# BCW2011 Hedging Walkthrough

Work through this page after [BCW2011 Refinancing Walkthrough](./bcw2011-refinancing-walkthrough.md).

The code discussed here is:

- `src/example/BCW2011Hedging.py`

## What To Watch

By the end of this page, you should understand:

- how BCW's hedging case modifies the HJB rather than just adding a plotted series,
- how Eq. (28)-(30) become a two-control FinHJB problem,
- why the hedge rule splits into maximum-hedging, interior, and zero-hedging regions,
- how the costly-margin solution differs from the frictionless comparison object.

## Reproduction

Run this example from the repository root:

```bash
MPLBACKEND=Agg uv run python src/example/BCW2011Hedging.py
```

## What Changes Relative To Refinancing

The hedging case keeps the same reduced state variable `w = W/K`, and it keeps the same issuance and payout logic as the refinancing case. The structural change is that the firm now chooses both:

- investment `i(w)`,
- hedge demand `\psi(w)`.

This means the value function still solves on one state dimension, but the policy problem is now genuinely multi-control.

## Paper Equations Used In This Case

### Costly-Margin HJB: Eq. (28)

BCW's HJB becomes:

$$
\begin{aligned}
rP(K,W) = \max_{I,\psi,\kappa} \;& (I-\delta K)P_K \\
&+ \left((r-\lambda)W + \mu K - I - G(I,K) - \epsilon \kappa W\right)P_W \\
&+ \frac{1}{2}\left(\sigma^2 K^2 + \psi^2 \sigma_m^2 W^2 + 2\rho\sigma_m\sigma\psi WK\right)P_{WW}.
\end{aligned}
$$

After homogeneity reduction, the repository solves the one-dimensional form in `w`.

### Margin Constraint: Eq. (29)

$$
\kappa = \min\left\{\frac{|\psi|}{\pi}, 1\right\}.
$$

With `\rho > 0`, BCW focuses on short futures positions, so `\psi \leq 0`.

### Interior Hedge Rule: Eq. (30)

$$
\psi^*(w) =
\frac{1}{w}
\left(
\frac{-\rho \sigma}{\sigma_m}
- \frac{\epsilon}{\pi}\frac{p'(w)}{p''(w)}\frac{1}{\sigma_m^2}
\right).
$$

This is the unconstrained interior hedge policy. The actual hedge rule is then clipped into the admissible regions:

- `\psi=-\pi` in the maximum-hedging region,
- Eq. (30) in the interior region,
- `\psi=0` in the zero-hedging region.

### Frictionless Comparison: Eq. (27)

The paper's no-margin benchmark fully eliminates systematic risk. In implementation, the repository does not solve a separate closed-form benchmark object outside FinHJB. Instead, it solves a comparison HJB with:

- `epsilon = 0`,
- very large `pi`,
- the same issuance/payout workflow,
- the same plotting interface as the costly-margin case.

This gives a directly comparable numerical object for Figure 6.

## How The Two-Control Problem Becomes FinHJB Code

| Economic object | FinHJB object | Repository role |
|---|---|---|
| hedging parameters | `Parameter` | adds `rho`, `sigma_m`, `pi`, `epsilon` to the refinancing baseline |
| controls | `PolicyDict` | stores `investment`, `psi`, and `psi_interior` |
| policy update | `Policy.cal_policy(...)` | computes both controls explicitly from the current grid |
| HJB residual | `Model.hjb_residual(...)` | implements Eq. (28) in reduced form |
| issuance and payout boundaries | `Boundary` + boundary targets | reused from the refinancing logic |

The design choice here is different from the single-control cases:

- `investment` and `psi` are updated together in one explicit policy step,
- `psi_interior` is stored separately so the code can diagnose `w_-` and `w_+` even though the actual hedge rule is clipped.

## Why The Solver Still Uses `boundary_search()`

Even though the policy problem is richer, the state dimension is still one. The outer numerical problem still asks for:

- the left issuance value,
- the right payout boundary.

So the workflow stays:

1. solve the interior HJB for the current boundary guesses,
2. recover issuance information from `p'(w)`,
3. update the boundary targets,
4. stop when issuance matching and payout super-contact both hold.

The script uses `method="hybr"` because these targets are coupled and the hedge control changes the curvature of the value function in a materially nonlinear way.

## The Three Hedge Regions

The repository extracts BCW's two endogenous cutoffs from `psi_interior`:

- `w_-` solves `\psi^*(w_-) = -\pi`,
- `w_+` solves `\psi^*(w_+) = 0`.

That gives the three-region interpretation:

1. `w \leq w_-`: maximum hedging, `\psi=-\pi`,
2. `w_- < w < w_+`: interior hedging, `\psi=\psi^*(w)`,
3. `w \geq w_+`: no hedging, `\psi=0`.

This is one of the cleanest examples in the repository of using an auxiliary policy series both for plotting and for economic diagnostics.

## Figure 6: How To Read The Comparison

![BCW hedging main figure](./assets/bcw2011-hedging-main.svg)

### Panel A: `\psi(w)`

The costly-margin solution shows BCW's three regions. The frictionless comparison is clipped for display, matching the paper's plotting convention.

### Panel B: `i(w)`

Hedging affects investment because better risk management changes both firm value and the marginal value of cash.

### Panel C: `p(w)`

The value-capital ratio is higher with better risk management, but the gain is strongest away from the most constrained states.

### Panel D: `p'(w)`

The marginal value of cash generally falls when the firm can hedge more effectively, except in the severe-constraint region where hedging capacity itself becomes liquidity-sensitive.

## Stable Quantitative Targets

Healthy runs usually show:

- costly margin: `w_- \approx 0.07`, `w_+ \approx 0.11`, `\bar w \approx 0.14`, `\psi \in [-5, 0]`,
- frictionless comparison: payout occurs earlier than under costly margin,
- the frictionless display line is clipped at `-10` for the figure.

These are the right economic checks before you compare cosmetic line shapes.

## Code Inspection Pattern

```python
from src.example.BCW2011Hedging import run_case

bundle = run_case(number=1000)
for label, result in bundle["results"].items():
    print(label, result["summary"])
```

The most informative outputs are:

- `psi`,
- `psi_interior`,
- `max_hedging_boundary`,
- `zero_hedging_boundary`,
- `return_cash_ratio`.

## How To Adapt This Pattern

Start from this case if your own model has:

- more than one control,
- a control-dependent diffusion term,
- an economically meaningful clipped interior control,
- boundary logic that still looks like refinancing.

It is the right template for one-dimensional models whose complexity comes from policies, not from extra state variables.

## Related Pages

- Continue to [BCW2011 Credit Line Walkthrough](./bcw2011-credit-line-walkthrough.md).
- Use [Results and Diagnostics](./results-and-diagnostics.md) when you want a solver-oriented way to inspect `psi`, `psi_interior`, and the inferred region boundaries.