Black Scholes

Black-Scholes-Merton Model and Extensions

> BSM derivation intuition, formula, put-call parity, Greeks, limitations, and extensions for dividends, American options, and implied volatility.

When to Activate

• User pricing European options or computing implied volatility
• Deriving or interpreting BSM Greeks for hedging
• Understanding BSM assumptions and when they break down
• Extending BSM for dividends, discrete hedging, or jumps
• Computing put-call parity relationships or identifying arbitrage
• Teaching or explaining option pricing fundamentals

Core Concepts

BSM Derivation Intuition

The Black-Scholes-Merton model rests on three pillars:

1. Geometric Brownian Motion (GBM): stock price follows dS = muSdt + sigmaSdW

• Log returns are normally distributed: ln(ST/S0) ~ N((mu - sigma^2/2)T, sigma^2T)
• Continuous paths, no jumps — a key limitation

1. No-arbitrage / risk-neutral pricing: construct a riskless portfolio of option + delta shares

• The portfolio earns the risk-free rate by no-arbitrage
• Under the risk-neutral measure, the drift of S becomes r (risk-free rate), not mu
• Option price = discounted expected payoff under risk-neutral measure

1. Continuous hedging: delta hedge is rebalanced continuously (no transaction costs)

• In practice, discrete hedging introduces hedging error proportional to gamma * (dS)^2

BSM Formula

European Call: C = S N(d1) - K exp(-rT) * N(d2)

European Put: P = K exp(-rT) N(-d2) - S * N(-d1)

Where:

• d1 = [ln(S/K) + (r + sigma^2/2)T] / (sigma sqrt(T))
• d2 = d1 - sigma * sqrt(T)
• N(x) = cumulative standard normal distribution
• S = spot price, K = strike, r = risk-free rate, T = time to expiry, sigma = volatility

Interpretation:

• N(d2) = risk-neutral probability that the option expires in-the-money
• S * N(d1) = present value of receiving the stock conditional on exercise
• K exp(-rT) N(d2) = present value of paying the strike conditional on exercise
• Delta of call = N(d1); Delta of put = N(d1) - 1

Put-Call Parity

For European options with same strike and expiry:

• C - P = S - K * exp(-rT)
• Holds by no-arbitrage regardless of model
• If violated, construct an arbitrage: buy the cheap side, sell the expensive side
• With dividends: C - P = S - PV(dividends) - K * exp(-rT)
• Put-call parity does NOT hold for American options (early exercise creates inequality)

BSM Greeks (Analytical)

Delta: dC/dS = N(d1) for calls; N(d1) - 1 for puts Gamma: d^2C/dS^2 = phi(d1) / (S sigma sqrt(T)) — same for calls and puts Theta: dC/dt = -(S phi(d1) sigma) / (2sqrt(T)) - r K exp(-rT) N(d2) for calls Vega: dC/d(sigma) = S phi(d1) sqrt(T) — same for calls and puts Rho: dC/dr = K T exp(-rT) * N(d2) for calls

Where phi(x) = standard normal density function.

BSM Assumptions and Limitations

1. Constant volatility: real markets show volatility smile/skew — OTM puts have higher implied vol
2. Log-normal returns: real returns have fat tails and negative skewness
3. No jumps: real prices jump (earnings, macro events) — jump-diffusion models address this
4. Continuous trading: discrete hedging introduces P&L noise
5. No transaction costs: in practice, hedging costs make BSM prices underestimates
6. Constant interest rates: relevant for long-dated options on bonds/rates
7. No dividends: base BSM assumes no dividends (correctable)
8. European exercise: BSM does not price American options directly

Methodology

Implied Volatility Computation

Implied volatility is the sigma that makes BSM price equal to the observed market price.

Newton-Raphson Method:

1. Start with initial guess sigma_0 (e.g., 20%)
2. Compute BSM price C(sigman) and vega V(sigman)
3. Update: sigma{n+1} = sigman - (C(sigman) - Cmarket) / V(sigma_n)
4. Converge when |C(sigman) - Cmarket| implied vol.

## Quality Gate

- BSM should only be used as a benchmark; always acknowledge smile/skew effects for real pricing
- Implied volatility must converge within 0.01% accuracy; verify with independent pricer
- Put-call parity must hold to within bid-ask spread; violations indicate data errors or arbitrage
- Greeks must be computed analytically from BSM for European options (not finite differences unless validating)
- For American options, do not use BSM — use binomial trees or finite difference methods
- Dividend treatment must match the actual dividend schedule (discrete dividends for single stocks, continuous yield for indices)
- When BSM implied vol varies significantly across strikes (>5 vol points), note that BSM is failing and a stochastic vol or local vol model is needed
- Validate BSM implementation against known test cases (e.g., ATM call approximately = S * sigma * sqrt(T) * 0.4)
- Document all input assumptions (rate curve, dividend yield, borrow cost) used in BSM calculation

## Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

- **Author:** [brainbytes-dev](https://github.com/brainbytes-dev)
- **Source:** [brainbytes-dev/everything-claude-trading](https://github.com/brainbytes-dev/everything-claude-trading)
- **License:** MIT

Install and usage instructions live in the source repository linked above.