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# What does the Black-Scholes equation tell you?

What does the Black-Scholes equation tell you?

##### The Black Scholes model is used to determine a fair price for an options contract. Moreover, this mathematical equation can estimate how financial instruments like future contracts and stock shares will vary in price over time.

Limitations of the Black-Scholes Model?

##### Assumes constant values for the risk-free rate of return and volatility over the option duration. None of those will necessarily remain constant in the real world. Assumes continuous and costless trading—ignoring the impact of liquidity risk and brokerage charges.

Firstly, the pricing of options depends on the model. (I.e., the prices obtained by different models become different). So the arbitrage strategy cannot become implemented, without the framework of the model. Only if the market meets the framework of this model can the arbitrage strategy become implemented. Furthermore, the basic assumptions of Black – Scholes model are the following:

(1) During the life of the option, the underlying stock of the option of the buyer will  not pay dividends or make other distributions;

(2) No transaction cost for buying and selling stocks or options;

(3) In addition, the short-term risk-free interest rate becomes known and remains unchanged over the life  span;

(4) Any purchaser of securities can borrow any amount of money at a short-term  risk-free rate;

(5) Short selling becomes allowed, and the short seller will immediately receive the money for the price of the stock sold short on that day;

(6) The option is A European option and can only be exercised at the expiration  date;(The American options in the data are temporarily treated as European options)

(7) All securities transactions occur continuously, and stock prices walk randomly.  (8) Stock prices are subject to lognormal distribution.

##### The Black-Scholes formula to value a European option written on a non-dividend paying stock.

Where d1 and d2 are preliminary calculations. N(x) represents the cumulative  probability function for a standardized normal variable. It is the probability that a  variable with a standard normal distribution will be less than x. N(d1) and N(d2) assess the probability that the stock price will exceed the strike price. So that the call option ends up becoming exercised at maturity. Specifically, N(d1) measures the  probability that the present value of future stock price will exceed the current stock  price; N(d2) measures the risk adjusted probability that the call option will become exercised.

##### Arbitrage strategy

Delta, defined as the rate of change of the option price with respect to the price of the underlying asset. Furthermore, the diagram below shows the relationship between a call price  and the underlying stock price.

The Black-Scholes Option Pricing Model can be used to calculate Delta. In addition, the Delta of a long European call option on a non-dividend paying stock, is N(d1). Furthermore, the Delta of a long European put option on a non-dividend paying stock, is N(d1) – 1.

Lastly, for a call option, if C > BS it is expensive, ignoring transaction fee, you need to sell the actual call and buy a synthetic call, or buy N(d1) shares.

If C < BS, it is cheap, you need to buy the actual call and sell synthetic call or sell  N(d1) shares.

For a put option, if P > BS is expensive, you need to sell the actual put and sell  N(d1) -1 of the stock.

If P < BS it is cheap, you need to buy the actual put and buy N(d1) -1 of the stock.