Option valuation : Method III
by J Victor on December 9th, 2012The Black Scholes Model
We learned from binomial model that price change in underlying stocks can be chopped to shorter periods like 6 months or even 3 months or 1 month and a chart can be drawn which would show the probable options values. Starting from the end of the tree, we worked backwards until we got the value of the option at the beginning. It may be practical to work back three or four steps using the binomial method. But, stock market is a place where the price changes keep occurring every minute, continually. Hence, a method will have to be found out where this chopping can extended ad infinitum.
Fisher black and Myron scholes came up with a solution to this, which later won them the Nobel prize. The method is called the Blackscholes model of option pricing.
The model has 5 inputs stock price, strike price, risk free return, tenure and volatility. Each of these inputs is very crucial in determining the option value. The formula involves a bit of mathematical calculations. So, for those who are not quite familiar with math and statistics may have a bit of trouble in understanding the whole thing.
The black scholes model is also made on the assumption that –
There are no dividends on the stock, any transaction costs, and taxes,
The short term risk free rate is known with certainty and will be constant during the lifetime of the option,
Short selling of stock is permitted,
Call option can be exercised only on expiry ie the option is European,
Stock prices move randomly and the prices of the underlying cannot be negative,
The volatility of the asset s known and is constant over the life of the option,
And that, trading takes place continuously.
The formula.
C (value of European option)
= Current market price x N (d1) – Present value of exercise price x N (d2)
We have to compute d1 and d2 separately, which is derived by the following formula.
Where,
σ = standard deviation of continuous compound rate
Ln = natural log
T = time remaining before expiry date (expressed in fraction of a year)
N = cumulative area of normal distribution evaluated at d1 and d2.
Although, the model is basically an improvement on the binomial model, it may not be easy to compute for everyone. Hence, a better option is use black scholes calculators that can be downloaded from many websites. The manual method of calculating black scholes using the above formula and websites from where you can download black scholes calculator will also be explained later in our article.
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