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Black-Scholes Options Pricing

Difficulty advanced

Overview

The Black-Scholes model provides a theoretical price for European-style options. It's the foundation of modern options pricing and risk management.

Payoff at expiration

K (strike) price → profit breakeven = K + premium −premium unlimited ↑ long call K (strike) price → profit breakeven = K − premium −premium max profit ↑ long put
payoff at expiration · ignores premium paid up front for purchase

Assumptions

  1. European options (exercise only at expiration)
  2. No dividends (can be adjusted)
  3. Constant volatility and risk-free rate
  4. Log-normal distribution of stock prices
  5. No transaction costs or taxes
  6. Continuous trading

The Model

Call Option Price

C = S × N(d1) - K × e^{-rT} × N(d2)

where: C European call price · S spot · K strike · r annualized risk-free rate · T time to expiry (years) · N(·) standard-normal CDF · e^{-rT} discount factor. does: prices a European call as (spot × prob-weighted in-the-moneyness) minus (PV of strike × risk-neutral exercise probability). Closed-form when underlying follows GBM.

Put Option Price

P = K × e^{-rT} × N(-d2) - S × N(-d1)

where: symbols as above. does: put price by reflecting the call formula across the strike. Equivalent to call via put-call parity: P = C − S + K·e^{-rT}.

d1 and d2

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

where: ln(S/K) log-moneyness · (r + σ²/2)T drift adjustment under risk-neutral measure (note Itô correction +σ²/2) · σ√T total volatility over T. does: d1 and d2 are the standardized log-moneyness terms. N(d2) is the risk-neutral probability the option finishes in-the-money; N(d1) is approximately the option's delta.

Where: - S = Current stock price - K = Strike price - r = Risk-free rate - T = Time to expiration (in years) - σ = Volatility (annualized) - N(x) = Cumulative normal distribution

The Greeks

Delta (Δ)

Sensitivity to underlying price change.

Call Delta = N(d1)
Put Delta = N(d1) - 1

Range: 0 to 1 (call), -1 to 0 (put)

where: N(d1) standard-normal CDF at d1 · range bounded by [0,1] for calls and [−1,0] for puts. does: ∂Option/∂S — dollar change in option per $1 move in underlying. ATM ≈ 0.5; deep ITM → 1 (or −1 for puts); deep OTM → 0.

Gamma (Γ)

Sensitivity of delta to underlying price change.

Gamma = N'(d1) / (S × σ√T)

where N'(x) = normal PDF = e^{-x²/2} / √(2π)

Gamma is highest for at-the-money, short-dated options

where: N'(·) standard-normal probability density function (PDF, not CDF) · same denominator scales as the option's total volatility exposure. does: ∂Delta/∂S — how fast delta changes with spot. Long gamma = "buying low, selling high" by mechanical hedging; short gamma = the opposite. Peak ATM near expiry.

Theta (Θ)

Sensitivity to time decay.

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

Theta is always negative (options lose value over time)

where: first term = time-decay of optionality (always negative); second term = interest-rate effect on the discounted strike (negative for call, positive for put). Theta is typically quoted per day (divide annualized by 365). does: ∂Option/∂t — the option's daily bleed from time passing alone, holding spot and vol constant. Long options pay theta; short options collect it.

Vega (ν)

Sensitivity to volatility change.

Vega = S × N'(d1) × √T

Vega is highest for at-the-money, long-dated options

where: N'(d1) standard-normal PDF · √T square-root-of-time scaling. does: ∂Option/∂σ — change in option price for a 1.0 (i.e. 100 percentage-point) move in implied vol. In practice, divide by 100 to get the "per-vol-point" sensitivity.

Rho (ρ)

Sensitivity to interest rate change.

Call Rho = K × T × e^{-rT} × N(d2)
Put Rho = -K × T × e^{-rT} × N(-d2)

where: product of present-value-of-strike and risk-neutral exercise probability. does: ∂Option/∂r — sensitivity to a 1.0 move in the risk-free rate. Mostly negligible for short-dated equity options; matters for long-dated options and fixed-income derivatives.

Implied Volatility

Find the volatility σ that makes the Black-Scholes price equal the observed market price. There is no closed form — solve numerically (Newton-Raphson on vega, or Brent's method when vega is small). Once you have it, IV is the option market's quote — traders price and risk options in vol terms, not dollar terms. Differences in IV across strikes give the smile; differences across expirations give the term structure. The whole derivative pricing toolkit (Greeks, hedge ratios, exotic pricing) is then expressed as functions of the IV surface.

Practical Applications

Volatility Trading

  • Buy options when IV < historical vol (underpriced)
  • Sell options when IV > historical vol (overpriced)

Model Limitations

Assumption Reality Impact
Constant volatility Volatility smiles/skews exist Model misprices OTM options
Normal distribution Fat tails observed Underestimates extreme moves
No dividends Most stocks pay dividends Adjusted BS needed
Continuous trading Markets gap overnight Model doesn't capture gap risk
Constant rates Rates change Minor for short-dated options

q&a

Black-Scholes assumes constant volatility — how can it still be useful?

It's a model, not the truth. Practitioners use Black-Scholes as a translation layer: market prices ↔ implied volatility. You don't price an option from σ; you read σ off the market price. The model's machinery (the Greeks, hedge ratios) remains the working language of options desks even though the assumptions are technically violated.

What's the difference between historical and implied volatility?

Historical (realized) vol is computed from past price moves — backward-looking. Implied vol is solved from current option prices — forward-looking, market's expectation of future realized vol. The gap between them is where vol trading lives.

Why use log-normal returns instead of normal?

Prices can't go negative; log-normal distributions are bounded at zero. Also, multiplicative price moves are roughly log-normal in practice (returns are approximately normal). Note that real returns have fatter tails than log-normal predicts — see Volatility Smile.

Can I use Black-Scholes for American options?

For non-dividend-paying stocks, American calls have the same value as European (never optimal to exercise early), so yes. For puts, or with dividends, you need numerical methods (binomial trees, Monte Carlo, or finite difference) because early exercise can be optimal.

What's the most common mistake when using Black-Scholes?

Treating the inputs as fixed when they aren't. σ moves around. r changes. Dividends arrive. Time decays non-linearly. The Greeks change as S moves. A static Black-Scholes price is a snapshot, not a prediction.

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