Black-Scholes Model

Important : In Black-Scholes model the main assumption is the fact that Brownian Motion process, i.e., log normal distribution, where  is the price of stocks.

The main parameters for the BS model are:

1. S_o = Stock price and this is known
2.  = strike price and this is known
3.  =  interest rate and this is known
4.   = time period and this is known
5. = volatility and this is unknown

One should remember that the main component in the model is , but the bottle neck is the fact that  which is the volatility is stochastic.

Two key assumptions is Black Scholes Model

1. Log normal prices
2. Volatility constant i.e., it is independent of time

Now recall that  and consider that
, if  which implies that

, assuming

Now generally if  is there, then we have the following

NOTE

• If there is no drift then the expected time to reach  is .
• If there is drift then the expected time to reach  is finite