# Pricing Options

Options values or prices are derived from theoretical models. The Black Scholes option pricing model is the most popular model which provides the theoretical price of options. The other popular models are the binomial model developed by Cox, Ross and Rubinstein and the Addison Whaley model. These are slightly more sophisticated than the Black Scholes model but option values derived from these models are not significantly different. Black Scholes option calculators are available websites, and spreadsheets come with a built in Black Scholes options pricing formula. Hence, an investor need not memorize the formula; he needs to just key in the basic parameters of the formula.

Black Scholes Option Pricing Model: This model was originally developed for European style options on on-dividend paying stocks by Fischer Black and Myron Scoles.

The Black Scholes pricing model assumes that percentage change in the price of underlying follows a normal distribution. Black Scholes model uses lognormal rather than normal distribution to avoid the possibility of negative stock process. A variable with a normal distribution can take any positive or negative value, while a lognormal distributed variable can take only positive values. A normal distribution is symmetrical, a lognormal distribution is skewed with the mean, median, and mode all different. According to Black and Scholes, stock returns have a lognormal distribution meaning that the logarithm of the stock’s return will follow the normal (bell-shaped) distribution.

Although not the first, the pricing model was the model to reduce the number of computations necessary for deriving the values.

The factors affecting the determination of options pricing are the underlying stock price, the exercise price, time, expiry, volatility, and the risk free rate.

1. Underlying stock price: The underlying stock price does not remain constant. Any future changes in the underlying stock price may affect the value of an option hence the underlying stock is important for determining the value of an option.
2. Strike/exercise price: The strike remains fixed throughout the life of an option and never changes.
3. Time until expiration: An option’s expiration date is fixed for the life of the option. There is a direct relationship between an option’s prices and amount of time until the option’s expiration. Options that have a distant expiry date trade at a premium relative to those that are approaching expiry. The theoretical value of the option decreases as the time to expiry decreases.
4. Interest rates: The interest rate is the cost of carrying the option. The higher the interest ate, the higher the call price and the lower the put price. The lower the interest rate, the lower the call price and the higher the put price. The theoretical value of an option is affected by the correlation between a change in interest rates and the amount of time until expiry.
5. Volatility is the fluctuation in the price of the underlying. It is a measure of the speed and magnitude at which the underlying stock price changes. If the movement in the price of a security is quite high as compared to the index, the security is more volatile than the index. Volatility is the standard deviation of the daily returns on an underlying. Volatility is a very variable affecting an options price. The higher the volatility, the higher the option premium.

The other inputs or factors are readily determinable but determining volatility is a difficult task.

There are different ways to measure volatility

i) Historical volatility is a measure of actual changes in an underlying over a specific period in the past
ii) Forecast volatility is an estimate of expected changes in an underlying over a specific period in the future.
iii) Implied volatility is the market’s assessment of the expected volatility of the underlying. Implied volatility can be derived by entering the current market price of an option into the pricing model along with other factors. The number that is derived is the volatility that the market is using to price the option—the implied volatility. Hence, an option trader will first of all find out the historical volatility of a stock and factor into the historical volatility his anticipations of the future to arrive at a best guess of future volatility of the stock.

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