Chapter 8 — Valuation on a Fixed Architecture
Part III · The Valuation Bench · Week 7
K→𝒢→Q→Ξ
What a fixed map lets you price. The chapter’s engine is the smallest problem exhibiting everything — two classes, one gated edge, closed form throughout — solved by a quasi-variational system and a verification theorem.
Learning objectives
- LOS 8.3 — State the quasi-variational system and the verification theorem for valuation on a fixed architecture.
- LOS 8.4 — Define gate pressure, reinterpret smooth pasting as zero institutional bindingness, and compute the smooth-pasting defect.
- LOS 8.5 — Price multi-step programs as compound exercises and quantify the staging premium.
The laboratory module
Module 8 — The Valuation Bench. Instruments on the gated listing problem: a free-boundary explorer, a gate bench, a program pricer, and a Ξ-blindness demonstrator.
The gated listing problem (Example 8.5). State \(k\) is geometric Brownian motion with drift \(\mu=0.02\), volatility \(\sigma=0.2\), rate \(r=0.06\), listing cost \(\gamma_L=10\). The homogeneous exponent \(\beta\) solves \(\tfrac12\sigma^2\beta(\beta-1)+\mu\beta-r=0\), which at these parameters reduces to \(0.02\beta^2=0.06\), so
\[\beta = \sqrt{3} \approx 1.732.\]
Value matching and smooth pasting give the free boundary
\[k^* = \frac{\beta}{\beta-1}\cdot\frac{\gamma_L}{0.25} = \frac{40\beta}{\beta-1} \approx 94.6.\]
Below \(k^*\) the private firm holds an option; at \(k^*\) the value pastes smoothly onto the list-now payoff. A gate above \(k^*\) binds and creates positive gate pressure — the smooth-pasting defect.
Guided experiments
- Verify \(\beta=\sqrt3\) and \(k^*\) at the text parameters, then find the volatility at which \(k^*\) doubles.
- Plot pressure against \(\theta\) for three \(k\) and explain the ordering.
- Run the Ξ-blindness demonstrator — Chapter 1’s Systems A and B priced by this chapter’s machinery return equal values with the germ frozen.
ch08 · 8/8 PASS