Solving a 37-Year-Old Structural Problem in Interest Rate Theory
How Shunya's research work resolved one of the quiet open questions behind Jamshidian's famous decomposition.
At Shunya, we care deeply about the difference between using a model and understanding its structure.
A model may work well in practice. A pricing shortcut may be widely used. A decomposition may become standard in the literature. But beneath all of that sits a harder question: when does the structure really hold, and when does it fail?
That question sits at the heart of a new research result developed as part of Shunya's research efforts: a structural resolution of the exact obstruction to the Jamshidian decomposition in multifactor affine models.
To understand why this result matters, it helps to start at the beginning with the man whose name the decomposition carries.
Farshid Jamshidian and the 1989 Breakthrough
Farshid Jamshidian, mathematician and quantitative finance researcher. PhD in Mathematics, Harvard University (1980).
Farshid Jamshidian is a mathematician and quantitative finance researcher whose work has shaped the field of interest-rate derivatives for decades. Born in Iran, he earned his PhD in mathematics from Harvard University in 1980 with a dissertation in integral geometry, before moving into the finance industry in the mid-1980s and joining Merrill Lynch Capital Markets as head of quantitative fixed-income research.
It was in this role, working at the intersection of rigorous mathematics and live markets, that he produced his most influential contribution. In 1989, while serving as Vice President in Merrill Lynch's Financial Strategies Group, Jamshidian published "An Exact Bond Option Formula" in the Journal of Finance, a four-page paper that would quietly become one of the most cited results in interest-rate theory.
The key observation was deceptively simple. In a one-factor model of interest rates, where a single scalar state variable drives the entire yield curve, every zero-coupon bond price at option expiry is a strictly monotone function of that same scalar. This shared monotonicity means that when the coupon bond equals its strike, each constituent zero-coupon bond simultaneously equals its own component strike. The entire coupon-bond option payoff can therefore be written pathwise as a sum of zero-coupon bond options, each with a deterministic component strike.
This is Jamshidian's decomposition: a European option on a coupon bond reduced, exactly, to a finite portfolio of options on individual discount bonds, each with a closed-form price in models like Vasicek and Hull–White.
Beyond Merrill Lynch, Jamshidian went on to hold positions at Fuji International Finance, Sakura Global Capital, and NIBC Bank, and served as Professor of Applied Mathematics at the University of Twente in the Netherlands. He also pioneered the use of the forward measure in derivatives pricing and contributed to the development of the LIBOR and swap market models, work that is now standard across the industry. He has published 28 research papers spanning interest rate modeling, risk management, and mathematical finance.
His 1989 result, however, remains the most widely applied. It is elegant, exact, and practically powerful.
But it also carried an implicit boundary, one that Jamshidian himself left open.
The Classical Insight, and the Missing Boundary
The decomposition works beautifully in one-factor models. The usual understanding in the literature has long been that it is "a one-factor trick" and that it breaks down in multifactor settings. But that description leaves an important question unanswered:
Exactly what structural condition makes deterministic-strike Jamshidian decomposition possible, and exactly what geometric obstruction destroys it?
In multifactor affine models (the Duffie–Kan class, which generalizes Vasicek and CIR to multiple state variables) the yield curve is driven by a vector of factors, not a single scalar. Bond prices remain exponential-affine in the state vector, but they are no longer necessarily monotone in any common scalar. The decomposition, as Jamshidian stated it, no longer obviously applies.
What the literature lacked was an exact characterization: not just an observation that the trick fails, but a precise statement of when it fails and why.
That is the problem Aryan Ayyar, Fellow-in-Residence at MAHE Bengaluru, set out to resolve as part of Shunya's ongoing research work.
What the Paper Proves
The core result is clean and exact.
Within the exponential-affine family of bond prices, where each zero-coupon bond price takes the form P(i) = A(i) × exp(−B(i) · x), with x the vector of state variables, A(i) a positive scalar, and B(i) the loading vector capturing that bond's factor sensitivity, the main theorem states:
An exact deterministic-strike Jamshidian decomposition exists if and only if all loading vectors
{B(i)}are positively collinear — that is, they all lie on a single positive ray in factor space.
In plain terms: the geometry of the factor loadings must collapse onto a common direction. When they do, all bond prices respond to the factors in lockstep, effectively reducing the multifactor model to a one-factor structure, and Jamshidian's original argument goes through verbatim. When they do not, the decomposition is provably impossible.
This is proved using a Simultaneous Threshold Lemma: the Jamshidian identity forces all bond prices to be simultaneously above or simultaneously below their component strikes at every state x. A Cauchy–Schwarz perturbation argument then shows that non-collinear loadings always violate this sign-coherence near the exercise boundary.
The immediate corollary: if the loading vectors span a subspace of dimension at least two, no exact deterministic-strike Jamshidian decomposition exists. In genuine multifactor models, this is the generic situation.
This sharpens what Jamshidian himself established, a sufficient condition within a specific parametric model, into a model-free necessary and sufficient characterization across the entire exponential-affine class.
Why This Matters
There are at least three reasons this result is significant.
1. It turns a classical intuition into a verifiable theorem
Much of mathematical finance rests on structural tricks that are known to work in special settings, with their exact frontier of validity left underdescribed. Here, that frontier is identified precisely: not approximately, not informally, but as a necessary and sufficient algebraic condition on the model's loading vectors. Instead of saying the trick is "one-factor in spirit," one can now check a concrete geometric property.
2. The obstruction is geometric, not computational
The decomposition does not fail in multifactor models simply because the algebra becomes messy. It fails because the exercise geometry itself changes. In one-factor models, the exercise boundary is a single scalar threshold. In genuinely multifactor models, it is a high-dimensional hypersurface. Once the loading vectors cease to be collinear, no choice of deterministic component strikes can make the payoff identity hold pathwise across the full state space.
3. It opens a disciplined framework for approximation
The paper does not stop at the impossibility result. It develops a near-collinearity theory: when loading vectors are close to a common direction, as often happens in practice especially at short maturities, a projected deterministic-strike approximation remains quantitatively controlled.
For a fixed reference direction u, one solves for projected strikes using a single shared threshold. The paper proves these projected strikes are minimax-optimal, minimizing worst-case strip width over all deterministic-strike choices. The pathwise approximation error is confined to a strip around a projected one-factor exercise hyperplane, and the pricing error scales as order δ² in the angular spread between loading vectors, with an explicit, finite constant under mild regularity conditions. This gives the theory both structural sharpness and practical relevance.
Three Escape Routes That Do Not Work
Part of what makes this result structurally satisfying is that it closes off natural attempts to evade the obstruction. The paper addresses three of them explicitly.
State-dependent strikes. Allowing component strikes to depend on the realized state restores the decomposition, but trivially. The proportional allocation κ(i) = K × P(i)(x) / C(x) always works, carrying no structural content. It is a tautology, not a decomposition.
Numéraire changes. The change-of-numéraire technique is indispensable in derivatives pricing. But it cannot help here. A change of numéraire leaves the exercise set {C(x) > K} exactly invariant, so the geometric obstruction in factor space is unchanged. The collinearity condition lives in the factor geometry, not in the choice of measure or pricing units.
Nonlinear scalar reparameterizations. Could some clever nonlinear function g(x) restore comonotonicity, making all bond prices monotone in a single transformed variable? The scalar-factor rigidity theorem answers this definitively: within the exponential-affine class, scalar-factor representability, pairwise pathwise comonotonicity, and positive collinearity of the loadings are mutually equivalent. No nonlinear reparameterization is more powerful than the simple linear projection g(x) = u · x.
The obstruction is not cosmetic. It is structural.
A Broader Lesson
Jamshidian's 1989 paper gave the field something rare: an exact result, proved cleanly, with immediate practical consequence. That kind of work sets a standard.
What Ayyar's paper does is take that standard seriously, asking not just whether the trick works in the one-factor case, but what exactly makes it work, and whether any trace of that structure survives in the multifactor world. The answer is a precise geometric condition, a quantitative approximation theory, and a set of rigidity results that close off the natural escape attempts.
Many important ideas in quantitative finance survive for decades in a partly informal state. People know how to use them, know roughly when they work, and know where the approximations begin. But there is a gap between practical familiarity and exact structural understanding, and that gap matters. It affects how confidently one can deploy an approximation, what its error looks like, and whether attempts to improve it are even well-posed.
Closing that gap is rarely glamorous. But it is where the most durable contributions are made.
The Paper
Title: On the Exact Obstruction to the Jamshidian Decomposition in Multifactor Affine Models
Author: Aryan Ayyar, Fellow-in-Residence, MAHE Bengaluru
Date: April 3, 2026
Available at: SSRN — Read the paper →
The paper establishes a necessary and sufficient condition for exact deterministic-strike Jamshidian decomposition in the exponential-affine bond-pricing class, develops a minimax-optimal near-collinearity approximation theory with explicit error bounds, and proves scalar-factor rigidity and numéraire invariance results that confirm the geometric character of the obstruction.
At Shunya, we want to do work that combines rigor with originality. Sometimes that means building tools. Sometimes it means framing problems better. And sometimes, as in this case, it means resolving a structural question that has remained quietly open for decades, and doing justice to the original insight that made the question worth asking. This result is part of that journey.