options-payoff

name	options-payoff
description	Option P&L analysis methodology: payoff diagrams, breakeven calculation, multi-leg strategy visualization, and Greeks-based scenario analysis.
category	asset-class

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.

---

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

Theta(Put)  = [-S × e^(-qT) × N'(d1) × σ / (2√T)
               + r × K × e^(-rT) × N(-d2)
               - q × S × e^(-qT) × N(-d1)] / 365

• Usually negative for option holders
• ATM options near expiration have the largest Theta magnitude, which benefits option sellers the most

Vega (Volatility Sensitivity, per 1% vol change)

Vega = S × e^(-qT) × N'(d1) × √T / 100

• Calls and puts have the same Vega
• ATM Vega is the largest, and Vega approaches 0 at expiration

Rho (Interest Rate Sensitivity, per 1% rate change)

Rho(Call) = K × T × e^(-rT) × N(d2) / 100
Rho(Put)  = -K × T × e^(-rT) × N(-d2) / 100

• The rate effect is usually small and often negligible for short-dated options

2.5 Implied Volatility Inversion (Newton-Raphson)

Given a market price P_market, solve for σ such that BS(σ) = P_market:

Iteration:
σ_{n+1} = σ_n - [BS(σ_n) - P_market] / Vega(σ_n)

Stopping condition: |BS(σ_n) - P_market| < 1e-6

Initial guess:
σ_0 = √(2π/T) × P_market/S  (Brenner-Subrahmanyam approximation)

Notes:
- If Vega is close to 0 (deep OTM / ITM), switch to bisection
- If the iteration does not converge (>100 rounds), return NaN and raise a warning
- IV > 500% is usually an outlier and should be filtered

---

3. Payoff Diagram Analysis

3.1 Expiry Payoff Curve

Calculation logic:

For each leg i (Call/Put, Long/Short, strike K_i, quantity n_i):
  Payoff_i(S_T) = n_i × direction_i × max(0, S_T - K_i)  # Call
  Payoff_i(S_T) = n_i × direction_i × max(0, K_i - S_T)  # Put

Where direction = +1 (Long) / -1 (Short)

Portfolio payoff = Σ Payoff_i - net premium cost
  (paid premium is positive, received premium is negative)

X-axis range: [min(K) × 0.7, max(K) × 1.3], step size 0.5 or 1

3.2 Theoretical Value Curve (Current Black-Scholes Pricing)

For each underlying price S, hold T, r, and σ constant and compute current theoretical PnL using the Black-Scholes formula:

TheoValue(S) = Σ n_i × direction_i × BS_price(S, K_i, T, r, σ, type_i) - net premium cost

The gap between the theoretical value curve and the expiry curve equals the remaining time value.

3.3 Break-Even Points

Numerically solve for the roots of Payoff(S_T) = 0:

• Use scipy.optimize.brentq to solve within adjacent intervals where the sign changes
• Single-leg strategies:
  • Long Call BEP = K + premium
  • Long Put BEP = K - premium
  • Short Call BEP = K + premium received
  • Short Put BEP = K - premium received
• Multi-leg strategies: solve numerically, possibly resulting in 0 to 2 BEPs

3.4 Max Profit / Max Loss

max_profit = max(payoff_curve)    # If inf, label as "Unlimited"
max_loss   = min(payoff_curve)    # If -inf, label as "Unlimited"

# Corresponding underlying price region
profit_range = S_range[payoff_curve > 0]

3.5 P&L Under Different Volatility Scenarios

Generate a σ scenario matrix using current IV × [0.5, 0.75, 1.0, 1.25, 1.5]. Plot one theoretical value curve for each σ and distinguish them by color to observe Vega sensitivity.

---

4. Python Code Templates

4.1 Black-Scholes Pricing Functions

import numpy as np
from scipy.stats import norm
from scipy.optimize import brentq
from typing import Literal

def bs_price(
    S: float,
    K: float,
    T: float,
    r: float,
    sigma: float,
    option_type: Literal["call", "put"],
    q: float = 0.0,
) -> float:
    """Black-Scholes option pricing.

    Args:
        S: Current underlying price
        K: Strike price
        T: Time to expiration in years
        r: Risk-free rate in annualized continuous compounding, e.g. 0.03
        sigma: Annualized volatility, e.g. 0.20
        option_type: "call" or "put"
        q: Continuous dividend yield, defaults to 0

    Returns:
        Theoretical option price

    Raises:
        ValueError: If sigma <= 0
    """
    if T <= 0:
        # After expiration, return intrinsic value directly.
        if option_type == "call":
            return max(0.0, S - K)
        return max(0.0, K - S)
    if sigma <= 0:
        raise ValueError(f"sigma must be > 0, got {sigma}")

    d1 = (np.log(S / K) + (r - q + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    if option_type == :
        price = S * np.exp(-q * T) * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)
    :
        price = K * np.exp(-r * T) * norm.cdf(-d2) - S * np.exp(-q * T) * norm.cdf(-d1)

     (price)


 () -> :
    
     T <= :
         {: , : , : , : , : }

    d1 = (np.log(S / K) + (r - q +  * sigma**) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    n_prime_d1 = norm.pdf(d1)
    exp_qt = np.exp(-q * T)
    exp_rt = np.exp(-r * T)

     option_type == :
        delta = exp_qt * norm.cdf(d1)
        rho = K * T * exp_rt * norm.cdf(d2) / 
        theta = (
            -S * exp_qt * n_prime_d1 * sigma / ( * np.sqrt(T))
            - r * K * exp_rt * norm.cdf(d2)
            + q * S * exp_qt * norm.cdf(d1)
        ) / 
    :
        delta = exp_qt * (norm.cdf(d1) - )
        rho = -K * T * exp_rt * norm.cdf(-d2) / 
        theta = (
            -S * exp_qt * n_prime_d1 * sigma / ( * np.sqrt(T))
            + r * K * exp_rt * norm.cdf(-d2)
            - q * S * exp_qt * norm.cdf(-d1)
        ) / 

    gamma = exp_qt * n_prime_d1 / (S * sigma * np.sqrt(T))
    vega = S * exp_qt * n_prime_d1 * np.sqrt(T) / 

     {
        : (delta, ),
        : (gamma, ),
        : (theta, ),
        : (vega, ),
        : (rho, ),
    }


 () -> :
    
    
    intrinsic = (, S - K  option_type ==   K - S)
     market_price < intrinsic - :
         ValueError()

    
    sigma = np.sqrt( * np.pi / T) * market_price / S
    sigma = (, (sigma, ))

     _  (max_iter):
        price = bs_price(S, K, T, r, sigma, option_type, q)
        vega = bs_greeks(S, K, T, r, sigma, option_type, q)[] *   

        diff = price - market_price
         (diff) < tol:
             (sigma, )

         (vega) < :
            
            :
                 (brentq(
                     v: bs_price(S, K, T, r, v, option_type, q) - market_price,
                    , , xtol=tol, maxiter=
                ))
             ValueError:
                 np.nan

        sigma -= diff / vega
        sigma = (, (sigma, ))  

     np.nan  

4.2 Multi-Leg Portfolio Payoff Calculation

from dataclasses import dataclass
import numpy as np

@dataclass
class OptionLeg:
    """Single option leg definition.

    Attributes:
        option_type: "call" or "put"
        K: Strike price
        direction: +1 for Long / -1 for Short
        quantity: Number of contracts, defaults to 1
        premium: Actual traded premium, positive when paid and negative when received
        T: Time to expiration in years, used for theoretical Black-Scholes pricing
        sigma: Volatility used in pricing
    """
    option_type: Literal["call", "put"]
    K: float
    direction: int  # +1 or -1
    quantity: float = 1.0
    premium: float = 0.0
    T: float = 0.25
    sigma: float = 0.20


def compute_expiry_payoff(
    legs: list[OptionLeg],
    S_range: np.ndarray,
) -> np.ndarray:
    """Calculate the expiry payoff curve.

    Args:
        legs: Option legs
        S_range: Array of underlying prices

    Returns:
        Payoff array aligned with S_range, including premium cost
    """
    total_payoff = np.zeros(len(S_range))
    net_premium = sum(leg.direction * leg.quantity * leg.premium for leg in legs)

    for leg in legs:
        if leg.option_type == "call":
            intrinsic = np.maximum(S_range - leg.K, 0)
        else:
            intrinsic = np.maximum(leg.K - S_range, 0)
        total_payoff += leg.direction * leg.quantity * intrinsic

     total_payoff - net_premium


 () -> np.ndarray:
    
    total_value = np.zeros((S_range))
    net_premium = (leg.direction * leg.quantity * leg.premium  leg  legs)

     leg  legs:
        prices = np.array([
            bs_price(S, leg.K, leg.T, r, leg.sigma, leg.option_type, q)
             S  S_range
        ])
        total_value += leg.direction * leg.quantity * prices

     total_value - net_premium


 () -> []:
    
    beps = []
     i  ((S_range) - ):
         payoff[i] * payoff[i + ] < :
            bep = brentq(
                 s: np.interp(s, S_range, payoff),
                S_range[i], S_range[i + ],
                xtol=
            )
            beps.append((bep, ))
     beps

4.3 Matplotlib Payoff Diagram

import matplotlib.pyplot as plt
import matplotlib.ticker as mticker

def plot_payoff_diagram(
    legs: list[OptionLeg],
    S_current: float,
    r: float = 0.03,
    q: float = 0.0,
    title: str = "Option Payoff Diagram",
    figsize: tuple = (10, 6),
) -> plt.Figure:
    """Plot the payoff diagram for an option portfolio.

    Args:
        legs: Option legs
        S_current: Current underlying price
        r: Risk-free rate
        q: Continuous dividend yield
        title: Chart title
        figsize: Figure size

    Returns:
        A matplotlib Figure object
    """
    K_values = [leg.K for leg in legs]
    S_lo = min(K_values) * 0.70
    S_hi = max(K_values) * 1.30
    S_range = np.linspace(S_lo, S_hi, 500)

    expiry_pnl = compute_expiry_payoff(legs, S_range)
    theo_pnl = compute_theo_value(legs, S_range, r, q)
    beps = find_breakeven_points(S_range, expiry_pnl)

    fig, ax = plt.subplots(figsize=figsize)

    # Shade profit and loss regions.
    ax.fill_between(S_range, expiry_pnl, 0,
                    where=(expiry_pnl >= 0), alpha=0.15, color="green", label="_nolegend_")
    ax.fill_between(S_range, expiry_pnl, 0,
                    where=(expiry_pnl < 0), alpha=0.15, color="red", label="_nolegend_")

    # Expiry payoff curve.
    ax.plot(S_range, expiry_pnl, color=, linewidth=, label=)

    
    ax.plot(S_range, theo_pnl, color=, linewidth=,
            linestyle=, label=)

    
    ax.axhline(, color=, linewidth=, linestyle=)

    
    ax.axvline(S_current, color=, linewidth=, linestyle=,
               label=)

    
     K  K_values:
        ax.axvline(K, color=, linewidth=, linestyle=, alpha=)
        ax.text(K, ax.get_ylim()[], , fontsize=,
                rotation=, va=, color=)

    
     bep  beps:
        ax.scatter([bep], [], color=, zorder=, s=)
        ax.annotate(, xy=(bep, ),
                    xytext=(bep, (expiry_pnl) * ),
                    fontsize=, ha=, color=,
                    arrowprops=(arrowstyle=, color=, lw=))

    
    max_p = (expiry_pnl)
    max_l = (expiry_pnl)
    stats_text = (
        
        
        
    )
    ax.text(, , stats_text, transform=ax.transAxes,
            fontsize=, va=, bbox=(boxstyle=, fc=, alpha=))

    ax.set_xlabel()
    ax.set_ylabel()
    ax.set_title(title)
    ax.legend(loc=)
    ax.yaxis.set_major_formatter(mticker.FuncFormatter( x, _: ))
    ax.grid(, alpha=)
    plt.tight_layout()

     fig

4.4 Plotly Interactive Payoff Diagram (Recommended for Frontend Display)

import plotly.graph_objects as go

def plot_payoff_plotly(
    legs: list[OptionLeg],
    S_current: float,
    r: float = 0.03,
    q: float = 0.0,
    title: str = "Option Payoff Diagram",
    sigma_scenarios: list[float] | None = None,
) -> go.Figure:
    """Generate a Plotly interactive payoff diagram with optional multi-sigma scenarios.

    Args:
        sigma_scenarios: For example [0.10, 0.15, 0.20, 0.25, 0.30].
            If None, use each leg's own sigma.
    """
    K_values = [leg.K for leg in legs]
    S_range = np.linspace(min(K_values) * 0.70, max(K_values) * 1.30, 500)
    expiry_pnl = compute_expiry_payoff(legs, S_range)

    fig = go.Figure()

    # Expiry payoff.
    fig.add_trace(go.Scatter(
        x=S_range, y=expiry_pnl,
        name="Expiry P&L", line=dict(color="steelblue", width=2),
        fill="tozeroy",
        fillcolor="rgba(70,130,180,0.1)",
    ))

    # Theoretical value under multiple volatility scenarios.
    if sigma_scenarios:
        colors = ["#FF6B6B", "#FFA500", "#4CAF50", "#2196F3", "#9C27B0"]
        for i, sigma in enumerate(sigma_scenarios):
            scenario_legs = [
                OptionLeg(
                    option_type=leg.option_type, K=leg.K,
                    direction=leg.direction, quantity=leg.quantity,
                    premium=leg.premium, T=leg.T, sigma=sigma
                )
                 leg  legs
            ]
            theo = compute_theo_value(scenario_legs, S_range, r, q)
            fig.add_trace(go.Scatter(
                x=S_range, y=theo,
                name=,
                line=(color=colors[i % (colors)], width=, dash=),
            ))
    :
        theo_pnl = compute_theo_value(legs, S_range, r, q)
        fig.add_trace(go.Scatter(
            x=S_range, y=theo_pnl,
            name=,
            line=(color=, width=, dash=),
        ))

    
    fig.add_hline(y=, line_dash=, line_color=, line_width=)
    fig.add_vline(x=S_current, line_dash=, line_color=,
                  annotation_text=, annotation_position=)

    
     K  (K_values):
        fig.add_vline(x=K, line_dash=, line_color=,
                      line_width=, opacity=)

    fig.update_layout(
        title=title,
        xaxis_title=,
        yaxis_title=,
        hovermode=,
        template=,
        legend=(orientation=, yanchor=, y=, xanchor=, x=),
    )

     fig

4.5 Greeks Profile vs Underlying Price

def plot_greeks_profile(
    legs: list[OptionLeg],
    S_current: float,
    r: float = 0.03,
    q: float = 0.0,
    greeks_to_plot: list[str] | None = None,
) -> go.Figure:
    """Plot portfolio Greeks as functions of the underlying price.

    Args:
        greeks_to_plot: Defaults to ["delta", "gamma", "vega", "theta"]
    """
    if greeks_to_plot is None:
        greeks_to_plot = ["delta", "gamma", "vega", "theta"]

    K_values = [leg.K for leg in legs]
    S_range = np.linspace(min(K_values) * 0.70, max(K_values) * 1.30, 300)

    # Compute portfolio Greeks.
    greek_values = {g: np.zeros(len(S_range)) for g in greeks_to_plot}
    for leg in legs:
        for j, S in enumerate(S_range):
            g = bs_greeks(S, leg.K, leg.T, r, leg.sigma, leg.option_type, q)
            for name in greeks_to_plot:
                greek_values[name][j] += leg.direction * leg.quantity * g[name]

    # Plot subplots.
    from plotly.subplots import make_subplots
    n = len(greeks_to_plot)
    fig = make_subplots(rows=n, cols=1, shared_xaxes=,
                        subplot_titles=[g.capitalize()  g  greeks_to_plot])

    greek_colors = {: , : ,
                    : , : , : }

     i, name  (greeks_to_plot, start=):
        fig.add_trace(
            go.Scatter(x=S_range, y=greek_values[name],
                       name=name.capitalize(),
                       line=(color=greek_colors.get(name, ), width=)),
            row=i, col=
        )
        fig.add_hline(y=, line_dash=, line_color=,
                      line_width=, row=i, col=)
        fig.add_vline(x=S_current, line_dash=, line_color=,
                      line_width=, row=i, col=)

    fig.update_layout(
        title=,
        height= * n,
        showlegend=,
        template=,
    )

     fig

---

5. Practical Usage

5.1 Strategy Selection Decision Tree by Market View

Market view
├── Strongly bullish
│   ├── Willing to pay premium → Long Call
│   └── Want lower cost → Bull Call Spread
├── Moderately bullish
│   ├── Already hold the underlying → Covered Call (income enhancement)
│   └── No existing position → Bull Put Spread (net credit)
├── Moderately bearish
│   ├── Already hold the underlying → Protective Put or Collar
│   └── No existing position → Bear Call Spread (net credit)
├── Strongly bearish
│   ├── Willing to pay premium → Long Put
│   └── Want lower cost → Bear Put Spread
├── Range-bound market (low-IV environment)
│   ├── Wide range → Short Strangle
│   ├── Narrow range → Short Straddle
│   └── Want limited risk → Iron Condor / Iron Butterfly
└── Large move expected (low-IV environment)
    ├── Direction unclear → Long Straddle / Long Strangle
    └── Slight directional bias → Call / Put Back Spread

5.2 Volatility Environment → Strategy Mapping

IV Regime	Rule of Thumb	Suitable Strategies	Strategies to Avoid
Low IV (< 20th percentile)	IV Rank < 20	Long Straddle, Long Strangle, Back Spread	Short strategies, because premium is too thin
Normal IV (20th to 80th percentile)	IV Rank 20 to 80	Vertical spreads, Calendar Spread, Diagonal	Single-leg positions with asymmetric risk
High IV (> 80th percentile)	IV Rank > 80	Short Straddle, Iron Condor, Covered Call	Long single-leg options due to rich premium

IV Rank formula:

iv_rank = (current_iv - iv_52w_low) / (iv_52w_high - iv_52w_low) * 100

IV Percentile: The historical percentile rank of current IV over the last 252 trading days.

5.3 When to Roll or Adjust

Rolling

• Trigger: Option Delta moves outside the target range, or time to expiration < 21 days
• Rolling Up / Down: Close the current leg and reopen at a higher / lower strike while keeping the same directional bias
• Rolling Out: Close the near-month leg and reopen further out on the curve to harvest additional time value
• Cost assessment: Compare the net debit / credit of the roll with the payoff from simply holding to expiration

Adjusting

• Delta-neutral rebalancing: Hedge with underlying or options when portfolio Delta deviates from target by more than ±0.10
• Gamma scalping: Under a Long Gamma portfolio, hedge Delta after large underlying moves to lock in gains
• Stop-loss rule: Force liquidation when losses reach 2× the initial premium received, a common rule for Iron Condors

Common Adjustment Examples

Iron Condor gets breached:

Underlying rallies above the short call:
1. Close the call spread and realize the loss
2. Reassess directional view:
   - Still bullish → reopen a higher put spread to preserve neutrality
   - Not bullish → close the entire portfolio

Covered Call faces assignment risk:

Underlying approaches the call strike:
1. Assess whether you are willing to sell the underlying at that price
   - Yes → allow assignment and keep premium + capital gain
   - No → Roll Up & Out to a higher strike and/or later expiration

---

Quick Usage Example

# Example: Iron Condor payoff diagram
legs = [
    OptionLeg("put",  K=90,  direction=-1, premium=1.5, T=0.083, sigma=0.20),
    OptionLeg("put",  K=85,  direction=+1, premium=0.5, T=0.083, sigma=0.20),
    OptionLeg("call", K=110, direction=-1, premium=1.5, T=0.083, sigma=0.20),
    OptionLeg("call", K=115, direction=+1, premium=0.5, T=0.083, sigma=0.20),
]

fig = plot_payoff_plotly(
    legs, S_current=100.0,
    title="Iron Condor (85/90/110/115, 1 month)",
    sigma_scenarios=[0.15, 0.20, 0.25, 0.30],
)
fig.show()

# Implied volatility example
iv = implied_volatility(
    market_price=5.0, S=100, K=100,
    T=0.25, r=0.03, option_type="call"
)
print(f"Implied volatility: {iv:.2%}")  # about 0.20