Question

black scholles model of derivatives

Answer 1

Black-Scholes Model:

The Black-Scholes model is used to calculate a theoretical price of an Option. The Black-Scholes price is nothing more than the amount an option writer would require as compensation for writing a call and completely hedging the risk of buying stock. The important point is that the hedger's view about future stock prices is irrelevant. Thus, while any two investors may strongly disagree on the rate of return they expect on a stock they will, given agreement to the assumptions of volatility and the risk free rate, always agree on the fair value of the option on that underlying asset. This key concept underlying the valuation of all derivatives -- that fact that the price of an option is independent of the risk preferences of investors -- is called risk-neutral valuation. It means that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.

The model is based on a normal distribution of underlying asset returns. The following assumptions accompany the model:

1. European Options are considered,
2. No transaction costs,
3. Short term interest rates are known and are constant,
4. Stocks do not pay dividend,
5. Stock price movement is similar to a random walk,
6. Stock returns are normally distributed over a period of time, and
7. The variance of the return is constant over the life of an Option.

The original formula for calculating the theoretical option price (OP) is as follows:

OP = S*N(d1) – X c^-rt N(d2)

Where:

d1 = [ln(S/X) + (r + v²/2) t] / v(t^-1/2)

d2 = d1 - v(t^-1/2)

The variables are:

S        = current stock price

X        = strike price of the option

t        = time remaining until expiration, expressed as a percent of a year r = current continuously compounded risk-free interest rate

v       = annual volatility of stock price (the standard deviation of the short-term returns over one year).

ln       = natural logarithm

N(x)   = standard normal cumulative distribution function e = the exponential function