# Black Scholes option pricing Model, Index Options

The Black Scholes Model is one of the most important concepts in modern financial theory. The Black Scholes Model is considered the standard model for valuing options. A model of price variation over time of financial instruments such as stocks that can, among other things, be used to determine the price of a European call option. The model assumes that the price of heavily traded assets follow a geometric Brownian motion with constant drift and volatility. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price and the time to the option’s expiry. Fortunately one does not have to know calculus to use the Black Scholes model.

### Black-Scholes Model Assumptions

There are several assumptions underlying the Black-Scholes model of calculating options pricing..

The exact 6 assumptions of the Black-Scholes Model are :

1. Stock pays no dividends.
2. Option can only be exercised upon expiration.
3. Market direction cannot be predicted, hence “Random Walk.”
4. No commissions are charged in the transaction.
5. Interest rates remain constant.
6. Stock returns are normally distributed, thus volatility is constant over time.

These assumptions are combined with the principle that options pricing should provide no immediate gain to either seller or buyer.

As you can see, many assumptions of the Black-Scholes Model are invalid, resulting in theoretical values which are not always accurate. Therefore, theoretical values derived from the Black-Scholes Model are only good as a guide for relative comparison and is not an exact indication to the over- or underpriced nature of a stock option.

### Limitations of the Black Scholes Model

The Black–Scholes model disagrees with reality in a number of ways, some significant. It is widely used as a useful approximation, but proper use requires understanding its limitations – blindly following the model exposes the user to unexpected risk.

Among the most significant limitations are:

1. The Black-Scholes Model assumes that the risk-free rate and the stock’s volatility are constant.
2. The Black-Scholes Model assumes that stock prices are continuous and that large changes (such as those seen after a merger announcement) don’t occur.
3. The Black-Scholes Model assumes a stock pays no dividends until after expiration.
4. Analysts can only estimate a stock’s volatility instead of directly observing it, as they can for the other inputs.
5. The Black-Scholes Model tends to overvalue deep out-of-the-money calls and undervalue deep in-the-money calls.
6. The Black-Scholes Model tends to misprice options that involve high-dividend stocks.

To deal with these limitations, a Black-Scholes variant known as ARCH, Autoregressive Conditional Heteroskedasticity, was developed. This variant replaces constant volatility with stochastic (random) volatility. A number of different models have been developed all incorporating ever more complex models of volatility. However, despite these known limitations, the classic Black-Scholes model is still the most popular with options traders today due to its simplicity. ### Variants of the Black Scholes Model

There are a number of variants of the original Black-Scholes model. As the Black-Scholes Model does not take into consideration dividend payments as well as the possibilities of early exercising, it frequently under-values Amercian style options.

As the Black-Scholes model was initially invented for the purpose of pricing European style options a new options pricing model called the Cox-Rubinstein binomial model is also used. It is commonly known as the Binomial Option Pricing Model or more simply, the Binomial Model, which was invented in 1979. This options pricing model was more appropriate for American Style options as it allows for the possibility of early exercise.

The Binomial Option Pricing Model (BOPM), invented by Cox-Rubinstein, was originally invented as a tool to explain the Black-Scholes Model to Cox’s students. However, it soon became apparent that the binomial model is a more accurate pricing model for American Style Options.

### Index Options

An index option is a financial derivative that gives the holder the right, but not the obligation, to buy or sell the value of an underlying index, such as the Standard and Poor’s (S&P) 500, at the stated exercise price on or before the expiration date of the option. No actual stocks are bought or sold; index options are always cash-settled, and are typically European-style options.

Index call and put options are simple and popular tools used by investors, traders and speculators to profit on the general direction of an underlying index while putting very little capital at risk. The profit potential for long index call options is unlimited, while the risk is limited to the premium amount paid for the option, regardless of the index level at expiration. For long index put options, the risk is also limited to the premium paid, and the potential profit is capped at the index level, less the premium paid, as the index can never go below zero.

Beyond potentially profiting from general index level movements, index options can be used to diversify a portfolio when an investor is unwilling to invest directly in the index’s underlying stocks. Index options can also be used in multiple ways to hedge specific risks in a portfolio. American-style index options can be exercised at any time before the expiration date, while European-style index options can only be exercised on the expiration date.

### Index Option Examples

Imagine a hypothetical index called Index X, which has a level of 500. Assume an investor decides to purchase a call option on Index X with a strike price of 505. With index options, the contract has a multiplier that determines the overall price. Usually the multiplier is 100. If, for example, this 505 call option is priced at \$11, the entire contract costs \$1,100, or \$11 x 100. It is important to note the underlying asset in this contract is not any individual stock or set of stocks but rather the cash level of the index adjusted by the multiplier. In this example, it is \$50,000, or 500 x \$100. Instead of investing \$50,000 in the stocks of the index, an investor can buy the option at \$1,100 and utilize the remaining \$48,900 elsewhere.

The risk associated with this trade is limited to \$1,100. The break-even point of an index call option trade is the strike price plus the premium paid. In this example, that is 516, or 505 plus 11. At any level above 516, this particular trade becomes profitable. If the index level was 530 at expiration, the owner of this call option would exercise it and receive \$2,500 in cash from the other side of the trade, or (530 – 505) x \$100. Less the initial premium paid, this trade results in a profit of \$1,400.

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