Option Pricing Binomial Black Scholes Greeks Calculator

Purpose and model boundary

This model prices a synthetic call or put through Black-Scholes or a recombining Cox-Ross-Rubinstein binomial tree. Black-Scholes provides analytic value and Greeks for European exercise. The tree supports European or American exercise at 3, 5, 10, or 12 steps and reports tree-derived price sensitivities. It is an educational model, not an executable quote, suitability analysis, implied-volatility service, or valuation opinion.

Inputs and units

Valuation model is Black-Scholes or CRR binomial tree, and option type is call or put. Exercise style and binomial steps become active only in tree mode; Black-Scholes always uses European exercise. Spot and strike are positive user-currency amounts per underlying, time to expiry is in years, risk-free rate and dividend yield are continuously compounded annual fractions, and volatility is an annual fraction per square-root year.

The workbook assumes constant rate, dividend yield, and volatility. All contract and market parameters are synthetic user inputs.

Governing relationships

Let S be spot, K strike, T time, r risk-free rate, q continuous dividend yield, sigma volatility, N(x) the standard normal CDF, and phi(x) its density:

d1 = [ln(S/K) + (r - q + sigma^2/2)T] / (sigma × sqrt(T))

d2 = d1 - sigma × sqrt(T)

Black-Scholes values are:

Call = S exp(-qT)N(d1) - K exp(-rT)N(d2)

Put = K exp(-rT)N(-d2) - S exp(-qT)N(-d1)

Call delta is exp(-qT)N(d1); put delta is exp(-qT)[N(d1)-1]. Gamma is exp(-qT)phi(d1)/(S sigma sqrt(T)). Vega per one volatility percentage point is S exp(-qT)phi(d1)sqrt(T) × 0.01. Theta per year uses the standard analytic calendar-time expression, including the rate and dividend terms for the selected call or put. Rho per one rate percentage point is K T exp(-rT)N(d2) × 0.01 for a call and -K T exp(-rT)N(-d2) × 0.01 for a put.

For a tree with n steps:

dt = T/n

u = exp(sigma × sqrt(dt)), d = 1/u

p = [exp((r-q)dt) - d] / (u-d)

discount = exp(-r dt)

Terminal node value is max(S_node-K,0) for a call or max(K-S_node,0) for a put. European nodes roll backward as discount × [pV_up + (1-p)V_down]. American nodes take the maximum of continuation and immediate intrinsic value at each node.

Tree delta uses the two first-step values. Tree gamma is the difference between the two second-step deltas divided by half the span between the second-step extreme stock prices. Tree theta is the central second-step value minus the root value divided by 2dt. Binomial vega and rho are not reported.

Calculation sequence

1. Validate model/type selectors and all common price, time, rate, yield, and volatility inputs. Validate style and step count only in tree mode.
2. Calculate immediate intrinsic value.
3. In Black-Scholes mode, calculate d1, d2, normal density/CDF terms, value, and analytic Greeks.
4. In CRR mode, calculate dt, u, d, p, and the per-step discount factor. Reject a probability outside [0,1].
5. Populate the recombining stock tree to the selected depth, set terminal payoffs, and roll value backward with the selected exercise rule.
6. Select analytic or tree value, delta, gamma, and theta. Expose vega/rho only for Black-Scholes and tree factors/probability only for CRR.
7. Build the expiry-payoff chart at nine workbook-generated underlying prices from 0.5K through 1.5K.

Outputs and interpretation

Option value is the selected model result. Intrinsic value is immediate exercise payoff, and time value is option value less intrinsic value. Delta, gamma, and theta use the selected model's analytic or finite-tree construction. Vega and rho are scaled to a one-percentage-point change and appear only in Black-Scholes mode. CRR up/down factors and risk-neutral probability appear only in tree mode. The route-used output records the effective model and exercise convention.

The chart shows payoff at expiry, not the current modeled option value across spot scenarios.

Validation and status logic

The workbook evaluates status in this order:

There is no CHECK branch. In CRR mode the derived gate requires 0 <= p <= 1, u > 1, and d > 0. Input failure takes precedence over the derived probability check.

Assumptions and limitations

• The underlying follows the selected lognormal or recombining-tree framework with constant parameters.
• Black-Scholes is European; exercise style is intentionally inert in that route.
• Dividend yield is continuous. No discrete dividend is modeled.
• A maximum 12-step tree is reviewable but may be too coarse for production valuation.
• There is no implied-volatility solver, term structure, smile, local/stochastic volatility, jumps, barrier, Asian, Bermudan, transaction-cost, liquidity, credit, or funding model.
• Model value and Greeks do not establish suitability, fair value, or investment merit.

Restrictions and non-computing states

Spot and strike must be 0.01 through 1,000,000,000; time 0.01 through 10 years; risk-free rate and dividend yield -0.95 through 1; and volatility 0.0001 through 5. Tree mode accepts only 3, 5, 10, or 12 steps and listed exercise styles. Black-Scholes ignores tree-only fields. A CRR state with invalid risk-neutral probability or factors prevents a result rather than clipping the probability.

Errors and warnings

This calculator refuses unknown options, missing active fields, or numbers outside their allowed ranges before calculation. Workbook NOT VALID distinguishes input-domain failure from an invalid or nonfinite selected valuation route. The workbook has no cautionary CHECK state. A connection or calculation-service failure is not a worthless option or a zero Greek.

Option-contract, call/put, strike, expiration, exercise, premium, and risk context follows the SEC's Investor Bulletin: An Introduction to Options and Investor.gov Options. The workbook independently implements the disclosed Black-Scholes and CRR relationships and embeds no security price, volatility surface, or market dataset.