1. Option portfolio and Black Scholes
2. European options and Black-Scholes
Now let's consider investments in European options. If you buy a European call option on a stock with strike price K and maturity time T in the future, this option gives you the right but not the obligation to buy that stock at the price K at time T. Obviously, if you think the stock price will rise above K by time T this could be a good buy.
There are also European put options. These give the right but not the obligation to sell a stock at a fixed strike and maturity. You can make money from a put if you think stock prices are going to fall.
The value of such an option at some time t < T will depend on a number of factors including the current stock price S, the time until the option matures big T minus small t, the rate of interest r, and also the annualized volatility sigma of the stock price. The annualized volatility is the standard deviation of an annual log-return.
A rational method of pricing options was proposed by Black and Scholes in the 1970s based on assuming that stock prices follow the geometric Brownian motion model.
3. Pricing a first call option
Let's price a simple European call option with Black-Scholes.
Assume the strike of the option K is 50 and the maturity T is two years. Assume the current time small t is 0, and the current stock price S is 40. Assume moreover that the interest rate r is half of one percent, and the volatility sigma is 25%. These are all quite typical values. Using the Black_Scholes function, the price of the option is two point six two, as shown.
When the volatility of the stock is increased by 20%, the price of the option is higher at three point six eight.
The cost of call options increases with volatility because it enhances the chances of the stock making large gains in value during the time to maturity.
The Black_Scholes() function can also price a put if the last argument is changed to the word "put".
There is some other terminology that is used in the exercise. An option is said to be in-the-money if its current price is in the range where the option would be exercised. That is the case for the option that was priced previously. Otherwise, it is said to be out-of-the-money.
Most of the quantities required to price an option are directly observable. The maturity and strike which describe the option are known. Values for the stock price and the interest rate can be easily obtained.
4. Implied volatility X needs change
The only parameter that poses more of a problem is volatility. While there are many methods of using historical stock prices to estimate volatility statistically, what is more common in financial markets is to use a concept called implied volatility.
If there is a market for options on a particular stock, then the prices that are quoted can be used to infer the implied values that would be used to price the options with the Black-Scholes formula.
To value a portfolio containing an option, you need a value for implied volatility. Moreover, to analyze possible losses on the portfolio over some time period, you need to consider possible fluctuations of the implied volatility parameter over that time period.
In other words, you need to consider implied volatility as a risk factor.
5. The VIX index
In the qrmdata library, one implied volatility dataset is included, and that is the VIX index provided by the Chicago Board of Exchange. The VIX index gives a measure of the implied volatility for options that are traded on the S&P 500 index.
Let's have a look at the time series of daily values.
The y-axis should be interpreted as a percentage value.
You can see that the implied volatility is itself quite a volatile measurement.
In the second R exercise, you will have a look at the returns of the VIX index and see whether they exhibit the typical stylized facts.
6. Let's practice!
OK. It's time to practice.