Across three OTC contracts, the paper never gives a number for the hedging error a desk would have to warehouse. The error appears only in scatter plots, despite the paper's promise to describe it explicitly along with the associated risk. The machinery is good, and the sparsity result deserves attention. One table of residual risk for the three hedges would change my view of the numerical section entirely.
One agent's break-even rate
Pennanen and Taoum price a two-month SOFR overnight index swap with a $500,000 notional on 28 August 2024. Their sell rate is 5.23544%, against a buy rate of 5.23177%. The calculation assumes one agent who holds $1 million in cash and has a risk aversion of 100.
The output is an indifference swap rate. At this break-even rate, the agent can re-optimise the hedge portfolio after taking the trade and remain exactly as well off as before. Preferences enter through the entropic risk measure V(u) = (1/rho) ln E exp(-rho u / B_I), applied to terminal wealth. The existing position is represented as a cashflow sequence, which means a different book can assign a different price to the same claim. The spread between the sell and buy rates reflects both the bid-ask cost function and the curvature of the value function in rho.
Convexity keeps the computation cheap
Trading is semi-static. At time zero, the optimiser selects one static buy-and-hold portfolio from quoted CME three-month SOFR futures and futures options. Later cashflows roll forward through the overnight money market at the simulated SOFR average.
Two features preserve convexity. A futures contract has an ask strictly above its bid, making its payout concave in position size. Positions are also capped at the best-level quantities, leaving a box-shaped feasible set. A decreasing convex risk measure composed with that concave payout produces a convex program, which MOSEK's interior point solver can handle directly.
The scenarios come from the authors' own earlier subjective SOFR term structure model. It places jumps on FOMC meeting dates and anchors its median path to the forward curve implied by futures quotes. The authors simulate 2^16 = 65,536 daily forward-curve paths over two years from 28 August 2024, then approximate the expectation with an integration quadrature.
Proposition 7 supplies the computational shortcut. Set B_I equal to the premium leg rolled forward through the money market to the terminal date, and three solves of the base problem produce both indifference rates. The reduction requires the rolled-forward premium leg to be almost surely strictly positive. Each pricing problem finishes in under a minute on a Dell Latitude laptop with an Intel Core i7 at 3.00 GHz x 8, 16 GB of RAM and 128 GB of virtual memory.
All market data comes from a single Bloomberg snapshot taken on 28 August 2024 at 15:30:00. CME lists 13,500 SOFR contracts, around 7,300 of them quoted, while almost half of the 13,500 show zero liquidity. Quotes existed for 24 of 46 listed three-month futures maturities. Within two years, CME listed 1,291 options. A filter retaining strikes inside the 90% confidence band of simulated futures rates leaves 619. Deep in-the-money strikes carry unit bid and ask sizes, described as CME market-maker "cabinets", and many out-of-the-money quotes have a zero bid.
Seven instruments do most of the work
Sparsity is the paper's memorable result.
For the two-month OIS, the optimiser selects about 7 of the 47 available instruments: one June three-month futures contract, plus calls and puts on September futures. The four-month call swaption has a strike of 3% and a $500,000 notional. Its underlying OIS expires one year after the option matures. The sell price is $243.11 and the buy price is $232.62, with about 10 of 116 instruments entering the hedge. The six-month caplet starts in one month with the same strike and notional. It prices at $96.35 to sell and $94.78 to buy. The optimiser uses 20 instruments from the 207 available at that horizon.
The hedging error stays in the plots
The paper states plainly that the available instruments cannot replicate any of the three payouts, whether the horizon contains 47 instruments or 207. Its abstract says the hedges "provide good approximations of the derivative payouts". For the OIS, the body says the hedge payout "approximately tracks the OIS payout". The swaption and caplet receive a narrower claim: their constructed hedges are optimal for the quoted prices, specified views and stated risk preferences.
Evidence for the approximation consists of scatter plots. They place hedge payout against the underlying variable leg and draw the contract payoff over the result. There is no residual variance and no comparison with a quadratic hedge using the same quotes.
The omission cuts directly across the paper's case against risk-neutral pricing. The abstract says that approach produces "approximate hedging strategies whose hedging error may be difficult to quantify". Its proposed alternative gives "an explicit description of the hedging error and the associated risk". The formulation does contain that description because terminal wealth z_I is explicit. Turning it into a reported number for three contracts observed on one afternoon would have required a table.
Small added costs erase the OIS hedge
For a practitioner, the sensitivity plots carry the most useful result. Add 0.025% in proportional cost, about two and a half basis points, and the OIS hedge collapses into cash. Below the threshold, its indifference rates decline roughly linearly as the added cost rises. The corresponding cutoff is about 0.02% for the swaption and about 0.35% for the caplet. Anyone who has hedged a swap using the wings of an options book will recognise the OIS figure.
Risk aversion produces an awkward fit
The product dependence surprised me.
For the swaption and caplet, the buy-sell spread widens monotonically with rho. The OIS behaves differently. The paper says that as rho rises, the rates decline and the spread "first widens before starting to narrow at approximately 4.83". It gives no units for 4.83.
Quoted two-month swap rates were 5.14789% and 5.14217%. Matching them requires risk aversion of roughly 3000, thirty times the illustrative 100. Raising rho lowers the model level by about 8.8bp until it meets the market.
Both discrepancies at rho = 100 run in revealing directions. The model bid-ask is 0.00367 percentage points, tighter than the quoted market spread of 0.00572, while its level is roughly 8.8bp above the market. The base case is therefore too tight on spread and too high on level, leaving one scalar to repair both.
The authors make no claim that they reproduce market quotes. In their words, Problem (OP) is "inherently subjective". They treat rho as a user input and argue that "subjectivity is the driving force behind all trading in practice". On those terms, the level gap becomes a calibration statement rather than a model error. The missing residual risk number remains, regardless of the choice of rho.
Liquidity stops at the discount curve
Zero-coupon bonds are assumed available at every daily maturity, trading at a single price without bid-ask. Those prices determine the OIS and swaption payoffs. The authors acknowledge the tension in Section 7.1, where they write that replicating a SOFR swap with bonds and overnight rolling "becomes impractical when ZCBs are illiquid". Yet the term structure model generating every scenario continues to use single-priced daily ZCBs after identifying their illiquidity. On the CME side, illiquidity supplies the paper's entire motivation.
Limits of one afternoon
The authors identify several simplifications. They omit margining and daily marking-to-market, arguing that a near market-neutral portfolio should generate small net daily flows. CME portfolio margining remains future work, with CME cited as reporting capital efficiencies of up to 100% for certain portfolios. The optimisation uses only the best bid and ask levels, although the paper notes that concavity survives the inclusion of deeper book levels. American exercise for the futures options is handled neatly. Exercise timing leaves the resulting futures position unchanged, allowing both sides to treat the options as European.
We could not test any of this. Pricing the two-month OIS requires the best bid and ask, including sizes, for the 47 instruments within that two-month window: one futures contract and options on the September futures. We hold no history of CME futures or futures-options quotes. Because the hedge set consists of CME three-month futures and options on those futures, quoted with sizes at the best level, another instrument set would not reproduce the exercise.
One scalar would move me. Report the residual risk in the paper's own entropic units or in dollars. Give it for the sparse hedge against every instrument available at the relevant horizon, then for a quadratic hedge on the same quotes. Use Ten dates rather than one. The framework already computes it.