Palmowski and Stępniak give a structuring desk a useful closed-form benchmark for a put settled on drawdown, and the paper supports no more than that. The formulas earn their place. The proofs leave more open than the theorem statements acknowledge. The numerical examples use hand-picked parameters. None are calibrated to market data.

The drawdown forces settlement

The contract is a perpetual American put. Its stock follows a log-price process with Brownian motion, drift and downward jumps. Those jumps have an exponential distribution with mean 1/ρ and arrive at rate λ. The holder may exercise at any time, but the drawdown trigger ends that choice: settlement occurs when the running maximum divided by the current price first reaches e^c. With the paper's illustrative e^c = 1.2, the stock triggers settlement 16.7% below its peak (1 − 1/1.2).

The holder receives (K − S)+ at that time. Forced settlement at intrinsic value matters here, since a knockout would pay zero. Both spot and the running maximum enter the value. The authors divide the maximum into three regimes: below a+c, between a+c and log K + c, and above log K + c.

Fluctuation theory for Lévy processes with only downward jumps supplies the pricing tools. With exponential jumps, the scale function W^(r), which enters the passage-time calculations, is a sum of exactly three exponentials. One has exponent γ_1 = 1. These terms and e^x make up every part of the value function, giving the closed form. A change of measure deals with the stock-price component of the payoff. It changes the jump rate to λρ/(ρ+1) and the jump parameter to ρ+1.

The risk-neutral drift is fixed at μ = r − σ²/2 + λ/(1+ρ). The authors explicitly acknowledge that this market has no unique martingale measure. They mention entropy minimisation as an option without selecting a measure. Their earlier report treated the contract under Black-Scholes; the contribution here is to add jumps.

Why can exercise use a fixed barrier?

The central claim puts optimal exercise at a fixed level: stop when log price first falls to a, the solution of a smooth-fit equation (eq. 40). Proposition 1 establishes a one-sided stopping region. The authors then postulate a boundary of a whenever the running maximum is below a+c. For a maximum between a+c and log K + c, the boundary becomes the drawdown level x̄ − c. The contract's trigger therefore handles the stopping in that regime.

The overshoot gives jumps their interest. Above a running maximum of log K + c, the drawdown trigger lies above the strike, so a payoff requires a jump through the trigger. The authors show a resulting kink in the price surface at x̄ = log(K)+c. In this regime, value increases with spot, reversing its behaviour below a maximum of log K + c. A higher spot puts the drawdown further away, allowing more time for a large downward jump. This sign reversal is the result a desk would be least likely to guess. It depends entirely on λ > 0.

Existence does not establish Theorem 3's uniqueness

Theorem 3 claims a unique a. Its proof invokes the intermediate value theorem, which establishes existence only. The proof also gives the limit as a → −∞ as rK W(c)²/W'(c) and describes it as negative. Since W and W′ are positive at c, the expression is positive. Equation (40) has a minus sign on that term, however, so the prose contains the slip and the existence argument remains intact.

Condition (6) is the more serious weakness. It requires the candidate value to exceed the payoff across the continuation region. Apart from invoking Proposition 1 and smooth fit, the authors explicitly check only (a+c, a+c). At that point, the value is KZ^(r)(c) − e^{a+c}, compared with a payoff of K − e^{a+c}. The inequality Z^(r)(c) > 1 settles that case. A check at one point cannot cover a two-dimensional region.

There are three smaller gaps. The verification lemma requires a C² value function, yet Figure 3 displays the kink at log(K)+c. The printed generator has drift r − σ²/2 and omits the λ/(1+ρ) compensator. And although the model permits σ ≥ 0, the expression W'(0) = 2/σ² excludes the pure-jump case.

Anyone using eq. (40) to quote should still check a* by simulation.

Parameters chosen for illustration

Figure 1 sets r = 0.1, σ = 0.2, e^c = 1.2, ρ = 3, λ = 0.2 and K = 100. These imply a 10% rate and, on average, a jump every five years. The average jump multiplies spot by ρ/(ρ+1) = 0.75, giving a 25% gap down. The corresponding risk-neutral drift is 0.13.

In the sensitivity plots, the exercise barrier e^{a*} increases with r and ρ, and decreases with σ and λ. Price is highest at the lowest r and σ. Along r = 0.5, however, it increases with σ. A 50% rate tests the formula under stress, well away from developed-market rates. The authors also say price grows "exponentially as ρ decreases and linearly as λ increases."

We did not find tables of values underlying these plots, nor a baseline e^{a*} reported in the text. The plots describe the formula's comparative statics. The paper provides no calibration to market data.

A benchmark for structuring

Given λ and ρ, the closed form calculates the discount the drawdown cap offers against an uncapped perpetual put. A desk can also solve for the e^c needed to reach a target premium, then read off the exercise level a*. These are practical structuring choices.

We could not backtest the contract. It is a bespoke perpetual instrument with no historical premiums or trades in listed-option data. Settlement also requires continuous first-passage information and the post-jump price. Neither minute bars nor end-of-day option marks can reconstruct those inputs. Substituting a finite-maturity listed put with a daily drawdown exit would change the stopping problem.

A calibration of λ and ρ to a real single-name skew would move me from benchmark to trade, if it showed a price worth selling in the jump-only regime above log K + c.