Question

How can i have i deep detailed answer about Black Scholes Formula ,with an example which detailed deep. I need more description about this formula about 10 pages (describe the way it works ) or a site that i can find some information.

Answer 1

Black-Scholes model, is a mathematical model of financial derivative markets from which the Black-Scholes formula can be derived. This formula estimates the prices of call and put options. The first widely adopted mathematical formula for pricing options. Some credit this model for the significant increase in options trading, and name it a significant influence in modern financial pricing. In their initial formulation of the model, Fischer Black and Myron came up with a partial differential equation known as the Black-Scholes equation, and later Robert Merton published a mathematical understanding of their model, using stochastic calculus that helped formulate what became known as the Black-Scholes-Merton formula.

The model determines the price of an option by calculating the return an investor gets less the amount that investor has to pay, using log-normal distribution probabilities to account for volatility in the underlying asset.

The Black Scholes Option Pricing Model:

The Model or Formula calculates an theoretical value of an option based on 6 variables. These variables are:

•Whether the option is a call or a put

•The current underlying stock price

•The time left until the option's expiration date

•The strike price of the option

•The risk-free interest rate

•The volatility of the stock

Assumptions of the Black-Scholes Model

1) No Dividends

The original Black-Scholes model did not take into account dividends. Since most companies do pay discrete dividends to shareholders this exclusion is unhelpful. Dividends can be easily incorporated into the existing Black-Scholes model by adjusting the underlying price input. You can do this in two ways:

1.Deduct the current value of all expected discrete dividends from the current stock price before entering into the model or

2.Deduct the estimated dividend yield from the risk-free interest rate during the calculations.

You will notice that my method of accounting for dividends uses the latter method.

2) European Options

A European option means the option cannot be exercised before the expiration date of the option contract. American style options allow for the option to be exercised at any time before the expiration date. This flexibility makes American options more valuable as they allow traders to exercise a call option on a stock in order to be eligible for a dividend payment. American options are generally priced using another pricing model called the Binomial Option Model.

3) Efficient Market

The Black-Scholes model assumes there is no directional bias present in the price of the security and that any information available to the market is already priced into the security.

4) Frictionless Markets

Friction refers to the presence of transaction costs such as brokerage and clearing fees. The Black-Scholes model was originally developed without consideration for brokerage and other transaction costs.

5) Constant Interest Rates

The Black-Scholes model assumes that interest rates are constant and known for the duration of the options life. In reality interest rates are subject to change at anytime.

6) Asset Returns are Lognormally Distributed

Incorporating volatility into option pricing relies on the distribution of the asset’s returns. Typically, the probability of an asset being higher or lower from one day to the next is unknown and therefore has a 50/50 probability. Distributions that follow an even price path are said to be normally distributed and will have a bell-curve shape symmetrical around the current price.

It is generally accepted, however, that stocks – and many other assets in fact – have an upward drift. This is partly due to the expectation that most equities will increase in value over the long term and also because a stock price has a price floor of zero. The upward bias in the returns of asset prices results in a distribution that is lognormal. A lognormally distributed curve is non-symmetrical and has a positive skew to the upside.

What you need to know about the Option Pricing Model

For the beginning call and put trader it is NOT necessary to memorize the formula, but it is important to understand a few implications that the formula or equation has for option pricing and, therefore, on your trading.

Here's what you need to know about the formula:

1.The formula shows the time left until expiration has a direct positive relationship to the value of a call or put option. In other words, the more time that is left before expiration, the higher the expected price will be. Options with 60 days left until expiration will have a higher price than options that only has 30 days left. This is because the more time that is left, the more of a chance the underlying stock price will move. But here is what you really need to understand--every minute that goes by, the cheaper the option price will become. Think of it this way. As time ticks by and as the days tick by, all things being equal, an option with 60 days left will lose about 1/60th of its value tomorrow when it only has 59 days left. That may not seem like a lot, but when we get to expiration week and as Monday changes to Tuesday, options lose 1/5 of their value. As Tuesday slips into Wednesday of expiration week, options lose 1/4 of their value, etc. so you must be careful! While nothing is certain in the stock market, there is ALWAYS one thing that is certain-time ticks by and options lose their value day by day. Please note:

2.The formula suggests the historical volatility of the stock also has a direct correlation to the option's price. By volatility we mean the daily change in a stock's price from one day to the next. The more a stock price fluctuates within a day and from day to day, then the more volatile the stock. The more volatile the stock price, the higher the Model will calculate the value of its options. Think of stocks that are in industries like utilities that pay a high dividend and have been long-term, consistent performers. Their prices go up steadily as the market moves, and they move small percentage points by week. But if you compare those utility stocks' price movements with bio-tech stocks or technology stocks, whose prices swing up and down a few dollars per day, you will know what volatility is. Obviously a stock whose price swings up and down $5 a week has a greater chance of going up $5 then a stocks whose price swings up and down $1 per week. If you are buying options, both put and calls, you Love volatility--you Want volatility. This volatility can be calculated as the variance of the the prices over the last 60 days, or 90 days, or 180 days. This becomes one of the weaknesses of the model since past results don't always predict future performance. Stocks are often volatile immediately after an earnings release, or after a major press release.

3.Watch out for dividends! If a stock typically pays a $1 dividend, then the day it goes ex-dividend the stock price should drop $1. If you have calls on a stock that you KNOW will drop $1 then you are starting off in the hole $1. Nothing is worse than identifying a stock you are confident will go up, looking at the call prices and thinking "boy those are cheap", buying a few contracts, and then finding the stock go ex-dividend and then you realize why the options were so cheap.

4.Beware of Earnings Releases and Rumors--You can calculate an option price all you want, but nothing can drive a stock price (and its call option prices as well) up more than a positive rumor or a strong earnings release. The Option Pricing Model simply cannot overcome the supply and demand curve of option traders hungry for owing a call option on the day of a strong earnings release or a positive press release.

The Option Pricing Model was developed by Fischer Black and Myron Scholes in 1973.

Why it Matters:

Empirical studies show that the Black-Scholes model is very predictive, meaning that it generates option prices that are very close to the actual price at which the options trade. However, various studies show that the model tends to overvalue deep out-of-the-money calls and undervalue deep in-the-money calls. It also tends to misprice options that involve high-dividend stocks. Several of the model's assumptions also make it less than 100% accurate. First, the model assumes that the risk-free rate and the stock's volatility are constant. Second, it assumes that stock prices are continuous and that large changes (such as those seen after a merger announcement) don't occur. Third, the model assumes a stock pays no dividends until after expiration. Fourth, analysts can only estimate a stock's volatility instead of directly observing it, as they can for the other inputs. Analysts have developed variations of the Black-Scholes model to account for these limitations.

Ultimately, however, the Black-Scholes model represents a major contribution to the efficiency of the options and stock markets, and it is still one of the most widely used financial tools on Wall Street. Besides providing a dependable way to price options, it helps investors understand how sensitive an option's price is to stock price movements. This in turn helps investors maximize the efficiency of their portfolios by giving them a way to calculate hedge ratios and more effectively implement portfolio insurance.

Despite the tremendous efficiencies created by the Black-Scholes model, many financial theorists claim the model's introduction indirectly increased the volatility of the stock and options markets by encouraging more trading (as investors sought to constantly fine-tune their hedge positions). Others claim the model actually steadies the markets because of its ability to measure equilibrium pricing relationships. When these relationships are violated, arbitrageurs are usually the first to discover and exploit mispriced options.