Derivative Pricing, Black Scholes Equation, Binomial Trees – Calculation reference

Black Scholes, Derivative Pricing and Binomial Trees

Black Scholes Formula for the value of an Option

Call option price (c)

Put option price (p)

Greeks (Delta, Gama, Vega, Theta, Rho)

Using Binomial Trees for pricing an option

Probability of an up move at a binomial tree node

Price of the option at a binomial tree node

The value of European Call option

price (c)

Price of European call option

price (p)

Price of european put option


N(x) is the cumulative probability distribution function (pdf) for a standardized normal distribution

S0 is the price of the underlyingasset at time zero

K is the strike or exercise price

r is the continuously compounded risk free rate

? is the volatilityof the asset price

T is the time to maturity of the option

q is the yield rate on the underlyingasset. Alternatively, if the asset provides cash income, instead of a yield, q will be set to zero in the formula and the present value of the cash income during the life of the option will be subtracted from S0.

d1 and d2 from Black Scholes


Calculating Delta from Black Scholes


Calculating Gamma from Black Scholes

Where N primeis the standard normal density function.

Theta (per year)

Calculating Theta

Vega(per %)


Rho(per %)


Binomial Trees - Probability

Where p is the probability that at the end of the time step, ?t, the price (S) will move up to Su. Alternatively 1-p is the probability that the price will move down to Sd.

r is the risk free rate

q is the dividend yield

? is the volatility of the prices

Binomial Trees - price at a node

Payoffs at nodes

Binomial Trees - Payoff at nodes

Price of Option

Price of the option = Expected present value of payoffs =

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