BBobop Black-Scholes

index tools
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Parameters
$100.00
$100.00
30d
30%
5.0%
0.0%
Option Prices
--
Call
--
Put
Intrinsic: -- / --   Time: -- / --
GreeksCall / Put
Put-Call Parity
Implied Volatility Solver
P&L
Greeks Surface
Vol & Time
Binomial Tree
Reference
P&L Diagram
Greek vs. Stock Price
Price Sensitivity
Binomial Tree (Cox-Ross-Rubinstein) Steps:
Comparison
Black-Scholes Formulas
Call Price:
C = S*e^(-qT)*N(d1) - K*e^(-rT)*N(d2)
Put Price:
P = K*e^(-rT)*N(-d2) - S*e^(-qT)*N(-d1)
d1 and d2:
d1 = [ln(S/K) + (r - q + v^2/2)*T] / (v*sqrt(T)) d2 = d1 - v*sqrt(T)
Where:
S = stock price, K = strike, T = time to expiry (years)
v = implied volatility, r = risk-free rate, q = dividend yield
N() = cumulative normal distribution

The Greeks:

Delta = rate of change in option price per $1 move in stock
Call Delta = e^(-qT) * N(d1) Put Delta = e^(-qT) * (N(d1) - 1)
Gamma = rate of change in Delta per $1 move in stock
Gamma = e^(-qT) * n(d1) / (S * v * sqrt(T))
Theta = time decay per day
Call Theta = -[S*e^(-qT)*n(d1)*v/(2*sqrt(T))] - r*K*e^(-rT)*N(d2) + q*S*e^(-qT)*N(d1) Put Theta = -[S*e^(-qT)*n(d1)*v/(2*sqrt(T))] + r*K*e^(-rT)*N(-d2) - q*S*e^(-qT)*N(-d1)
Vega = sensitivity to 1% change in volatility
Vega = S * e^(-qT) * n(d1) * sqrt(T)
Rho = sensitivity to 1% change in interest rate
Call Rho = K * T * e^(-rT) * N(d2) Put Rho = -K * T * e^(-rT) * N(-d2)

Put-Call Parity:
C - P = S*e^(-qT) - K*e^(-rT)

Binomial Tree (CRR):
u = e^(v*sqrt(dt)) (up factor) d = 1/u (down factor) p = (e^((r-q)*dt) - d) / (u - d) (risk-neutral prob)

Assumptions:
- European-style options (no early exercise)
- Log-normal stock price distribution
- Constant volatility, rate, dividend yield
- No transaction costs or taxes
- Continuous trading possible