Cotton's order-four series takes 4687 ms to price a Vasicek swaption; a numerical solve of his ODE takes 418 ms. The regime-switching pricer is carefully built, with an unusually candid validation suite. Its headline asymptotic expansion has less to offer anyone pricing options through a Fourier integral. The series explains the source of the switching premium. For Fourier-priced options, though, it runs slower than the numerical solve it approximates. Cotton acknowledges this himself, while calling the series the fastest route for a bond.
A common system for affine and Lévy models
A hidden Markov chain with generator Q switches the parameters of a standard model, such as the Vasicek mean level or Black-Scholes volatility. In affine and exponential-Lévy models, the state dependence factors out if switching is confined to drift levels, volatilities, correlations or jump intensities. The remaining calculation is a linear ODE with an entry for each regime: a' = (Q + diag g(τ)) a.
Only the forcing g changes from model to model. Cotton supplies it for Vasicek, CIR, Hull-White, G2++, Black-Scholes, Heston, Merton and Bates, variance gamma, geometric Asians, intensity baskets and equity with stochastic rates. Start the system from the unit vector e_j rather than all ones, and it prices a payoff conditional on the chain ending in regime j. This allows the Gil-Pelaez integrals to be conditioned on the expiry regime. Jamshidian's decomposition can then operate regime by regime for coupon-bond options and swaptions. Caps follow as puts on the period bonds.
The series expands in ε, the chain's mean holding time. At order zero, the model uses the stationary average of the forcing. The first correction splits into two parts. A Green-Kubo matrix measures the parameters' integrated autocovariance along the chain and supplies the extra convexity from switching. A memory term, built from the group inverse of Q, distinguishes the price starting in regime i from the price starting in regime j. That difference fades as the chain forgets its starting point. An initial layer enforces the initial condition separately in each regime; omitting it leaves a constant-forcing residual of 0.043ε.
The cost of averaging is visible in Cotton's 2x5 payer swaption. His two-regime Vasicek model uses mean levels of 6% and 2%, with vols of 1.5% and 0.8%. At ε = 0.333, the averaged model's relative error is 5.3e-02. Order one reduces it to 5.9e-03. Using averaged parameters costs about 5% on this swaption.
Does the series save time?
Barely, for a bond: order four takes 7.7 ms, compared with 8.3 ms for the numerical solve. Fourier pricing reverses that ranking. The numerical engine handles every quadrature node in a single batched call. For the Vasicek swaption, the expansion takes 4687 ms, the numerical solve 418 ms and the grid 55 ms. The Black-Scholes call needs 233 ms versus 10 ms; Heston needs 10273 ms versus 9930 ms. Cotton's own description is direct: the expansion supplies "the formula, the diagnostics and the greeks under the integral, not speed."
Nor does a higher order reliably improve accuracy. Order three performs worse than order two at every holding time in the swaption table. At ε = 0.05, their errors are 1.4e-07 and 1.1e-07; at ε = 0.333, they are 3.8e-04 and 1.6e-04. Order six subsequently gets to 8.5e-11 at ε = 0.05 and 2.6e-10 at ε = 0.2. The engine uses the right stopping rule: stop when the last term ceases to shrink. Asked to do so, it stops at the best truncation.
Exact sensitivities to the chain are a stronger reason to use the series. For the first-order Vasicek bond, analytic derivatives through the group inverse agree with central differences to relative errors between 1.6e-11 and 5.2e-09. The formula gives, for example, ∂P/∂Q12 = 0.0028187193, a direct sensitivity to a switching rate.
High Fourier frequencies break the expansion
The effective expansion parameter is ε multiplied by the size of the forcing. In a characteristic function, that forcing increases with frequency, roughly as ½σ²u² in Black-Scholes. Hold ε fixed and the terms eventually stop shrinking with order. Hold the order fixed and they diverge beyond some frequency.
Cotton identifies the problem and supplies a repair. Whenever the last term at a node fails to shrink, the engine switches to a numerical solve of the reduced system. The option price therefore comes from a hybrid calculation. The geometric Asian already needs this treatment at moderate holding times. Substituting the averaged value at diverging nodes would leave a 1e-3 error. At a switching rate of 50, the Asian closed form and the engine agree to under 1e-6 for strikes of 95 and 105.
The evidence ends at correct prices
The frozen-limit tests keep a live chain running between two identical regimes. The results are clean: the Vasicek bond agrees exactly with QuantLib, the Heston put to 1.1e-11 and the Jamshidian swaption to 2.2e-07. Grid engines retain their grid error, including 3.3e-05 for the American put. For the CDS spread, the difference is 8.9e-05. Cotton attributes that gap to QuantLib rounding the schedule to whole days.
Cotton also identifies both blind spots in these frozen-limit checks. They cannot detect the direction of time in a time-dependent forcing. Once switching is active, that changes the Asian price by 7%. A reversible test chain also produces a symmetric Green-Kubo matrix. Detecting the antisymmetric part requires a non-reversible chain; it contributes a 1e-4 effect to the first-order coupon-bond formula.
The certificates therefore test active switching too, comparing prices with the numerical ODE, a coupled pricing PDE on a grid or Monte Carlo. These checks use different methods, although they share the same repository. Some reductions are inexact, including Heston with a switched vol of variance. There, the first-order price is within 1e-4 to 7e-4 of the coupled PDE, while the correction amounts to 0.3 to 3% of the price.
A modelling choice produces the figure most relevant to a trading desk. When only a drift level switches, observing the regime and inferring it through the Wonham filter produce prices about 15% apart. The example is a one-into-five payer swaption with a two-year holding time, and the inferred price is lower. The mark depends on what information is available. More accurate quadrature cannot settle that choice.
Every parameter in the paper is synthetic.
We did not find a calibration to market quotes or a hedging test. The paper gives no grounds for calling a listed option mispriced. We could not test it ourselves. Our coverage extends only to the equity-option component on US ETF options. A relative-value implementation would need bid and ask quotes that we do not have; using closing prices would test a different strategy.
A timing table in which the order-four series beats the batched solve on a Fourier instrument would change my view of the expansion. Until then, use the numerical solve for Fourier-priced European option prices and the series to explain them.