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The Black-Scholes model is used to calculate a theoretical call price, ignoring dividends paid during the life of the option, using the five key determinants of an option’s price viz., stock price, strike price, volatility, time to expiration, and short-term risk free interest rate.

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

Where:

The variables are:

OP       = theoretical option price
S          = stock price

X         = strike price
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

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

Assumptions underlying the above formula:

• It is possible to short sell the underlying stock.
• There are no arbitrage opportunities.
• Trading in the stock is continuous.
• There are no transaction costs or taxes.
• All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share).
• It is possible to borrow and lend cash at a constant risk-free interest rate.

Except the volatility factor all the other parameters used in this model viz., strike price, time remaining till expiration, the risk-free interest rate, and the current underlying price are objective and are observable. Hence we can conclude that there is a direct relationship between the option price and the volatility. By observing the option price and pegging the other parameters in this formula it is possible to arrive at the volatility that is implied by the market. By applying such derived volatility implied by the market over the other strike prices and expiry we can test the validity of the Black-Scholes option pricing model. It would be observed that the implied volatility tends to be higher for lower strike prices, and lower for higher strike prices.

It is interesting to note that currencies tend to have more symmetrical curves, with implied volatility lowest at-the-money, and higher volatilities for deep in-the-money and out-of-the-money strike prices, while commodities have higher implied volatility for higher strike prices and lower implied volatility for lower strike prices, exactly the opposite of equities.

Lognormal distribution

The Black-Scholes pricing model assumes a lognormal distribution. A lognormal distribution is skewed to the right, which means it has a longer tail towards it right as compared with a normal distribution that is bell-shaped. The lognormal distribution allows for a stock price distribution of between zero and infinity and has an upward bias. This is because while a stock price can only drop 100%, it can rise by more than 100%.

• It enables one to calculate a very large number of option prices in a very short time.

Limitations of Black-Scholes model:

• Black-Scholes model cannot be used to accurately price options with an American-style exercise as it calculates the option price at expiration only. Early exercise as in the case of American option cannot be priced correctly using this model which is a major limitation of this model.
• All exchange traded equity options (ETO) have American-style exercise as against the European options which can only be exercised at expiration. That means this model cannot be used for pricing most ERO options. The exception to this is an American call on a non-dividend paying asset as the call is always worth the same as its European equivalent since there is never any advantage in exercising early. 