Call Warrant Pricing

免責事項: このホワイトペーパーは英語版が正式な文書となります。他言語の翻訳は参照用です。

Pricing Philosophy

PIPO Call Warrants are priced through a combination of:

AMM Market Price: Real-time supply/demand via CPMM
Oracle Valuation: Fundamental anchor from oracle committee
Time Value: Decay toward expiry
Implied Probability: Market’s expectation of outcome

Price Interpretation

What Does the Price Mean?

A Call Warrant price of $0.40 can be interpreted as:

Interpretation	Explanation
Implied Probability	~40% chance underlying exceeds strike
Expected Return	If ITM, expect ~40% of max payout
Risk/Reward	Risk $0.40 to potentially gain $0.60

Price Bounds

Price	Market Sentiment
$0.05-0.15	Deep OTM, low probability
$0.20-0.35	OTM, speculative
$0.40-0.60	ATM, balanced
$0.65-0.80	ITM, high confidence
$0.85-0.98	Deep ITM, near certainty

Pricing Factors

1. Intrinsic Value

Intrinsic Value = max(0, (Current Valuation - Strike) / Strike)

Example:
Current Valuation: $200B (Oracle)
Strike: $180B
Intrinsic Value = ($200B - $180B) / $180B = 11.1%

Minimum warrant price ≈ $0.111 (plus any time value)

2. Time Value

Time value reflects the probability of favorable price movement before expiry:

Time Value = f(Time to Expiry, Implied Volatility, Interest Rate)

General relationship:
- More time → Higher time value
- Higher volatility → Higher time value
- Higher rates → Slightly higher call value

3. Implied Volatility

Since pre-IPO companies don’t have historical price data, implied volatility is derived from:

Source	Weight
AMM price movements	40%
Comparable public companies	30%
Recent funding round frequency	15%
News sentiment volatility	15%

4. Supply/Demand

AMM prices reflect actual market supply and demand:

High demand (more buyers) → Price increases
High supply (more sellers) → Price decreases

Pricing Model

CPMM Foundation

PIPO uses Constant Product Market Maker (CPMM):

x × y = k

Where:
x = Warrant token reserve
y = USDC reserve
k = Constant product (invariant)

Price Calculation

Spot Price = y / x (USDC per Warrant)

After buying Δx warrants:
New price = y / (x - Δx) × (1 + slippage)

Price Impact = Δx / x × 100%

Example Trade

Pool State:
- Warrant reserve (x): 100,000 tokens
- USDC reserve (y): 40,000 USDC
- k = 4,000,000,000
- Spot price = $0.40

User buys 1,000 warrants:
- Receives: 1,000 tokens
- Pays: 40,000 / (100,000 - 1,000) × 1,000 = 404.04 USDC
- Effective price: $0.404 (1% slippage)
- New spot price: 40,404 / 99,000 = $0.408

Oracle Price Anchor

Purpose

The oracle provides a fundamental anchor to prevent manipulation:

Oracle Role:
├── Establishes "fair value" reference
├── Used for ITM/OTM determination
├── Triggers circuit breakers if AMM deviates too far
└── Final settlement price source

Deviation Limits

Deviation from Oracle	Action
< 10%	Normal trading
10-20%	Warning displayed
20-30%	Increased fees (2x)
> 30%	Trading paused, review

Time Decay (Theta)

Decay Pattern

┌─────────────────────────────────────────────────────────────────┐
│                   TIME DECAY CURVE                              │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  Time Value                                                     │
│      │                                                          │
│  100%│●                                                         │
│      │  ●                                                       │
│   80%│    ●                                                     │
│      │      ●                                                   │
│   60%│        ●                                                 │
│      │          ●                                               │
│   40%│            ●                                             │
│      │              ●                                           │
│   20%│                ●                                         │
│      │                  ●  ●                                    │
│    0%│────────────────────●●●●●                                 │
│      └────────────────────────────────────────────────────────  │
│       12mo   9mo    6mo    3mo    1mo   1wk  Expiry            │
│                                                                 │
│  Note: Decay accelerates as expiry approaches                  │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘

Decay Formula (Approximation)

Theta ≈ -0.5 × σ × S × N'(d1) / √T

Where:
σ = Implied volatility
S = Current underlying value
N'(d1) = Standard normal PDF
T = Time to expiry

Simplified daily decay:
Daily Theta ≈ Time Value / Days to Expiry × Acceleration Factor

Pricing Examples

Example 1: ATM Call, 6 Months to Expiry

Underlying: SpaceX
Oracle Valuation: $180B
Strike: $180B (ATM)
Time to Expiry: 6 months
Implied Volatility: 50%

Components:
├── Intrinsic Value: $0.00 (ATM)
├── Time Value: ~$0.38
└── Total Price: ~$0.38

Interpretation:
- 38% probability of being ITM at expiry
- Or: Expected 38% of max payout

Example 2: OTM Call, 3 Months to Expiry

Underlying: SpaceX
Oracle Valuation: $180B
Strike: $220B (22% OTM)
Time to Expiry: 3 months
Implied Volatility: 50%

Components:
├── Intrinsic Value: $0.00 (OTM)
├── Time Value: ~$0.12
└── Total Price: ~$0.12

Interpretation:
- 12% probability of SpaceX reaching $220B by expiry
- High risk, high reward speculation

Example 3: ITM Call Near Expiry

Underlying: SpaceX
Oracle Valuation: $200B
Strike: $180B (11% ITM)
Time to Expiry: 2 weeks
Implied Volatility: 50%

Components:
├── Intrinsic Value: ~$0.11
├── Time Value: ~$0.03
└── Total Price: ~$0.14

Interpretation:
- High probability of profitable exercise
- Minimal time value remaining
- Price closely tracks intrinsic value

Dynamic Pricing Adjustments

News Events

Major news can cause rapid price movements:

Event: SpaceX successfully lands Starship
├── Pre-event price: $0.35
├── Volume surge: 5x normal
├── Post-event price: $0.52
└── Duration: Price stabilizes in 24-48 hours

Liquidity Depth Impact

Deep liquidity pool (high k):
- Large trades have small price impact
- Prices more stable

Shallow liquidity pool (low k):
- Large trades have significant price impact
- Prices more volatile

Fair Value Estimation

For Users

PIPO provides a “Fair Value Estimate” widget:

┌─────────────────────────────────────────────────────────────────┐
│                   FAIR VALUE ESTIMATE                           │
├─────────────────────────────────────────────────────────────────┤
│                                                                 │
│  SPACEX-CALL-200B-Q42025                                       │
│                                                                 │
│  Current AMM Price:     $0.32                                   │
│  Fair Value Estimate:   $0.35 ± $0.03                          │
│  Status:               UNDERVALUED (buy signal)                │
│                                                                 │
│  Components:                                                    │
│  ├── Intrinsic:        $0.00                                   │
│  ├── Time Value:       $0.28                                   │
│  ├── Volatility Adj:   $0.04                                   │
│  └── Liquidity Adj:    $0.03                                   │
│                                                                 │
│  Confidence: Medium (limited historical data)                  │
│                                                                 │
│  [View Methodology] [Set Price Alert]                          │
│                                                                 │
└─────────────────────────────────────────────────────────────────┘

Price Discovery Quality

Metrics

Metric	Target	Current
Bid-Ask Spread	< 2%	TBD
Price Impact (10K trade)	< 1%	TBD
Oracle-AMM Deviation	< 15%	TBD
Time to Price Recovery	< 1 hour	TBD

Monitoring

Price Discovery Dashboard (Internal)
├── Real-time bid-ask spreads
├── Trade size vs price impact
├── Oracle deviation alerts
├── Manipulation detection flags
└── LP health metrics

API Reference

Get Current Price

const price = await pipo.getPrice('SPACEX-CALL-200B-Q42025');

// Response
{
  warrantId: 'SPACEX-CALL-200B-Q42025',
  spotPrice: 0.32,
  bidPrice: 0.315,
  askPrice: 0.325,
  oracleValuation: 180000000000,
  intrinsicValue: 0,
  timeValue: 0.32,
  impliedProbability: 0.32,
  lastUpdate: '2025-10-15T12:00:00Z'
}

Get Price Quote

const quote = await pipo.getQuote({
  warrant: 'SPACEX-CALL-200B-Q42025',
  side: 'BUY',
  amount: 1000
});

// Response
{
  inputAmount: 1000, // tokens
  outputAmount: 325.50, // USDC
  effectivePrice: 0.3255,
  priceImpact: 1.5, // percent
  fee: 0.975, // USDC (0.3%)
  minimumReceived: 318.99, // with 2% slippage tolerance
  route: 'AMM_DIRECT'
}