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

Black Scholes, Derivative Pricing and Binomial Trees

1. Black Scholes Formula

a. Call Option price (c)

Price of European call option

b. Put Option price (p)

Price of european put option

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.

d1 and d2 from Black Scholes

2. Greeks

a. Delta

Calculating Delta from Black Scholes

b. Gamma

Calculating Gamma from Black Scholes

Where
N primeis the standard normal density function.

c. Theta (per year)

Calculating Theta

d. Vega (per %)

Vega

e. Rho (per %)

Rho

3. Binomial Tree

a. Probability

Binomial Trees - 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

b. Price at node

Binomial Trees - price at a nodec. Payoffs at nodes (1 step tree) – European Call Option

Binomial Trees - Payoff at nodes

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}