options-payoff

# Options Payoff — Option P&L Analysis Methodology

## Overview

This skill is designed for option strategy analysis scenarios within the Vibe-Trading quantitative framework, covering:
- P&L curve generation for single-leg and multi-leg option portfolios
- Black-Scholes pricing and Greeks calculation
- Implied volatility inversion
- Strategy selection decision support

**Constraint**: For research and backtesting only. Do not output live trading instructions, in line with the project's guardrails.

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## 1. Supported Strategy Types

### 1.1 Single-Leg Strategies

| Strategy | Bias | Premium | Max Profit | Max Loss |
|------|------|--------|----------|----------|
| Long Call | Bullish | Paid | Unlimited | Premium |
| Long Put | Bearish | Paid | Strike - premium | Premium |
| Short Call | Neutral / mildly bearish | Received | Premium | Unlimited |
| Short Put | Neutral / mildly bullish | Received | Premium | Strike - premium |

### 1.2 Vertical Spreads

| Strategy | Structure | Market View | Net Premium |
|------|------|----------|----------|
| Bull Call Spread | Long Call (lower K) + Short Call (higher K) | Moderately bullish | Net debit |
| Bear Put Spread | Long Put (higher K) + Short Put (lower K) | Moderately bearish | Net debit |
| Bull Put Spread | Short Put (higher K) + Long Put (lower K) | Moderately bullish | Net credit |
| Bear Call Spread | Short Call (lower K) + Long Call (higher K) | Moderately bearish | Net credit |

### 1.3 Straddles / Strangles (Volatility Strategies)

| Strategy | Structure | Market View |
|------|------|----------|
| Long Straddle | Long Call (ATM) + Long Put (ATM) | Large move up or down, low volatility |
| Short Straddle | Short Call (ATM) + Short Put (ATM) | Range-bound market, high volatility |
| Long Strangle | Long Call (OTM) + Long Put (OTM) | Large move, lower cost than a straddle |
| Short Strangle | Short Call (OTM) + Short Put (OTM) | Tight range, collect two-sided premium |

### 1.4 Butterflies / Iron Butterflies

| Strategy | Structure | Feature |
|------|------|------|
| Long Butterfly (Call) | Long Call (K1) + 2× Short Call (K2) + Long Call (K3) | Low-cost bet that the underlying expires near K2 |
| Long Butterfly (Put) | Long Put (K3) + 2× Short Put (K2) + Long Put (K1) | Same logic, built with puts |
| Iron Butterfly | Short Call (K2) + Short Put (K2) + Long Call (K3) + Long Put (K1) | Net credit, max profit at K2 |

### 1.5 Condors / Iron Condors

| Strategy | Structure | Feature |
|------|------|------|
| Long Condor (Call) | Long Call (K1) + Short Call (K2) + Short Call (K3) + Long Call (K4) | Bet that the underlying stays between K2 and K3 |
| Iron Condor | Short Put (K2) + Long Put (K1) + Short Call (K3) + Long Call (K4) | Most common neutral strategy with capped risk on both sides |

Here K1 < K2 < K3 < K4, and K2 / K3 are usually OTM.

### 1.6 Calendar Spreads (Time Spreads)

| Strategy | Structure | Market View |
|------|------|----------|
| Calendar Spread | Short near-month Call/Put (K) + Long far-month Call/Put (K) | Short-term range-bound market + rising forward volatility |
| Diagonal Spread | Short near-month Call/Put (K1) + Long far-month Call/Put (K2) | Calendar spread with mild directional bias |

Calendar spreads profit because near-month Theta decay is faster than far-month Theta decay.

### 1.7 Ratio Spreads

| Strategy | Structure | Feature |
|------|------|------|
| Ratio Call Spread | Long 1× Call (K1) + Short N× Call (K2), N>1 | Limited upside profit, losses if the upside move becomes extreme |
| Ratio Put Spread | Long 1× Put (K2) + Short N× Put (K1) | Limited downside profit, losses if the downside move becomes extreme |
| Call Back Spread | Short 1× Call (K1) + Long N× Call (K2), N>1 | Profits from extreme upside, loses on a modest rally |
| Put Back Spread | Short 1× Put (K2) + Long N× Put (K1), N>1 | Profits from extreme downside, loses on a mild decline |

### 1.8 Protective / Hedging Strategies

| Strategy | Structure | Use Case |
|------|------|------|
| Covered Call | Long underlying + Short Call (K) | Generate income on an existing position, give up gains above K |
| Protective Put | Long underlying + Long Put (K) | Downside protection on an existing position, pay an insurance premium |
| Collar | Long underlying + Long Put (K1) + Short Call (K2) | Lock the position into a zero-cost / low-cost range |

---

## 2. Black-Scholes Pricing Model

### 2.1 Core Assumptions

- The underlying price follows geometric Brownian motion (lognormal distribution)
- Risk-free rate `r` is constant
- Volatility `σ` is constant (historical or implied)
- No dividends, or adjust with a continuous dividend yield `q`
- European options only (exercise at expiration)

### 2.2 Full Formula

```
S  = current underlying price
K  = strike price
T  = time to expiration (years)
r  = risk-free rate (annualized continuous compounding)
q  = continuous dividend yield (commonly used for China A-share / index options)
σ  = annualized volatility
N  = standard normal CDF

d1 = [ln(S/K) + (r - q + σ²/2) × T] / (σ × √T)
d2 = d1 - σ × √T

Call = S × e^(-qT) × N(d1) - K × e^(-rT) × N(d2)
Put  = K × e^(-rT) × N(-d2) - S × e^(-qT) × N(-d1)
```

### 2.3 Put-Call Parity

```
Call - Put = S × e^(-qT) - K × e^(-rT)
```

Use this to verify pricing consistency and detect arbitrage. When dividends exist, replace `S` with `S × e^(-qT)`.

### 2.4 Greeks Calculation

#### Delta (Price Sensitivity)
```
Delta(Call) = e^(-qT) × N(d1)
Delta(Put)  = e^(-qT) × (N(d1) - 1)
```
- Range: Call [0, 1], Put [-1, 0]
- ATM ≈ ±0.5, deep ITM → ±1, deep OTM → 0

#### Gamma (Rate of Change of Delta)
```
Gamma = e^(-qT) × N'(d1) / (S × σ × √T)

N'(x) = (1/√(2π)) × e^(-x²/2)  [standard normal PDF]
```
- Calls and puts have the same Gamma
- Gamma is highest near ATM and explodes as expiration approaches

#### Theta (Time Decay, per day)
```
Theta(Call) = [-S × e^(-qT) × N'(d1) × σ / (2√T)
               - r × K × e^(-rT) × N(d2)
               + q × S × e^(-qT) × N(d1)] / 365