Question

How can i find more information about Black Schole model ( Nyron Scholes) . I Want more details about this model . More details about the cases and the describe.

Answer 1

Black Schole Model (also called as Black-Scholes-Merton Model) is a pricing model used to determine the fair value for a put and call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility.

Assumptions and Limitations

The model is based on following assumptions-

1. Interest rate remains constant
2. Options can only be exercised upon expiration
3. Stock pays no dividends
4. Market direction cannot be predicted
5. volatility is constant
6. No commissions are charged in the transaction
7. Type of option - put or call

The model has few limitations too-

1. The models considers that volutality and risk free rate are constant
2. It considers prices are continuous and large changes don't take place
3. It assumes that stock don't pay dividends until expiration.
4. It overvalues out-of-money calls and undervalues deep in-the-money calls

Black-Scholes Formula

C= SN(d1) - N(d2)Ke^-rt

• C= Call Premium
• S= Current Stock price
• t = Time until expiration
• K= Option Striking price
• r= risk free rate
• N = cumulative standard normal distribution
• e= exponential term

d1= [ ln ( S / K ) + ( r + s² / 2 ) t ] / s . $\sqrt{t}$

d2= d1 - s . $\sqrt{t}$

• s= Standard Deviation
• ln= Natural log

For Example

6-month call option ; exercise price of $50 on a stock ; trading at $52 costs $4.5 ; annual risk-free rate is 5% ; the annual standard deviation of the stock returns is 12% . Determine whether you should buy the option?

Need to calculate fair price of option ; compare it with current price of the option and purchase if fairly priced

Need to find d1 and d2

d1= [ ln 60/50 + (5% + 12% ² / 2 ) * 0.5 ] / under root of 12% ² * 0.5 = 0.7993

d2 = [ ln 60/50 + (5% - 12% ² / 2 ) * 0.5 ] / under root of 12% ² * 0.5 = 0.7144

Next, we can find the standardized normal distribution probability using "The Standard Normal Distribution Table". N(d₁) and N(d₂) equal 0.7879 and 0.7625 respectively.

Once we have N(d₁) and N(d₂), we can put the relevant numbers in the Black-Scholes formula:

C=$52×0.7879 −$50×e−0.05×0.5×0.7625=$3.788

The option value as per the model is lower than the premium on the call options currently traded. It might be because the option is overvalued or because our estimate of the volatility is lower.

If we have current value of call premium C, stock price S, exercise price X, time to maturity t and risk-free rate r, we can work back to find out the implied volatility. In the above example, the market estimate of annual standard deviation of return based on call premium of $4.5 is 18.06%.