FIN.8:5 - Archetypal Grounding

A constructed European call has exercise price 100. Its tradable underlying is worth 100 now and can be worth 120 or 80 in one period, with no interim payout. Risk-free growth is 1.05; the stated idealized model permits replication without transaction frictions. Payoffs are 20 and zero. The risk-neutral up weight is (1.05−0.8)/(1.2−0.8) = 0.625, so option value is 0.625×20/1.05 = 11.90. The weight is a pricing device under these assumptions, not a forecast that the up state occurs with probability 62.5%. To see the replication, buy 0.5 units of the underlying and borrow 40/1.05 = 38.10. The initial outlay is 50 − 38.10 = 11.90; at expiry the position pays 60 − 40 = 20 or 40 − 40 = 0. If those trades or terms are unavailable, this particular replication no longer establishes a price.

For a different, nontraded expansion, suppose the decision date has two stated scenarios: an expansion costing 60 then produces value 90 in the favorable case and 40 in the adverse case. If the corporation may choose at that date, net exercise values are 30 and zero, instead of 30 and −20 under an unavoidable commitment. That shows the consequential branch. A price today additionally needs the cost of preserving the choice and justified timing and risk grounds; the two scenario numbers alone do not establish it.

To complete a present decision, now stipulate a one-year waiting period, a nonrefundable fee of 8 paid today, and information that reveals which of those two values applies before exercise. The values and costs are after tax. Suppose the favorable probability is 0.5 and the qualified valuation basis assigns no risk premium to either strategy’s incremental payoffs before that decision: the uncertainty has no priced systematic component, and no additional nonmarket risk charge is required for this use. The applicable one-year risk-free return is 5%. These are case assumptions; an ordinary nontraded project must establish its own risk basis.

The extra value of waiting before its fee is 14.29 − 4.76 = 9.52. A fee of 12 therefore makes waiting worth only 2.29: committing is preferable at the stated probability, even though the net waiting strategy remains positive. With fee 8 and favorable probability p, waiting beats declining when p > 0.28 and beats committing when p < 0.58; at a boundary the relevant strategies tie. The changed probability must retain the same unpriced-risk assumption for these thresholds to apply. Outside that basis, keep the exercise comparison and obtain supported pricing grounds. In every case the 60 must be available before exercise; a favorable value cannot supply the money by itself.

Now replace perfect revelation with two equally likely signals, whose conditional favorable probabilities are 0.65 and 0.35. The overall favorable probability stays 0.5. Retain the stipulated absence of an additional priced-risk adjustment for this signal-conditioned strategy and its residual uncertainty. At exercise, conditional net values are 0.65×30 + 0.35×(−20) = 12.5 and 0.35×30 + 0.65×(−20) = −2.5. Exercise only after the first signal; its favorable indication does not guarantee the favorable final outcome. Today’s value is 0.5×12.5/1.05 − 8 = −2.05, below the fixed commitment’s 4.76 and declining at zero. Substituting the unchanged unconditional probability into the earlier perfect-information formula would incorrectly retain 6.29. The information available before action has changed the strategy, even though the final-state frequencies have not.

Combining switching and abandonment. In another constructed case, an existing operation must continue unless a package bought for 5 today secures both a switching capability and an exit arrangement. At year 1 an equally likely high or low state becomes known. Continuing produces net cash 12 or −12 at year 2. Switching costs 3 at year 1 and changes year-2 cash to 18 or −4. Exit instead produces net cash 5 or −5 at year 1 and irreversibly removes the operating and switching asset. These are complete after-tax cash consequences, including the relevant obligations. Funds and implementation are available at the required dates; every strategy has a qualified zero-premium valuation basis of 5% per year.

At year 1 compare the available actions at that same date:

Mandatory continuation has value zero today because the equally weighted year-2 cash is zero. The combined policy, before its purchase cost, is worth 0.5 × [(18/1.05 − 3) − 5]/1.05 = 4.35 today. Paying 5 makes its incremental NPV −0.65, so the package is not worth buying on these grounds. At the low node, exit still costs 5; it is preferred because the remaining loss is smaller than under either operating action.

To see why separate option values cannot be added, first value each permission alone against the same mandatory continuation, excluding the purchase fee. With switching alone, switch in both states: today’s value is 0.5 × [(18/1.05 − 3) + (−4/1.05 − 3)]/1.05 = 3.49. With exit alone, continue in the high state and exit in the low: 0.5 × (12/1.05 − 5)/1.05 = 3.06. Their sum 6.55 would falsely justify paying 5. In the low state both separate calculations credit an improvement over the same continuation loss, although exit destroys the asset that could switch. The feasible joint policy takes the best available action once at each observation.