Introduction
The Black Scholes model is one of the significant concepts in the modern financial world. It is a very well known options pricing model. The model is basically used in calculating the fair price of a call or put option while ignoring any dividends which may be paid during the life the option.
Brief History
The model was developed by three economists namely – Fischer Black, Myron Scholes and Robert Merton, hence the name Black Scholes Model. The formula was a significant innovation in the option pricing domain. A Nobel Prize was awarded to its founders during the year 1997.
The world of Black Scholes Model (Assumptions)
The Black Scholes Model has some underlying assumptions which can be listed as below:
- No Dividends: One of the prime assumptions used in this model is that the underlying security does not pay any dividends during its life.
- Efficient Market: The market is assumed to be liquid has price continuity and all the players have equal access to all the information available.
- No commission or transaction cost: The model assumes that there is no transaction cost involved in buying and selling the asset.
- Constant Volatility: Here volatility is the measure of how much the stock is expected to move in the near future and is considered to be constant. This is one of the major constraints in this model. Volatility can be constant relatively in the short run but it can never be constant in the long run.
- As the options are considered European, they can only be exercised at their expiration. The model ignores the American style options which can be exercised at any point in time during the life of the option.
- The risk-free rate of investment and underlying volatility is assumed to be constant.
- The returns are distributed evenly throughout the period of the option.
It is quite obvious that none of these assumptions can be satisfied entirely. Transaction cost cannot be eliminated and they exist in all markets. Also, interest rates will vary with time and most of the stock prices do not follow the geometric movement precisely.
Yet, it does provide a very strong basis to determine a fair value of the option and provides a fundamental framework to reach an optimum pricing.
Black Scholes Model’s Input
The model consists of five major inputs which are:
- Spot Price: The market price of the asset as on the day of valuation is the spot price. This price is difficult to estimate however for the simplicity of use, the closing market price is used.
- Strike price: This is the price at which the option holder has the right to buy or sell the underlying security. It is not difficult to get this input as it is mentioned in the option contract.
- Time to maturity: The time (years) till the option expires and after which the option holder do not have any right to exercise it.
- Risk-free interest rates: This rate is normally the risk-free rate of the zero coupon government bond.
- Volatility: It is one of the most important inputs in the option pricing model. There are several ways to estimate volatility. The most common method to estimate volatility is to collect data from the previous term of the security.
Black Scholes Model’s Formula
The Black Scholes equation is calculated in two different parts viz. SN(d1) and N(d2)Ke^(-rt). The first part SN (d1) multiplies the price by a change in the call premium with relation to the change in its underlying price. This part explains and shows the benefit of purchasing at the underlying price.
The next part, N(D2) Ke^(-rt) shows what would be the current value for paying the exercise price upon expiration. The value of the option is then calculated by deducting the second part from the first part.
Here is a link to a very relevant video I found to understand Black Scholes equation, N(d2) to be specific. This should help you clearly understand basics of the formula.
Also find here an excel file you can download which will be your Black Scholes calculator.
Example of fair pricing
Let’s say you are participating a simple game. One guy will toss a coin. If it’s a tail, he will pay you 100 rupee, nothing is paid if it’s a head.
Result | Pay-out |
Probability |
Heads |
0 | ½ |
Tails |
100 |
½ |
What would be the fair price of this game? It would be the expected pay-off out of this game. Right?
Expected payoff = Sum of(probability*payout) = (1/2)*0 + (1/2)*100 = 50
So if you have an option to play this game priced at 50 rupees, there is most probably no profit or loss to you. If it is priced lower than this, you have an advantage since you have chance to win more. If it is priced more than this, you are paying a premium here. You have less chance of recovering the money you will pay to play this game.
Here did you notice that we assumed that the coin is a fair coin and the probability of both head and tails is equal to half each. This generally doesn’t happen in the real world scenario, as you might have encountered this thing multiple times by now.
The same is done by Black Scholes Model for option pricing.
Relief for Traders and Investors
It is tedious task to use Black Scholes equation for option pricing and sometimes can be intimidating for traders and investors. There is now a variety of Black-Scholes Model Calculators available online. Now-a-days, most of the trading platforms have robust options analysis tools.
Loopholes in Black-Scholes
- Random price movements: The model assumes that the prices of the security move randomly. It implies that the price of the underlying asset can go up or down with same probability in a given time frame. This assumption does not hold good in the realistic world. The prices are always determined by various factors which cannot be assigned a same probability in the way they affect the movement of the prices.
- Constant Volatility: As mentioned before, volatility can be relatively constant for a short term but can never be constant in the longer run. Measures of volatility are negatively correlated with asset price returns while being positively correlated with volumes or number of trades. Hence the volatility cannot be held constant in the longer run.
- No dividend payout: the model assumes any dividend payout during the life of the asset. This assumption does not apply in most of the cases since most of the public companies pay a dividend to their shareholders.
- No fees: The model assumes that there are no barriers to trade and no transaction cost. It is not realistic since stock brokers charge based on the spreads and other criteria.
- Constant interest rates: The model also assumes that the interest is constant and always known which is hardly the case in a real scenario. Even the risk-free government zero coupon yields do fluctuate with time with the change in volatility.
- Even distribution of returns: The model assumes that the returns from the asset are distributed evenly across the life of the asset which is not the case as compared to actual scenario.
Conclusion
Black-Scholes option pricing model has been a significant contribution in articulating the prices of options and corporate bonds. It is still used today for estimating the worth of options by most of the institutional portfolio managements. In spite of several loopholes in the model, there are multiple reasons for its wide use.
One of the biggest attributes of Black Scholes equation is that it provides a framework for option pricing. The Black Scholes Model has now become the standard for option pricing. Most of the new theories and approaches in option pricing are based on the principles provided by the model.
If you are trading in options, it is recommended to start investing in stocks/mutual funds as well to generate long term wealth. We recommend you to read these investing books to become a pro investor.
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