13 Session 4: The OU Process for Liquidity (Intuition Only)
| Unit | 1 — Why DCF Fails |
| Book Chapter | 2 (sections 2.8–2.9) |
| Track | Common core (both tracks) — TRACK DIAGNOSTIC SESSION |
This session is deliberately positioned as a self-test for track selection. The OU process is taught at the intuition level only — no derivations, no proofs, no measure theory. Students who find this content easy and want more depth are Track 2 candidates. Students who find this content appropriate and don’t want proofs are Track 1 candidates.
The full mathematical derivation of the OU process appears in Session 25 (Track 2 only), drawing on the original Class 5 slides.
13.1 Learning Objectives
By the end of this session, students will be able to:
- Describe an Ornstein-Uhlenbeck process in plain English and write its SDE form.
- Identify the three OU parameters \(\kappa\), \(\bar{L}\), \(\sigma\) and explain what each controls economically.
- Compute the OU stationary distribution given the parameters.
- Simulate an OU path in their head (or on paper) using the discretized recursion.
- Connect OU dynamics to interest rate models (Vasicek) and to mean-reverting volatility.
- Assess their own readiness for Track 2 based on comfort with this material.
13.2 Pre-Class Assignment
- Read: Book Chapter 2, sections 2.8–2.9 (~12 pages)
- Watch: 8-minute video introducing mean reversion (link on course site)
- Optional (Track 2 candidates): Skim Karatzas-Shreve Section 5.6 (Ornstein-Uhlenbeck process)
13.3 In-Class Outline (75 minutes)
| Time | Segment | Format |
|---|---|---|
| 0:00–0:05 | Recap Session 3 · Why we need a process model | Lecture |
| 0:05–0:20 | What is a stochastic process? (intuition: BM, random walk) | Lecture |
| 0:20–0:40 | The OU process: SDE form, what each parameter does | Lecture + live simulation demo |
| 0:40–0:55 | Stationary distribution and what it tells us | Lecture |
| 0:55–1:05 | Connection to Vasicek and other mean-reverting models | Lecture |
| 1:05–1:15 | Self-diagnostic for track selection | Activity |
13.4 Discussion Questions
- The OU process is Gaussian (stationary distribution is normal). The empirical secondary market discount data has fat tails (more extreme observations than Gaussian predicts). How much does this limitation matter for valuation?
- The half-life of liquidity shocks is ~18 months at \(\kappa = 0.45\). What’s the implication for an LP who plans to exit in 12 months vs. 36 months?
- Vasicek (1977) introduced the OU process to model interest rates. Why did it take ~50 years for the same framework to be applied to liquidity in private markets?
13.5 Worked Example: OU Path Simulation in Plain English
Setup: Start at \(L_0 = 0.85\) (slightly stressed, like Q3 2024). Use \(\kappa = 0.45\), \(\bar{L} = 1.0\), \(\sigma = 0.32\). Simulate quarterly (\(\Delta t = 0.25\)).
Step 1 (next quarter): Mean reversion pulls toward 1.0: - Drift contribution: \(0.45 \times (1.0 - 0.85) \times 0.25 = 0.017\) - Random shock: \(\sigma \sqrt{\Delta t} \cdot Z = 0.32 \times 0.5 \times Z = 0.16 Z\) - If \(Z = +1\) (one std above): \(L_{0.25} = 0.85 + 0.017 + 0.16 = 1.03\) - If \(Z = -1\): \(L_{0.25} = 0.85 + 0.017 - 0.16 = 0.71\) - If \(Z = 0\): \(L_{0.25} = 0.867\)
Step 2 (second quarter): Repeat. Now starting from a new value.
Step 3 (long-run): After many quarters, the distribution of \(L_t\) converges to \(\mathcal{N}(1.0, 0.1138)\) regardless of starting point.
13.6 What to Expect Next Session
Session 5 covers the term structure of private capital returns — how the OU dynamics interact with investment horizon to produce horizon-dependent valuation effects. Key topics:
- The empirical term structure of PE returns by vintage and asset class
- How Jensen bias grows with \(T\)
- Why long-horizon assets (infrastructure, core real estate) are most affected by the liquidity illusion
- DLOM (Discount for Lack of Marketability) vs. GE-LAV — comparing methodologies
Reading: Book Chapter 3, all sections (~25 pages).
Track declaration: End of Session 5 is the track selection deadline. If you’re undecided, schedule a 15-min meeting with me this week.