid, computed against the forward vol surface. The vega profile is far from constant. It tracks the curvature of the P&L: negative where the P&L is concave down, positive where it is concave up, crossing zero where curvature changes.

When the SPX sells off, implied vols typically rise; when it rallies, they typically fall. Set that against the vega profile above:

• DOWN MOVE. The portfolio is short vega in that region and vol is rising. The rising vol works against the benefit of dropping underlying prices.
• UP MOVE. The portfolio is long vega in that region and vol is falling. The falling vol works against the benefit of rising underlying prices.

The portfolio has undesirable vega in both directions of spot move. This isn't exposure to a parallel shift in vol — it's conditional vol risk: exposure to the particular combination of spot direction and the conditional vol response.

Figure 4 above carries a toggle between value and vega; switch it to vega and the profile in question is on screen. It is worth looking at directly before reading any further, because everything that follows is an attempt to change its shape.

Two shapes are worth trying before drawing anything of your own, and the difference between them is the whole point of the section. Flip the sign, where the figure opens, sets the target to the exact negative of the current profile — not a dampening of the exposure but a reversal of it, long vega where the book was short and short where it was long. Flatten sets the target to zero: the conventional hedge, neutralising the exposure so the book is insensitive to the forward volatility wherever spot lands. Then draw something the book never had: monotonically long vega below the forward, monotonically short above it, so that a sell-off with vol rising and a rally with vol falling both work for the position rather than against it. None of these costs a penny of premium, and none disturbs a single spot position — the strip is built entirely from instruments that are, today, worth exactly nothing.

Take the default: a parallel shock of two vol points on 2026-02-02, T₁ = 46 days, with the sign flipped. Raise the forward smile two points and the unhedged book loses 222 five per cent below the forward; lower it two points — enter −2 — and it loses 198 five per cent above. Those are the two combinations the equity market tends to deliver, vol up in a sell-off and vol down in a rally, and the book is on the wrong side of both because its vega has the wrong sign in both regions. With the flipped overlay on, the same two shocks are worth +219 and +213. The framework does not merely dampen the loss; it turns it round, putting the book long volatility where a sell-off raises it and short where a rally lowers it. Flatten instead and the overlay simply neutralises: +1.4 and +8.5 against the unhedged −222 and −198. Both are free. Nothing was paid, nothing was received, and no spot position was touched.

4.3 · Example 2: a portfolio spanning four expiries

The first example lives at a single expiry, which makes its vega profile easy to read but hides the problem that motivated the framework in the first place. Consider a second portfolio — the one carried in Master Portfolio — spread across four consecutive expiries. Each expiry holds the same nine-pair risk-reversal ladder, long puts against short calls at mirrored moneyness, delta-hedged with forwards. What differs is the sign: the ladder is put-long in the first and third expiries and put-short in the second and fourth.

Aggregate its vega and you get a number close to zero, and that number is worthless. It is worthless for the reason the desk already half-knows and the risk report does not: adding a 53-day vega to a 144-day vega is a category error, the volatility analogue of adding the DV01 of a two-year note to that of a thirty-year bond. The two numbers move on different underlyings. Netting them produces a figure that is neither a hedge ratio nor a risk measure.

The forward volatility surface is what makes the decomposition well-posed. Mark every expiry at a common horizon \(T_1\), each against its own locked-in forward volatility \(\sigma_{12}^{(n)}\), and the book's exposure separates cleanly into one curve per expiry. Below, those four curves and their sum. The sum is small. The curves are not.

Read the curves rather than the total. On 2026-07-27, marked at the 25-day horizon, the four expiries run to $13,600, $20,200, $25,700 and $28,300 of vega per vol point across the spot range, with adjacent expiries carrying it in opposite directions. At the spot where the book is worst, it holds $85,900 of gross vega across the four and reports −$9,100 net — a factor of nine and a half concealed by a single number. A desk running this book on aggregate vega believes it is close to flat. It is not flat; it is crossed, and it will stay quiet only for as long as the four forward volatilities move together. They do not have to. Any steepening or flattening of the volatility term structure — which is to say, any change in the forward volatilities themselves — hits the four expiries differently and the cancellation stops.

Now tick Flatten all four expiries. For each expiry the overlay is a strip of forward volatility contracts \([[c_{12}(k)]]\) struck across the horizon's spot range, weighted to cancel that expiry's own vega curve. Because vega is additive once every expiry is marked on the same horizon spot grid, zeroing each expiry individually zeroes the whole book — and because each contract in each strip costs nothing, so does the entire overlay. This is the operational form of the claim: the book's conditional volatility exposure is removed expiry by expiry, at zero premium, without touching a single spot position.

With five contracts per strip the peak total vega falls from $11,980 to $814 — 93% removed. Nine takes it to $200 and eleven to $78; past about fifteen it settles near $25, 99.8% removed, and further contracts buy nothing. That floor is not a shortage of instruments but the fine structure of a four-expiry vega surface, which no finite strip reproduces exactly. The sequence is not quite monotone — seven does worse than five, at $1,202 — because the strikes are spaced across a fixed span and an odd count lands them differently against the book’s own kinks; that is a property of this particular fit, not of the method. Appendix A.3 takes up how far the discretisation has to be pushed before it stops mattering.

Figure 7 puts all four expiries, their total and the flattened total on one chart. Figure 8 takes the same book one expiry at a time: choose an expiry to see its own vega curve, the strip fitted to cancel it and what is left once it is flattened — or choose the total.

The framework, complete.

Paper I established that forward volatility is a model-free market observable: extractable from two implied volatility smiles, lockable through a static hedge in currently traded options, and structurally identical in its risk-management role to forwards in any other asset class. Paper II has shown what this means in practice.

We took a portfolio of five options at a single expiry, plus forwards, and asked what its risk profile looks like at the 46-day horizon. Pricing each option at the locked-in forward volatility \(\sigma_{12}\) gave a unique, model-free Unrealized P&L profile across realized spot scenarios. We observed that the position carries an opposing sign of vega in both directions of spot move — combined with the spot-vol correlation of equity markets, a recipe for adverse outcomes regardless of which way spot moves. Exposure that aggregate vega calculations entirely conceal.

The Fundamental Equation gave us the means to neutralize, even flip that exposure. By assembling a strip of forward-vol-contracts at quantities chosen to flatten the portfolio's forward-vega profile across spot, and discretizing the continuum-of-\(T_1\)-butterflies component into a small basket of liquid options, we produced a fourteen-instrument hedge that costs nothing today and meaningfully changes the conditional-vol exposure at horizon. Under adverse vol scenarios — spot down with vol up, spot up with vol down — the hedged portfolio's P&L actually benefits. The vol exposure has been flipped at zero cost.

Four points are worth emphasizing.

BEYOND HEDGING

The framework generalizes far beyond hedging. The same machinery that flattens vega across spot can be used to take a specific conditional-vol exposure: a trader who believes the market is mispricing the vol response to an upside breakout can buy that exposure, in isolation, without unwanted side bets on other parts of the surface. Forward-vol-contracts are a true instrument for expressing views on the forward volatility surface, not merely a hedging primitive.

VEGA IS INDETERMINATE

Vega — as conventionally aggregated — is an indeterminate risk measure. The portfolio in this paper carries a small, near-zero aggregate vega today, and yet its true exposure at horizon is a strongly spot-dependent function with values up to $100 in either direction. The forward-vol framework reveals this exposure; the standard framework hides it. For any options book of meaningful size, this matters.

CROSS-FIRM CONSISTENCY

The consistency across firms that has long characterized risk management in equities, currencies, rates, and commodities can now be extended to options. Two institutions holding the same options portfolio can, for the first time, agree on what its forward Greeks are — because they agree on the forward volatility surface, which is determined by current option prices alone.

A NATURAL DEALER PRODUCT

This framework points toward a natural product: forward-vol-contract market making. A trading desk could quote a forward-vol-contract as a packaged instrument. The client specifies the forward-vega profile they want — a vector of (spot, target vega) points like the (6,000, +$50), (6,600, +$25), (6,900, −$125) used here — and the desk delivers exactly that profile for a fee above the zero-cost theoretical value. The desk's economics improve as the book grows: a long book of client positions and a short book of others gets netted at the level of the continuous \(T_1\) strip, and only the residual netted strip needs to be discretely replicated and hedged. A desk with a thousand client trades on its book may need to hold no more \(T_1\) hedges than a desk with twenty. The forward-vol-contract is structurally well-suited to being a dealer-intermediated product.

The framework is complete. The forward volatility surface is observable, lockable, and tradable — three properties that, taken together, restore options to the same risk-management footing as every other asset class.

From a target profile to fourteen tradeable instruments.

The body of this paper shows what the overlay does: Figure 5 fits it to whatever vega profile you draw, Figure 6 puts it under a shock, Figure 7 applies it expiry by expiry. What it does not show is how the strip is assembled from instruments a desk can actually trade, or why the whole construction costs nothing. That is set out here, so the argument is available without interrupting the worked examples. Nothing in this appendix is needed to read § 04; everything in it is needed to implement what § 04 describes.

A.1 · The forward-vol-contract strip

Recall from Paper I that the forward-vol-contract centered at log-moneyness \(k\) is the doubly bracketed quantity \([[c_{12}(k)]]\), defined by

which is a zero-cost static position whose value at \(T_1\) is exactly \(c_{12}(k - k^*)\) — a clean exposure to the forward volatility surface evaluated at the realized log-moneyness. Its forward vega is \(\partial c_{12}/\partial \sigma_{12}\) evaluated at \(k - k^*\), a sharp bell-shaped function of the realized spot peaked near \(K = F_1 \cdot e^k\).

To deliver the target forward-vega profile \(v_\text{target}(S^*)\), form a strip of forward-vol-contracts at strikes \(K_1, \dots, K_N\) with quantities \(q_1, \dots, q_N\), and choose the \(q_i\) so that the strip's net forward vega plus the portfolio's existing forward vega equals \(v_\text{target}\) across spot, in a weighted least-squares sense:

Each contract \([[c_{12}(K_i)]]\) has zero cost today, so any linear combination of them is also zero-cost. The hedge is, by construction, free.

A.2 · The output: a strip of N = 5 forward-vol-contracts

Each \([[c_{12}(K_i)]]\) in the strip is itself constructed from a \(T_2\) call minus a continuum of forward-weighted \(T_1\) butterflies. Substituting these into the strip,

we see that the full hedge is:

• Long \(\sum_i q_i \cdot [c_2(K_i)]\): a position in \(T_2\) calls (long if \(q_i > 0\), short if \(q_i < 0\)) at the five strikes \(K_i\).
• Short a strip of forward-weighted \(T_1\) butterflies at every \(k'\), weighted by \(\sum_i q_i \cdot c_{12}(K_i - k')\).

The \(T_2\) piece is a finite set of liquid instruments — five calls, directly tradable. The \(T_1\) piece is the continuum of forward-weighted butterflies that the Fundamental Equation always introduces, which we must discretize.

A.3 · Replicating the T₁ leg with a discrete basket

Paper I's Numerical Considerations treated the continuum-of-\(T_1\)-butterflies issue: replace \(\int w(k') \cdot [e^{k'} p_1(k')\, dk']\) with a finite-strike basket of \(N_1\) \(T_1\) options whose combined payoff at \(T_1\) matches \(e^{k^*}w(k^*)\) as closely as possible, in a \(p_1\)-weighted least-squares sense.

For our hedge, the relevant weight function is \(g(k^*) = e^{k^*}\sum_i q_i \cdot c_{12}(K_i - k^*) = \sum_i q_i \cdot c_2\bigl(K_i - k^*;\,\sigma_{12}\bigr)\) — the locked-in payoff that all five of the \(T_1\)-butterfly strips collapse to at \(T_1\), summed. The Lab's "Replicate Combined \(T_1\) Strip" performs this discretization using \(N_1 = 7\) \(T_1\) calls. The \(N_1 = 7\) calls track the continuous strip closely across the spot range, with the largest residuals near the boundary of the fitting window.

The final implementable hedge is:

The counts and strikes above are read from Figure 5's strip and the lab's basket, so they follow whatever you set there. All traded, all currently liquid, and zero net cost at inception.