options-pricing

Options Pricing

Status: STUB — This skill provides a basic Black-Scholes implementation and an overview of planned capabilities. Full implementation is awaiting community contribution.

Options pricing is the quantitative foundation of derivatives trading. For crypto markets, options on BTC and ETH trade actively on Deribit, Lyra, and Aevo, while Solana options are emerging on platforms like Zeta Markets and PsyOptions. Understanding pricing models, implied volatility surfaces, and Greeks is essential for hedging, volatility trading, and constructing structured products.

This skill is informational and analytical only. It does not provide financial advice or trading recommendations.

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Current Capabilities

This stub includes a working Black-Scholes calculator with Greeks computation and a basic implied volatility solver. See scripts/black_scholes.py for the implementation.

import math
from scipy.stats import norm

def black_scholes_call(S: float, K: float, T: float, r: float, sigma: float) -> float:
    """Price a European call option using Black-Scholes.

    Args:
        S: Current underlying price.
        K: Strike price.
        T: Time to expiration in years.
        r: Risk-free rate (annualized).
        sigma: Volatility (annualized).

    Returns:
        Theoretical call option price.
    """
    d1 = (math.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    return S * norm.cdf(d1) - K * math.exp(-r * T) * norm.cdf(d2)

Run the demo:

python scripts/black_scholes.py --demo

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Planned Capabilities

When fully implemented, this skill will cover:

Pricing Models

Model	Option Style	Use Case
Black-Scholes	European	Vanilla calls/puts, quick Greeks
Binomial Tree	American	Early exercise, dividend-paying assets
Monte Carlo	Exotic	Path-dependent, barrier, Asian options
Black-76	Futures	Futures options on crypto perpetuals

Greeks

Greek	Measures	Formula Basis
Delta	Price sensitivity to underlying	dC/dS
Gamma	Delta sensitivity to underlying	d²C/dS²
Theta	Time decay per day	dC/dT
Vega	Sensitivity to volatility	dC/dσ
Rho	Sensitivity to interest rates	dC/dr

Implied Volatility

• Newton-Raphson and bisection IV solvers
• Volatility smile and skew analysis
• IV surface construction (strike x expiry)
• IV term structure analysis
• Vol-of-vol estimation

Crypto Options Platforms

Platform	Chain	Assets	Style
Deribit	Off-chain	BTC, ETH	European
Lyra	Optimism/Arbitrum	ETH, BTC	European
Aevo	Ethereum L2	BTC, ETH, alts	European
Zeta Markets	Solana	SOL, BTC	European
PsyOptions	Solana	SOL, various	American

Structured Products

• Covered calls and protective puts
• Straddles and strangles for volatility trading
• Vertical spreads for directional exposure
• Iron condors for range-bound markets
• Calendar spreads for term structure trades

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Prerequisites

# Core (for full implementation)
uv pip install numpy scipy

# Optional (for visualization)
uv pip install matplotlib

The included scripts/black_scholes.py uses only the Python standard library (math module) and runs without any dependencies.

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Use Cases

Hedging

Compute delta-neutral hedge ratios for crypto spot positions using options. Calculate the number of put contracts needed to protect a portfolio against downside moves.

Volatility Trading

Compare implied volatility to realized volatility to identify over/underpriced options. When IV significantly exceeds realized vol, selling premium may be favorable (and vice versa).

Structured Products

Price structured products that combine options at different strikes and expirations. Analyze payoff profiles and breakeven points before execution.

Risk Assessment

Use Greeks to understand portfolio-level exposure to price moves (delta), acceleration (gamma), time decay (theta), and volatility changes (vega).

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Quick Reference: Black-Scholes Formulas

Call price:

C = S * N(d1) - K * e^(-rT) * N(d2)

Put price:

P = K * e^(-rT) * N(-d2) - S * N(-d1)

Where:

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

Put-call parity:

C - P = S - K * e^(-rT)

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Files

File	Description
references/planned_features.md	Planned features, formulas, data sources, and implementation priorities
scripts/black_scholes.py	Black-Scholes calculator with Greeks and implied vol solver

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Contributing

This skill is a stub awaiting full implementation. To contribute:

1. Implement binomial tree pricing for American-style options
2. Add Monte Carlo simulation for exotic payoffs
3. Build IV surface construction from market quotes
4. Integrate Deribit API for live options chain data
5. Add portfolio Greeks aggregation

See references/planned_features.md for the full feature list and implementation priorities.

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This skill provides analytical tools and mathematical models for informational purposes only. It does not constitute financial advice. Options trading involves substantial risk of loss.