The cap drives the economics of this collar, while the walk-away right moves far less value. Pricing the contract takes two exchange options and a barrier rescaling. The hard judgment lies in sigma_A, sigma_T, rho and the monitoring window.
Start from the rights being traded. In a stock-for-stock bid, the target holder may swap one target share at completion for q acquirer shares. Haug's framing is that "a stock-for-stock takeover bid is naturally an exchange option", with relative offer value X_t = qA_t/T_t. Event-driven funds already use this structure to price deal spreads, collar provisions and walk-away rights. It produces the value of the consideration conditional on closing, along with the price of the covenant allowing the acquirer to leave. Completion probability remains outside the model.
Haug and Haug (2002) extend Margrabe by placing a knock-in or knock-out barrier on the ratio of two prices instead of a single asset. That extension fits merger agreements because a walk-away covenant can void the bid after the target falls far enough relative to the acquirer. This paper supplies the collar. Its contribution has four parts: a Margrabe call-spread decomposition for the collared bid, per-leg barrier rescaling through H_K = H/K, the initial-trigger conditions required by the closed forms, and a barrier-contingent target-share floor component when 1 < F < C.
There is no data.
"No datasets were generated or analysed during the current study." The figures all come from a hypothetical calibration. It sets A0 = 50, T0 = 45 and q = 1, giving X0 = 1.1111. The remaining assumptions are tau = 0.5, r = 5%, b_A = 2%, b_T = 1%, sigma_A = 35% and sigma_T = 30% at rho = 0.40. Together they produce sigma_X = 0.3585. The collar uses F = 0.90 and C = 1.30, with the barrier at H = 0.85. The headline values are 7.7178 without the collar, 5.4542 after the cap and 5.2500 after adding the walk-away.
For a collar on the same ratio, with F <= 1 < C, the terminal payoff reduces to (qA_T - T_T)+ minus (qA_T - C·T_T)+. It is a call spread on the ratio. The remaining regimes are compact. When 1 < F < C, the payoff includes a guaranteed (F - 1)T_T. When F < C <= 1, the collared consideration never rises above the target share value and the payoff is zero. Once a walk-away covenant voids the offer when X touches H, each leg becomes a knock-in or knock-out Margrabe option under the Haug-Haug ratio-barrier formulas.
The cap owns the value change
With F = 0.90 at or below the exchange threshold of one, flooring the ratio leaves (X_T - 1)+ unchanged. The floor never enters the payoff. Haug confines this statement to the terminal payoff identity, which does not erase the pre-maturity time value of an out-of-the-money option. For anyone reading the term sheet, the implication is direct: under the standard collar regime, the floor allocates no value and the cap supplies the entire collar effect.
The exchange offer is worth 7.7178 before the collar. Its 1.30 cap leg has a value of 2.2636, leaving 5.4542 for the collared offer. The cap therefore removes about 29.33% of the uncollared value. Add the walk-away at H = 0.85 and the uncapped leg becomes 7.5006, while the capped leg becomes 2.2506. Their down-and-out value is 5.2500. The covenant removes 0.2042, roughly 3.74% of the collared value.
The denominators matter. The 29.33% uses the uncollared 7.7178, while the 3.74% uses the collared 5.4542. In this calibration, the cap shifts roughly eleven times as much value as the termination right: 2.2636 removed by the cap versus 0.2042 by the walk-away.
That ordering belongs in a term-sheet negotiation. Haug also observes that many transactions discuss the economic value of these provisions only informally.
Barrier bookkeeping can produce nonsense
Every leg requires a separately transformed barrier H/K. The cap-struck leg is an exchange option on qA_t/(C·T_t). A contractual barrier at X = 0.85 therefore becomes 0.85/1.30 = 0.6538 for that leg, and C rescales the underlying ratio as well. Haug shows that this transformation preserves each leg's initial trigger status. Y0 = X0/K crosses H/K at exactly the same point that X0 crosses H. The legs cannot enter different knock-in states.
The up-barrier case contains a numerical trap. For H <= C = 1.30, the capped up-and-out leg is exactly zero, although the closed form can return a negative value. The displayed up-and-out expression assumes an untriggered contract, requiring S < H and H > 1. With S < H <= 1, every path ending in the money must cross the barrier first, making the correct value zero. Blind substitution can still generate a number, including a negative one. Haug calls it "an extrapolation outside its admissible domain, not an option value."
The same reasoning applies to the capped leg when H <= C. Its up-and-out value is identically zero, while its up-and-in value equals the vanilla cap option. Table 3 considers initially untriggered up barriers with X0 < H and runs H between 1.12 and 1.50. In every row where H <= 1.30, the capped leg follows the piecewise rule. The leg struck at one continues to use the closed form. Once H > C, the standard up-and-out formula applies to the capped leg again.
Across both barrier tables, in-out parity must add to 5.4542 in every row, and the printed digits do. At rho = 0.95, sigma_X falls to 0.1140. The down-and-out and no-barrier values then coincide at 5.2449 because the barrier at 0.85 is effectively unreachable from 1.1111.
Unsourced inputs decide the price
Haug does not assign an a priori sign to the volatility and correlation statics. He does sign two comparative statics: value rises with the exchange ratio q and with the cap C. Table 4 shows why the other signs remain unsettled. Its no-barrier collared value is non-monotone in relative volatility, starting at 5.2527 for rho = -0.50, reaching 5.4581 at rho = 0.75 and declining to 5.2449 at rho = 0.95.
Since dsigma_X/drho = -sigma_A·sigma_T/sigma_X < 0, correlation has the opposite sign from the spread's net vega. Higher rho reduces value where net vega is positive, then reverses direction where the short cap leg dominates. Longer completion horizons also lower the no-barrier collar slightly. Its value moves from 5.4569 at tau = 0.25 to 5.0847 at tau = 1.50. Over the same range, the down-and-out value with H = 0.85 drops much faster, from 5.4343 to 3.9257.
The price consequently depends on sigma_A, sigma_T and rho in a direction that cannot be inferred beforehand. Those are exactly the inputs the paper leaves unsourced. It describes the calibration as hypothetical and chooses b_A = 2%, b_T = 1% without connecting either assumption to a transaction.
My objection concerns the target process. During a live bid, the target trades near the offer. A 30% marginal volatility combined with a freely diffusing T_t does not describe that post-announcement behavior. Haug gives no discussion of post-announcement volatility. Feeding post-announcement vols into the model reduces sigma_X, which Table 4's no-barrier column shows could move the collared value in either direction.
Haug identifies several boundaries himself. A collar tied to the absolute acquirer price creates a two-dimensional numerical problem. An average-price exchange ratio produces an Asian exchange option. Completion risk arising from antitrust, financing or the shareholder vote requires an added survival process or jump-to-failure intensity. The clean decomposition also requires the collar and barrier to reference the same ratio qA/T.
He follows that list with his answer: "These extensions are important in applications, but the closed-form collar decomposition remains a useful benchmark." I agree. Two omissions still matter for actual use: the source of the marginal volatilities and the assumption of continuous barrier monitoring, given that real walk-away tests are usually discrete-window VWAP averages.
Every table holds the floor at or below one, using F = 0.90 in the base case. None supplies a printed value for the binding-floor regime. The paper completes that regime by combining an established asset-or-nothing barrier claim with two transformed exchange-option legs, yet it leaves no numerical result against which an implementation can be checked.
We could not test this on our data. Pricing a real transaction requires the merger agreement, including the exchange ratio, collar levels, walk-away trigger and monitoring window, together with any amendments and the eventual completion or termination outcome. We hold none of those deal terms, and an equity price database does not provide them.
A desk can use this as a benchmark for the state-contingent value of consideration when a deal closes, then apply its own completion probability. The pair I would carry into a negotiation is 0.2042 against 2.2636.