McCloud's four-part P&L ledger tells a hedger which residual loss a better hedge can remove. That is the paper's trading claim, and its closed-form algebra makes the claim unusually precise.
The investor ranks a payoff X using the entropy-adjusted mean, −(1/α) log E exp[−αX], the certainty equivalent under exponential utility. McCloud gives final prices a quadratic Gaussian form: a constant, a linear term in independent standard normal factors and a quadratic term. He calls the quadratic term's matrix the convexity. Completing the square then evaluates the Gaussian integral in closed form. Because the exponent is quadratic, the change of measure alters factor covariance as well as mean. Effective underlying volatility is σ̄(1 + αλγ̄)^(−1/2), so derivative notional λ changes it.
The hedge is built in stages. Optimising the underlyings alone establishes a funding rate r and the investor's base portfolio. McCloud then adds the derivative and funds it with underlyings. The resulting hedge is δ* = δσ + sδs: δσ offsets derivative risk spanned by the underlyings, while δs covers the difference between the derivative price and the cost of that hedge, scaled by funding spread s. A quadratic determines the indifference price and s together.
Four non-negative penalties account for hedged P&L, with each penalty paired to realised P&L attribution. The abstract counts three incompleteness penalties. Strategy is the fourth and the one the hedger controls. Convexity captures quadratic exposure beyond a linear hedge; dimension captures factors outside the underlying span. Funding arises from the spread leg and alone depends on the derivative's initial price. Strategy records how far the position held lies from δ*. McCloud says it "is the only penalty that can be controlled by the investor, and is eliminated when the funding strategy matches the optimal hedge portfolio". The paper contains no market data. Its illustration uses a synthetic bond, swap and derivative, followed by a multi-interval sketch driven by an Ornstein-Uhlenbeck process.
What happens in the bond-swap case?
The case study's derivative is quadratic in a swap. The swap starts at q = 0 with normal volatility σq = 0.2, and independent noise has coefficient ρ = 0.5. A bond supplies funding, with b = 1 and σb = 0.1; its lognormal price is approximated as normal. Payoff coefficients are η = 0.5, ε = 1 and γ = 2. Risk aversion is α = 1, and baseline notional is λ = 0.
Here δ = ε/(1 + αλγρ²σq²). At mid, δ = 1, the value McCloud highlights on a grid of −1 ≤ δ ≤ 3. Every penalty is multiplied by αλ = 0 at mid, leaving the price untouched. The mid indifference price is η plus the convexity adjustment ½γ(1+ρ²)σq². The penalties remain defined: dimension χσ = ε²ρ²σq², funding χs = σb²p², and convexity by its finite limit.
At the ends of the notional sweep, −1 ≤ αλ ≤ 1, the hedge at αλ = ±1 scales by 1/(1 ± γρ²σq²). It declines on the long side and rises on the short. The long indifference price is below the short, opening a bid-offer around mid.
The case study gives a checkable prediction. Convexity γ affects its hedge through unspanned noise ρ. Set ρ = 0 and the hedge stays at ε for any notional, even with a convex payoff. With no underlying portfolio (ω* = 0), a convexity-heavy position without unspanned noise therefore hedges at ε regardless of size. Outside that case, convexity can still change the hedge through −γ̄σ̄†φ.
A listed-option test requires choices
We are building a daily end-of-day test on US equity options. Both the asset class and daily frequency are ours; McCloud uses no data. Each day, we use the option mid for p and fit the next-day option price as a quadratic in the next-day stock move. The fit supplies the paper's ε̄ and γ̄. We then compare McCloud's δ* with a Black-Scholes delta on realised one-day P&L.
Two distributional checks come before any capital allocation. The residual from the quadratic fit should follow the unspanned quadratic Gaussian form. In McCloud's case study, that component scales with the stock move and is not normal when γ ≠ 0, so a normality test would miss the stated assumption. The one-day stock move must also look Gaussian.
McCloud acknowledges the Gaussian assumption's cost: "every non-trivial underlying portfolio has a normal final price that is negative with positive probability, which renders the no-arbitrage principle ineffective as a guide". He argues that entropic pricing determines discounting and the change of measure. For the derivative, no-arbitrage still acts through 1 + r > 0.
Several implementation choices remain. We found no guidance for selecting α, which matters because every penalty scales with αλ. The underlying covariance must be invertible under the paper's assumption. McCloud says the singular case requires regularisation, introducing a gauge choice. His formulas apply only within an interval around αλ = 0. For the case study, the penalties require αλγ(1+ρ²)σq² > −1 and the price requires αλζσb² ≥ −1/2. Past the square-root boundary, the paper says there is no indifference price at that notional. We also found no transaction-cost term; a desk test must supply its own costs.
One interval at a time
The multi-period extension repeats the single-period calculation across intervals whose factors come from an Ornstein-Uhlenbeck process. McCloud identifies the obstacle to carrying the closed form backward: a quadratic Gaussian final price "does not necessarily imply a quadratic Gaussian initial price for the derivative". Funding rate r and funding spread s* introduce non-linearity on each interval. Backward induction from expiry may cease to be closed after one step.
For our daily backtest, a fresh fit supplies the next-day option price each day. We use the closed form for the one-day hedge only.
We did not find two abstract promises carried through in the body. The abstract mentions fractional Ornstein-Uhlenbeck processes, while the body treats the standard process driven by Brownian motion. Its multi-interval construction requires independent normal variables on distinct intervals and obtains them from Brownian increments over disjoint intervals. Fractional increments lack that property. The paper also proposes hedge ratios as regressors for deep hedging, yet we found no regression exercise. The algebra is exact. Our view would improve if, across names, the ex ante dimension and funding penalties ranked realised residual hedge variance better than the gamma term in a delta hedge does.