# Options Pricing Tree

> A free, in-browser option pricer. It values American and European calls and puts on a
> binomial or trinomial tree and compares the result with Black-Scholes-Merton. No sign-in,
> no server: every number is computed on the visitor's device.

URL: https://options-pricing-tree.skillsafe.ai/

## Inputs
- Spot price S, strike K, time to expiry T in years (1 day to 50 years)
- Volatility, risk-free rate (may be negative) and continuous dividend yield, in percent
- Call or put; American or European
- Tree: CRR binomial (u = 1/d), Jarrow-Rudd binomial (drift in the moves, probability solved
  so the tree is arbitrage-free) or trinomial (up, flat, down; probabilities from two binomial
  half-steps). 1 to 2,000 steps.

## Outputs
- Tree price, Black-Scholes price of the European option, the difference, and the
  early-exercise premium (American minus European on the same tree)
- Tree diagram of the first steps: stock price, option value, hold value, exercise value,
  decision and risk-neutral probability of reaching each node; early-exercise nodes marked
- Greeks: Delta, Gamma, Theta (per day), Vega (per volatility point), Rho (per rate point),
  on the tree and in closed form
- Convergence chart of all three trees as steps grow, against Black-Scholes (European) or a
  2,000-step reference (American)
- Option value against spot with the payoff at expiry; early-exercise boundary over time
- Implied volatility from a market price, on the tree and with Black-Scholes
- Exports: Markdown summary, tree CSV, convergence CSV, JSON; a shareable link that encodes
  the inputs in the URL fragment

## Checks
The engine matches an independent reference (closed-form binomial and multinomial sums,
separately written American trees, erfc-based Black-Scholes) to about 1e-12, and gives 4.4885
for the textbook 5-step American put (S = K = 50, r = 10%, volatility 40%, 5 months).

Prices are model outputs for learning and checking, not trading advice.