# | The Black-Scholes Option Essay

The science of trade has evolved over the past few decades to include financial markets. A financial market is a place where business entities and individuals can trade financial commodities, securities and other fungible commodities at a reasonable transaction costs. The items are also sold at prices that reflect the supply and demand of these commodities. Examples of securities sold in the financial markets include stocks, bonds and precious commodities like metals._x000D_

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Like many other markets the prices of commodities are determined by various factors. Among these factors, demand and supply play a major role in fixing the price of a financial commodity. Unlike the other markets which are composed of manufacturers and consumers, financial markets are composed of investors and entities who are disposing their investments._x000D_

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A call option is a financial contract between two persons; buyer and seller, for the purchase of stock or whatever basis of an underlying asset. This agreement gives the investor (buyer) the right but not the obligation to purchase a bond, stock, commodity or any other instrument within a specific period of time (Errera 2002). The investor of a call option hopes that the price of the underlying financial instrument will rise in the near future. This becomes profitable if the price of the financial instrument rises above the strike price. The writer or the seller hopes that the price of the financial instrument will not rise and sells it benefiting from the premium paid by the seller._x000D_

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Finance is the most dynamic areas of corporate world. Since financial are becoming more complex various models have been put forward to determine the call option of a financial instrument. In 1970, Robert Merton, Fisher Black and Myron Scholes came up with financial instrument pricing model which came to be known as Black-Scholes model._x000D_

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Black-Scholes Option Pricing Model_x000D_

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The Black-scholes model is used to determine the price of a call option of financial instruments (Fischer & Black 1973). This model ignores the dividend paid on the financial instrument during the period._x000D_

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The interest rate is constant over the lifetime of the option. This means that the interest rates associated with the return of the capital is constant throughout the period of existence of the call option._x000D_

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The market is of high efficiency such that the market movements cannot be predicted. In this case the prices of the underlying assets are determined by purely the forces of demand and supply. In this particular case the movements of these prices cannot be determined by a single factor. Consequently the market becomes very efficient (Errera 2002)._x000D_

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The stock returns follow a log normal distribution. This type of distribution is assumed because the stock prices cannot fall by more than 100% while the same stock price can rise by more than 100%. In this case the graph is skewed towards the right (Fischer & Black 1973)._x000D_

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The model ignores any divided that are paid out during the option life. If there are any returns associated with the underlying asset they are ignored in the model. This becomes one of the main disadvantages of the model._x000D_

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Lastly the model assumes that the call option is exercised only on the expiry date. This limits the application of the model to the European call option because the American option allows the exercise of the call option at any time._x000D_

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Determinants of the option price_x000D_

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The price of an option in Europe is determined by many factors. This paper is going to look at how the decrease in each of these factors will affect the prices of a call option._x000D_

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Market price of the underlying asset/stock_x000D_

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In financial markets, different assets have varying prices. Moreover, even a single asset can have different prices at different time depending on market forces of demand and supply (Errera 2002). For a given activity price of an option, the higher the market price of an asset the higher the value of the call option. On the other hand, the lower the market prices of an asset the lower the value of a call option (Fischer & Black 1973). For example a call option involving an asset costing $12,000 will be priced higher than the call option of the same asset costing $9,000 in some future time._x000D_

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It is also good to note that, an option’s value is composed of two factors i.e. intrinsic value and the time value. Intrinsic value of an option is the real money portion of the option’s premium. In this case an investor analyses the fundamental value of an asset both qualitatively and quantitatively. The qualitative determinant of intrinsic value of an asset will take into consideration to factors like governance, business model, and target market factors. While the quantitative will consider financial statement analysis, ratios etc aspects of the business to determine whether a business is currently in favor or out of favor with the market. The time value is composed of the expected value of an asset or an option at a given time in the future._x000D_

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If all the other factors are held constant, a decrease in the price of any underlying asset is going to lead to reduction in the call option price. The decrease in price may be caused by many factors. These include fall in the value of returns associated with the asset, fall in demand of the asset, expected fall in returns of the asset, economic crises etc. The low price of the underlying asset is associated with less profit and hence the buyer will not be willing to pay a high price. Therefore, as the price of the underlying asset decreases the price of the call option decreases. These two factors are directly proportional (Errera 2002) ._x000D_

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In cases where a stock pays divided, the options are explicitly not payout protected. This implies that when the stock goes ex-divided, there will be a decrease in the stock prices by the divided amount declared. This decrease in the price will result to a decrease in the price of a call option._x000D_

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The exercise price_x000D_

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The exercise price also known as the strike price is the price at which a specific derivative contract can be exercised before the expiry date. In call options, the strike price involves case where a security can be bought up to the date of expiration. The difference between option’s strike price and the underlying security’s current market price represents the profit gained upon the sale of an option. At this instance the maximum loss that can be incurred is the amount of the premium paid. Therefore, the strike price is the key determinant of the premium. Strike price are usually determined when the contract is made. According to Kumar (2007), the strike prices are in increments of $5 and $2.5. The maximum profit associated with a security in unlimited since the profit will be directly proportional to the rise in the price of the security. However the loss is only limited to the premium paid upon the signing of the contract._x000D_

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The strike price of a call option is fixed throughout the existence of the life option (Errera 2002). When all other factors are kept constant, a decrease in the exercise price will lead to an increase in the price of a call option. For example; if a person chooses to go for a long call option, the person is buying the right to purchase the underlying instrument at an agreed exercise price before the expiration of the contract. The premium of a long call indicates a debit balance in the person’s trading account. This premium represents the maximum risk which the call option can experience. Profit can only be made if the price of the underlying asset goes above the exercise price of the asset. If this so happens, a person can either exercise the call or offset the call option by selling the call to a third party with the same expiration date and the exercise price. This raises the call option price of a financial instrument in the market since the strike price is relatively lower than the price of the asset. Therefore low strike prices are associated with high profit expectation in future. As the exercise price reduces the call option price increases since this offer comes with high returns of the risk involved_x000D_

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Time to expiration of the option_x000D_

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A call option, like any other contract, has an expiration date. Expiration time is the period between when the contract is signed and the time when the call option is no longer valid. Before the expiration date, a buyer has a number of options he or she can make. Foremost, he or she can execute the call and buy the underlying asset at a price either making a profit or a loss of the premium (Fischer & Black 1973). Another option is to sell the call option to a third party at the current market price of the financial instrument if he or she is in a position to make the profit. Again, it is always possible to wait for the prices of the security to go up before executing the contract hence increasing the probability of making profit._x000D_

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A short time contract will increase the probability of the underlying asset of having market price equal to the exercise price. On the other hand a long time contract will decrease the probability of the underlying asset having a market price equal to the exercise price. In this case a long time call option is preferred than a short time call option._x000D_

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The period after the expiration date, a call option has no value. If the buyer fails to execute the contract then he or she loses the premium. If all the other factors remain constant, the shorter the period of expiration of an option the lower the price of a call option. The reason for this is that, as the period to expiration decreases, there is less time left for the underlying asset’s price to rise (Newman 2007). Therefore, therefore the probability of increase in price of the underlying asset decreases. When these circumstances prevail and the time left until expiration decreases, the option price approaches its intrinsic value. Through this reasoning longer expiration periods are more favorable to the buyer than shorter expiration period although they do not guarantee a better outcome (profit)._x000D_

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Since the short expiration time is not favorable for the buyer, he or she only be willing to buy a call option at a low price. It is also favorable for the seller since he or she can sell the call option after the expiration of the previous one generating premium. In such a scenario any decrease in the expiration time of a call option will attract a low option price._x000D_

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The expected volatility of the underlying asset_x000D_

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Volatility is the measure of how swift the prices of an underlying asset change. It remains a key to understanding the reason why the call option fluctuates and why they act it that way. There are two types of volatility; implied volatility and historical volatility._x000D_

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Implied volatility is the estimated volatility of a financial instrument in real time (Kevin 2010). The value for implied volatility is obtained from market expectation formulas. This offers volatility prediction of the underlying asset over the period of the call option. Another way of arriving at implied volatility is by use of the market price of a call option based on an option pricing model. In this case it is the volatility that yields a theoretical value for an option which is equal to the current market price given a particular pricing model. This volatility goes up when the market is id downtrend and falls when the market is in uptrend (Kevin 2010)._x000D_

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The historical volatility is also referred to as statistical volatility. This is a measure of how much the prices have been changing over time through the use of past data. Since this volatility is dynamic, it is calculated daily and summarizes in form an index or a percentage._x000D_

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Volatility of the underlying asset over the period of existence of a call option remains the most subjective and difficult in quantification (Newman 2007). In the present, software’s have been developed to quantify the volatility of the price of an underlying asset in financial markets. This volatility has a significant effect on the time value of an option’s premium. It measures the risk or simply the uncertainty of the price of an option’s underlying asset. Securities that are more volatile have uncertain profit expectation in future. Example of these securities is the money. The price of a dollar can change frequently within a short period of time even within a day. Hence the market price of this underlying asset may not be predictable in near future hence the profit associated with it is very uncertain._x000D_

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Volatility is measured by standard variance in price of the underlying asset (Kumar, 2007). If all the other factors are held constant, a decrease in the volatility in price of the underlying asset will result to a decrease in the price of the call option. This is because the maximum that a buyer can lose is the premium. The low volatility reduces the chances of the market price of the underlying asset (Kumar, 2007). This leads to reduction in probability of the buyer of a call option to make profit from the difference between the exercise price and the market price of the underlying asset. Therefore, given a list of call options of different underlying asset, a buyer who wants to make profit in the near future or time would choose a call option whose volatility is high so as to increase the chances of making profits. These lenders the call options of low volatile options unpopular with buyers decreasing their selling price._x000D_

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Current Continuously Compounded Risk-Free Interest Rate_x000D_

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Risk free rate is the theoretical rate of return on any investment without exposure to risk of a financial loss. It represents the interest that an investor would expect the rate of investment to be if the investment is absolutely risk-free over a given period of time._x000D_

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As the interest rates rises, the asset prices tend to increase due to the expected growth in prices of the assets. Nevertheless, the current value of any future cash flow to be received by an individual owning the call option decreases. If the interest rates are rising continuously the expected market price of the underlying asset will be higher (Newman 2007). Consequently, the profit associated with a call option upon disposing to a third party or exercising the contract is high._x000D_

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This leads us to the fact that if all other factors are held constant, the current compounded risk free interest rates decreases over the time; they will have the following effects._x000D_

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The probability of the market price of the underlying asset to increase in the contract time period will be small. This means that the buyer of a particular call option will not be able to make profit (if any) from either selling the call option to a third party or exercising the contract. This will discourage buyers from purchasing those particular call options with low risk free rate. Consequently, investors will be interested with those call options with high risk free rate. At the end of the day, the prices of call options with low risk free rate will eventually decrease due to low demand._x000D_

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For example, if we assume that a stock is selling at $100 per share and a person wants to purchase a call option that expires in one year and the risk free interest rate is 5%. At maturity the stock will be priced at 105$ per share. But if the risks free interest rate is at 3%. At maturity the stock will be priced at 103$ per share. Therefore a buyer will opt to purchase a call option with higher rate of interest. And if the interest falls the price of the call option will also fall._x000D_

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Conclusion_x000D_

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Except the volatility of the market price of the underlying asset, all other parameters in Black-scholes option pricing model are observable. These are the strike price, risk free interest rates, time remaining till the expiration and the current price of the underlying asset (Newman 2007) . From these facts it is possible to conclude that there is a direct relationship between the volatility and the option price. Through observation of the call option price and pegging other parameters in the Black-Scholes option pricing model it is possible to arrive at the volatility rate in the market for a particular asset in the market._x000D_

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The Black-Scholes pricing model follows a lognormal distribution. This distribution is skewed to the right, meaning that it has a longer tail in the right compared to the normal distribution which is usually bell- shaped (Newman 2007). The reason for this is that while the stock prices in Europe can drop 100%, they are capable of rising by more than 100%._x000D_

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The main advantage associated with the black-Schole model is that it can be used to calculate the very large numbers of call options prices in a very short time._x000D_

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Like other models Black-Schole model has its own limitations. Foremost, it cannot be used to calculate price options with American-style exercises. This is because the American option price at expiration only.

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