# Market Example

### Example: Market Creation and Swap

Let's walk through a concrete example to see how everything fits together.

#### Market Creation

**Input Parameters:**

* Asset: sUSDe (yield-bearing)
* Expiry: 1 year from now (31,536,000 seconds)
* Rate range: 3% to 7% APY (rate\_min = 0.03, rate\_max = 0.07)
* Initial fee rate: 1% (fee\_rate = 1.01)

**Step 1: Calculate Initial Anchor**

```
years_to_expiry = 31,536,000 / 31,536,000 = 1.0
implied_rate_min = (1 + 0.03)^1 = 1.03
implied_rate_max = (1 + 0.07)^1 = 1.07
initial_anchor = (1.03 + 1.07) / 2 = 1.05
```

The initial anchor (1.05) represents the midpoint of the expected rate range.

**Step 2: Calculate Scalar Root**

```
rate_diff = 1.07 - 1.05 = 0.02
LN_MAX_PROPORTION = ln(0.96/0.04) ≈ 3.178
scalar_root = (3.178 * 1.0) / 0.02 ≈ 158.9
```

The scalar root controls how sensitive the exchange rate is to pool imbalances. Higher values = more capital efficient but narrower tradable range.

**Step 3: Calculate ln\_fee\_rate\_root**

```
ln_fee_rate = ln(1.01) ≈ 0.00995
ln_fee_rate_root = 0.00995 * 31,536,000 / 31,536,000 = 0.00995
```

This is the "root" fee rate that will be scaled by time to expiry for each swap.

**Market Parameters Summary:**

* `scalar_root`: 158.9
* `initial_anchor`: 1.05 (5% APY midpoint)
* `ln_fee_rate_root`: 0.00995

#### Example: SY → PT Swap

Now let's see how fees work on an actual swap. Assume:

* Current time: 6 months before expiry (15,768,000 seconds remaining)
* Market state: 1,000 PT and 1,000 SY in the pool (balanced)
* PY index: 1.0 (no yield accrued yet)
* User wants to swap: SY for 100 PT

**Step 1: Calculate Current Fee Rate**

```
time_to_expiry = 15,768,000 seconds (6 months)
fee_rate = e^(ln_fee_rate_root * time_to_expiry / seconds_per_year)
fee_rate = e^(0.00995 * 15,768,000 / 31,536,000)
fee_rate = e^(0.00995 * 0.5)
fee_rate = e^(0.004975)
fee_rate ≈ 1.00499
```

The fee rate is \~0.499% (half of the 1% annual rate, since we're halfway to expiry). This is the multiplier applied to the trade.

**Step 2: Calculate Exchange Rate**:

```
proportion (p) = 1,000 / (1,000 + 1,000) = 0.5
ln(p/(1-p)) = ln(0.5/0.5) = ln(1) = 0
rate_scalar = scalar_root * seconds_per_year / time_to_expiry
rate_scalar = 158.9 * 31,536,000 / 15,768,000 = 317.8
rate_anchor ≈ 1.05 (calibrated from last_ln_implied_rate)

exchange_rate = ln(p/(1-p)) / rate_scalar + rate_anchor
exchange_rate = 0 / 317.8 + 1.05 = 1.05
```

This means 1.05 PT = 1 SY (PT is at a discount, as expected for positive yield).

**Step 3: Execute Swap with Fee** User wants to swap SY for exact PT amount (e.g., wants 100 PT):

```
Pre-fee exchange rate: 1.05 (from Step 2)
Pre-fee SY needed: 100 / 1.05 = 95.23 SY

Fee calculation (for swap_sy_for_exact_pt):
fee = pre_fee_asset_to_account * (fee_rate - 1)
fee = 95.23 * (1.00499 - 1)
fee = 95.23 * 0.00499
fee ≈ 0.475 SY

Total SY needed (including fee):
net_asset_to_account = 95.23 + 0.475 = 95.705 SY
```

**Alternative way to think about it:** The post-fee exchange rate is: `1.05 / 1.00499 ≈ 1.0448` At this rate, to get 100 PT you need: `100 / 1.0448 ≈ 95.71 SY` 

**Step 4: Fee Distribution**

```
Total fee: 0.475 SY
Protocol fee (if 50%): 0.2375 SY → goes to protocol reserve
LP fee (50%): 0.2375 SY → goes to liquidity providers
```

**Final Result:**

* User wants: 100 PT
* User pays: \~95.705 SY (includes fee)
* Fee paid: \~0.475 SY (\~0.5% of trade value)
* Effective rate: 95.705 SY = 100 PT (includes fee)
* Pre-fee rate would have been: 95.23 SY = 100 PT

**Key Takeaway:** The fee decreases as expiry approaches. If this same swap happened 1 month before expiry, the fee would be \~0.08% instead of \~0.5%, because `time_to_expiry` is smaller in the fee calculation.


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