29 Session 20: Stochastic Control · HJB Equation (Concepts Only)
| Unit | 4 — Math Intuition Bridges |
| Book Chapter | 11 (concepts only — full derivation in Session 25 Track 2) |
| Track | Common core (both tracks) |
29.1 Learning Objectives
By the end of this session, students will be able to:
- State the optimal control problem in plain language — what is being optimized, over what choices, subject to what constraints.
- Articulate the dynamic programming principle (Bellman’s logic) and why it works.
- Describe what the HJB equation is — a PDE that the value function must satisfy — without solving it.
- Apply the HJB intuition to the LP exit timing problem from Sessions 11–13.
- Identify the smooth pasting condition as the boundary condition that determines the optimal exit threshold.
29.2 Pre-Class Assignment
- Read: Book Chapter 11 (read for concepts; skim proofs)
29.3 In-Class Outline (75 minutes)
| Time | Segment | Format |
|---|---|---|
| 0:00–0:05 | Recap Session 19 · Today: optimal decisions under uncertainty | Lecture |
| 0:05–0:20 | The control problem — what we’re solving for | Lecture |
| 0:20–0:35 | Dynamic programming and the value function | Lecture + intuitive proof sketch |
| 0:35–0:55 | The HJB equation — what it says and what it doesn’t | Lecture |
| 0:55–1:10 | Smooth pasting and the exit boundary | Lecture (revisit Session 12 with new lens) |
| 1:10–1:15 | Bridge to Session 21 (mean-field games) | Lecture |
29.4 Discussion Questions
- The HJB equation is a PDE. Most LPs don’t solve PDEs. Is the framework’s reliance on this math a barrier to adoption? How can it be mitigated?
- Real LPs face liquidity, regulatory, tax, and behavioral constraints in addition to the value-maximization problem. How does HJB extend to handle these?
- The smooth pasting condition assumes the exit payoff is smooth. What if the secondary market has discrete pricing tiers (e.g., -5%, -10%, -15% bid levels) creating jumps? Does the boundary still exist?
29.5 Worked Example: Sketch of the GE-LAV HJB
The GE-LAV exit problem in HJB form:
State: \(L_t\). Time horizon: \([0, T]\). Liquidity dynamics: \(dL_t = \kappa(\bar{L} - L_t) dt + \sigma_L dW_t\).
Value function: \(V(L, t)\) = expected discounted hold value from \((L, t)\).
HJB equation (continuation region \(L > L^*(t)\)):
\(\dfrac{\partial V}{\partial t} + \kappa(\bar{L} - L) \dfrac{\partial V}{\partial L} + \dfrac{1}{2} \sigma_L^2 \dfrac{\partial^2 V}{\partial L^2} - r(L) V + C(L, t) = 0\)
Where \(C(L, t)\) is the expected cash flow rate from the position.
Terminal condition: $V(L, T) = $ terminal payoff (final distribution).
Boundary conditions at the free boundary \(L^*(t)\): - \(V(L^*, t) = P_\text{secondary}(L^*)\) (value match) - \(\dfrac{\partial V}{\partial L}\Big|_{L^*} = \dfrac{dP_\text{secondary}}{dL}\Big|_{L^*}\) (smooth pasting)
Numerical solution:
Discretize \(L\) on a grid, \(t\) on a time-step grid. Back-step from \(t = T\) to \(t = 0\) using finite differences. At each step: - Compute \(V\) assuming continuation - Compare to \(P_\text{secondary}(L)\) - Take maximum (American option style) - Record the boundary where the max switches
Output: The \(L^*(t)\) chart from Session 12.
29.6 What to Expect Next Session
Session 21 covers mean-field games (MFG) — the framework for when many agents are each solving their own HJB and influencing the aggregate state. This is the general equilibrium layer of GE-LAV.
Key concepts: - McKean-Vlasov SDEs (the OU process when drift depends on the population distribution) - The fixed-point structure of MFG equilibrium - The collective externality and why it matters for private market valuation
Reading: Book Chapter 13 (concepts only).