docs/volume-iii/pricing-models.md

# GRU Bond Pricing Models

## Overview

GRU bond pricing models provide comprehensive valuation frameworks accounting for perpetual bond structures, index adjustments, and multi-reality settlement considerations.

## Base Pricing Model

### Formula
```
Price = PV(Coupons) + PV(Perpetual Component) + Index Adjustment(LiXAU/LiPMG/etc.)
```

### Components

#### Present Value of Coupons
```
PV(Coupons) = Σ(Coupon[i] / (1 + r)^t[i])
```
Where:
- `Coupon[i]` = Coupon payment at period i
- `r` = Discount rate
- `t[i]` = Time to coupon payment i

#### Present Value of Perpetual Component
```
PV(Perpetual) = Principal / r
```
For perpetual bonds, the principal value is discounted at the required rate of return.

#### Index Adjustment
```
Index Adjustment = Principal × (IndexValue / BaseIndexValue - 1) × IndexWeight
```
Where:
- `IndexValue` = Current index value (LiXAU, LiPMG, etc.)
- `BaseIndexValue` = Index value at bond issuance
- `IndexWeight` = Weight of index in bond pricing

## Discounted Acquisition Model

### Formula
```
Acquisition Price = Nominal / 0.15
```

### Purpose
Used for reserve expansion and sovereign acquisition of GRU bonds.

### Characteristics
- High discount (85% off nominal)
- Reserve expansion mechanism
- Sovereign access pricing
- Long-term holding incentive

## GRU Liquidity Loop-Linked Yield

### Formula
```
Yield = f(7→10→9.55 cycles, Index Volatility, Sovereign Risk)
```

### Components

#### Liquidity Loop Cycles
The 7→10→9.55 cycle represents:
- Initial capital: 7 GRU
- Quantum mint: 10 GRU
- FX/spread deduction: 9.55 GRU
- Reinjection into next cycle

#### Index Volatility
```
Index Volatility = σ(IndexReturns) × VolatilityWeight
```

#### Sovereign Risk
```
Sovereign Risk = SRI_Score × RiskWeight
```

### Yield Calculation
```
Yield = BaseYield + LoopAdjustment + VolatilityAdjustment - RiskPenalty
```

Where:
- `BaseYield` = Base GRU bond yield
- `LoopAdjustment` = f(cycle efficiency, loop iterations)
- `VolatilityAdjustment` = Index volatility impact
- `RiskPenalty` = Sovereign risk premium

## Index-Linked Pricing

### LiXAU (Gold Index)
```
Price Adjustment = Principal × (XAU_Price / XAU_Base) × XAU_Weight
```

### LiPMG (PGM Basket Index)
```
Price Adjustment = Principal × (PGM_Index / PGM_Base) × PGM_Weight
```

### LiBMG (BMG Basket Indices)
```
Price Adjustment = Principal × Σ(BMG[i]_Weight × (BMG[i]_Price / BMG[i]_Base))
```

## Perpetual Bond Specific Pricing

### Perpetual Component Valuation
For 99-year perpetual bonds:
```
Perpetual Value = Annual Coupon / Required Yield
```

### Buy-Back Option Pricing
For 10-year buy-back option:
```
Option Value = PV(Principal at Buy-Back) - PV(Coupons Lost)
```

## Multi-Reality Pricing

### Quantum State Pricing
```
Price_Quantum = Σ(Probability[i] × Price[i])
```

### Parallel Reality Pricing
```
Price_Parallel = Average(Price[Reality[i]])
```

### Holographic Pricing
```
Price_Holographic = Project(Price_Classical, Holographic_Field)
```

### Merged Pricing
```
Price_Merged = Merge(Price_Classical, Price_Quantum, Price_Parallel, Price_Holographic)
```

## Pricing Service Integration

### Real-Time Pricing
- Continuous price updates
- Index value feeds
- Market data integration
- Automated recalculation

### Historical Pricing
- Price history tracking
- Performance analytics
- Volatility calculations
- Risk metrics

### Pricing Validation
- Model validation
- Backtesting
- Stress testing
- Regulatory reporting