A mortgage desk should keep Imagawa's exact three-part decomposition of a mortgage-backed security (MBS) price. It separates the curvature of principal returned early from that of scheduled payments lost to prepayment. Those effects can pull in different directions. The headline phase diagrams offer less: their parameters are hand-picked, and the strongest driver of the boundary, the long-run mean ȳ of refinancing sensitivity, has no link to an actual pool. The abstract says the negative region expands with ȳ. The author also says the parameters illustrate the mechanism. Until someone measures ȳ, how much can a desk do with the map?
The paper uses simulations of its own model to ask which MBS price components generate negative convexity, and where rates and refinancing sensitivity change its sign.
Inside the model
The instrument is a 30-year fixed-rate pass-through, with mortgage rate m = 0.035. Pricing is in continuous time and ignores fees. Prepayment intensity combines a baseline λc, which follows a CIR process (λc(0) = 0.02, long-run 0.04), with a refinancing term y(t)·(m − a − b·R10)+. All numerics use a = 0 and b = 1 and begin with a flat initial zero curve. R10 is the 10-year zero yield in a one-factor Hull-White model (a_r = 0.03, σ_r = 0.01). Refinancing sensitivity y follows a Gaussian mean-reverting process: κ_y = 0.5, σ_y = 0.03, y(0) = 0, with correlation −0.6 to the short rate.
The exact price split has three components. V0 values cash flows that ignore rates. Vr values the principal prepaid when rates fall. Vop is a short position in options on the baseline cash flows lost through those prepayments. At first order, Vr resembles a strip of floorlets on R10. Vop resembles a strip of Asian-style options because its payoff integrates put-type payoffs along the path.
Imagawa approximates the central survival expectation through a cumulant expansion under forward measures, following the spirit of Ozeki et al. (2009). This produces closed forms for each component's price, duration and convexity. The exercise is entirely simulated. As the author discloses, nothing is fitted to TBA, pool or IO/PO prices.
Prepayment pulls curvature in different directions
Calling prepayment a single short call conceals the contest between Vr and Vop. In the paper's condition for negative convexity, the second R-derivative of Vop must exceed the combined second derivatives of V0 and Vr. Curvature decides the sign. The components' price levels and durations cannot answer that question.
Compare the runs at different ȳ. With ȳ = 1.0, convexity remains positive throughout R − m from −3.5% to +3.5%; Vr contributes positive convexity everywhere. With ȳ = 5.0, total convexity becomes negative at low rates. Over roughly −1.0% to 0.0% of R − m, Vop's curvature exceeds the positive curvature of V0 and Vr combined. Further into the low-rate region, Vr itself contributes negative curvature, even as Vop's negative contribution diminishes. Part of the deepest low-rate region's negative convexity therefore comes from the prepayment component that adds value. That result is new to me. It is the strongest finding in the paper.
Duration gives a related warning. At ȳ = 1.0, falling rates first lengthen duration and then shorten it. Imagawa gives the relation ∂D/∂R = D² − C: convexity below D² is enough to shorten duration. Negative convexity, in the paper's words, is "a sufficient but not a necessary condition for the shortening of duration." A hedge based solely on convexity's sign misses that distinction.
Where the closed form loses ground
Adding the second cumulant improves prices. For ȳ = 5 and R − m = −0.035, the deviation from Monte Carlo simulation of the same model moves from −0.62403% with one cumulant to −0.48781% with two. At the same rate point with ȳ = 1, it improves from −0.11609% to −0.10323%. At R − m = +0.035, the error for ȳ = 5 is −0.00309%.
The approximation underprices the simulation at all 30 points in the table. Every deviation is negative, and larger ȳ brings a wider gap. The paper acknowledges that Monte Carlo sampling and time-discretization error account for part of its residual series. That makes the benchmark imperfect. The bias nevertheless runs systematically in one direction.
Convexity fares worse. At low rates with ȳ = 5, the closed form diverges noticeably from pathwise Monte Carlo. Imagawa acknowledges the discrepancy and changes methods for the phase diagrams. Those use pathwise Monte Carlo, including a conditional Gaussian-path trick that captures the delta-function terms at the refinancing kink. All discussion of sign boundaries below concerns that Monte Carlo version. According to the paper, the closed form broadly follows the sign changes while drifting in level at low rates. Monte Carlo supplies the plotted boundaries.
When does the sign turn?
The phase diagrams span R in [0, 0.07], y(0) in [0.01, 50] and ȳ in {1, 5, 10, 20}. Raising ȳ pushes the zero-convexity boundary toward higher R. Negative convexity occurs even at small y(0) for ȳ = 5, 10 and 20. For ȳ = 10 and 20, the boundary is almost vertical at small y(0). It also extends above R = m = 3.5%. With a = 0 and b = 1, there is no refinancing incentive at inception in that region.
Much of this follows from the parameter choices. At κ_y = 0.5, the distance between y and ȳ halves in about 1.4 years (ln 2 / 0.5). At σ_y = 0.03, y has stationary standard deviation 0.03 / sqrt(2 × 0.5) = 0.03, set against a mean of 1 to 20. These runs make refinancing sensitivity close to deterministic. The nearly vertical boundary thus largely expresses how quickly y returns to the chosen ȳ. The out-of-the-money result holds within the model; its size rests on a ȳ that the paper does not estimate.
A simple toy model explains the nonmonotonic magnitude. For a perpetual bond with λ = y(m − R)+, the curvature is ∂²V/∂R² = −2m(y − 1)/{R + y(m − R)}³. It is negative when y > 1, and its magnitude decreases as R falls further below m. Imagawa presents this explicitly as intuition. The example has neither stochastic rates nor mean reversion.
Calibration still stands between the maps and a trade
The author identifies the omissions: no calibration to market prices, a single factor with parallel shifts, and no loan age, seasonality, burnout or borrower heterogeneity. The tests also begin with a flat curve. Burnout is the omission that matters most here, because it directly changes refinancing sensitivity, a variable against which every boundary is plotted.
We could not test this mechanism on our data. A faithful test requires pool-level prices and prepayment histories, which we do not have. MBS ETFs combine many pools and turn over, erasing the pool-specific refinancing sensitivity on which the mechanism depends. Treasury futures have no prepayment exposure at all.
The diagrams become tradeable when ȳ has a pool-level estimate, inferred from TBA coupon stacks or IO/PO pairs as the paper itself suggests.