# Black Scholes, Derivative Pricing and Binomial Trees

## 1. Black Scholes Formula

### b. Put Option price (p)

Where

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

S0 is the price of the underlying asset at time zero

K is the strike or exercise price

r is the continuously compounded risk free rate

σ is the volatility of the asset price

T is the time to maturity of the option

q is the yield rate on the underlying asset. 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.

## 2. Greeks

### b. Gamma

Where
is the standard normal density function.

## 3. Binomial Tree

### a. Probability

Where p is the probability that at the end of the time step, Δt, 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

σ is the volatility of the prices

### d. Price of a European Call option (1 step tree)

Price of the option (value at time 0) = Expected present value of payoffs = C=e-rΔt{pCu + (1-p)Cd}