Three exercise rights separated by a mandatory wait are worth less than three American puts. With spot at the strike of 100, the paper's finest grid values three rights at 28.13, against 29.61 for three single-right puts. The exercise threshold can fall discontinuously at the fixed dates T - jδ, within (m-1)δ of expiry. A smooth single-right boundary cannot generate those drops.
Why do the rights interact?
The holder can use up to m rights, each paying (K - x)^+ on exercise. Consecutive exercises must be separated by at least δ. Any unused rights expire.
Exercising today blocks another exercise for δ, tying the rights together. The stopping reward combines the put payoff with the discounted value of the remaining m - 1 rights, available δ later. Ano calls this second component the delayed residual-right value.
The wait also sets fixed feasibility deadlines, d_j = T - jδ. After d_j, there is room for only j more exercises before maturity. A holder carrying more than j rights therefore loses the use of one right as each deadline passes, whether or not it was exercised.
This distinguishes the contract from the swing put studied by De Angelis and Kitapbayev, which requires every right to be used by maturity. That obligation produces a stopping region bounded by two time-dependent curves for each right except the last. Ano allows rights to lapse. The contract therefore gives no reason to exercise an out-of-the-money right merely to preserve later rights.
A threshold for each right, with breaks at the deadlines
With m rights remaining, exercise is optimal exactly when the price reaches or falls below bm. This boundary is strictly below K. The boundaries satisfy b[1] ≤ b[2] ≤... ≤ b[m]. More rights mean earlier exercise at a higher price. The optimal exercise time is the first time the price hits the boundary.
Between deadlines, each boundary is locally Lipschitz on the open strip, with smooth fit holding away from the deadlines. Each also solves a recursive Volterra integral equation, stated for t < T - δ.
At a deadline, the m-right price boundary drops to the boundary of the lower-right problem. The value function can fall at that point too. At d_j, a max condition replaces ordinary time-continuous matching: the holder takes the larger value from exercising immediately or holding j rights. Smooth fit is not imposed at the deadline.
The paper gives a lower bound on the last jump. For two rights, the value immediately before d_1 exceeds the one-right value by at least (K e^{-rδ} - x)^+. Under the illustration's parameters, K e^{-rδ} is about 99.50. We applied that bound ourselves: with two or more rights and spot at 70 at d_1 = 0.9, the value drops by at least 99.50 - 70 = 29.5 across the deadline.
Monotonicity in time within a strip remains unproved. Ano explicitly says the paper does not establish it. The numerical boundaries are nondecreasing on every open strip for m up to 3. As the paper acknowledges, those computations support the property without replacing a proof.
The paper also discloses that exchanges with ChatGPT helped develop the proof strategy for the Lipschitz result. The author states that all arguments were independently reviewed and verified.
What do the second and third rights buy at spot = strike = 100?
The numerical illustration uses a single at-the-money parameter set: K = 100, T = 1, r = 0.05, σ = 0.30, δ = 0.10.
With 2000 time steps and 2600 space steps, the finest grid gives V[1] = 9.868892, V[2] = 19.254787, V[3] = 28.128712.
The second right adds 9.39; the third adds 8.87. Treating two rights as separate puts overstates their value by about 0.48. For three rights, the overstatement reaches about 1.48.
Across the full refinement, V[1] changes only from 9.863938 to 9.868892.
Initial exercise boundaries are 69.17, 71.48 and 73.33. Their gaps, 2.31 and 1.85, are about 16 to 20 times the finest spatial step of 0.115385.
The individual boundary estimates are less stable. Across the five grids, b3 moves between 73.33 and 73.75 without a monotonic pattern. On the three finest grids, it remains within about 0.1, roughly one mesh step. The ordering reflects a real economic effect. The second decimal of the reported boundaries falls inside the finest mesh step of 0.115385.
The figure shows the deadline jumps at 0.8 and 0.9.
Implementation choices left open
Deadline alignment matters. In every run reported in the paper, δ/Δt is an integer, placing each deadline on a time node. The Bermudan convergence argument relies on that alignment, which the finite-difference runs also follow. Ano leaves convergence for a general binomial grid at the deadlines to a companion paper and makes no claim for it here.
The spatial grid appears uniform in x: 300/600 gives the coarsest Δx of 0.5. The refractory expectation, which values the rights after the mandatory wait, is evaluated by Gauss-Hermite quadrature with 256 nodes. Since those nodes fall off the grid, the calculation needs interpolated V[m-1] values at t + δ. We did not find a stated interpolation scheme, time-stepping scheme or SOR tolerance. The reported boundary is the midpoint between the last stopping node and the first continuation node. That limits its precision to one mesh step.
The recursion determines the computation order. Solving V[m] requires V[m-1] across the full time grid first. At each d_j, the solver uses max(g[m], V[j]) in place of ordinary time continuity.
The verification theorem comes with a condition that matters for certifying a solver's output. A candidate boundary must itself be continuous and nondecreasing on the main strip. Output with that shape qualifies for verification, and passing would also establish monotonicity there. This route cannot certify a non-monotone output.
We could not test the contract on market data. Listed American puts have a single right. Our options data include neither contracts with multiple rights and refractory periods nor their premiums. Substituting ordinary puts would strip out the deadline mechanism studied in the paper.
Random termination has a narrower result
The extension introduces an independent random time at which the contract disappears. For a general survival law satisfying a multiplicative survival condition, Ano establishes only that the direct value is at most the recursive value.
The two values are proved equal for Poisson termination with intensity λ. The contract then has discount rate r + λ, while the asset drift remains r. Remark 7.7 interprets this as a multiple-exercise put on an asset with dividend yield λ in a market whose interest rate is r + λ.
The paper does not extend its single-crossing and boundary results to random termination, including the Poisson case. It leaves open whether termination risk raises the m-right boundary before the last deadline T - δ.
For a desk, the paper supplies a specification for an exercise engine on a deadline-aligned grid, supported by one synthetic parameter set.