About Options!

Table of contents

  1. A little bit about options....
  2. Who buys options?
  3. Who sells options?
  4. How are these things priced?
  5. How do I use the pricer?
  6. How do I compute volatility?
  7. Where can I get volatilities from the net?
  8. What are delta, gamma, and theta?
  9. What else can I use the pricer to value?
  10. Shares are call options?
  11. Buying debt means selling put options?
  12. What next?

A little bit about options....

A call option gives the buyer of the option, the right but the not the obligation to buy a fixed number of stock at a fixed price at a fixed time in the future. A put option gives the buyer of the option, the right but not the obligation to sell a fixed number of stock at a fixed price at a fixed time in the future. For example, consider the case where someone had bought a call option on IBM at $60 expiring on 21 December 1995. On the expiration date, if the price of IBM was above $60 then it would be worth the buyers time to ``exercise'' or take up his/her right to buy IBM at the prearranged price of $60 and sell it into the market at a higher price. An european option can only be ``exercised'', or delivery of the stock forced by the buyer, at expiration. An american option can be exercised at any time up to and including expiration.

Who buys options?

There are several types who like to buy options. People who would like to participate in movements in the underlying asset but lack the resources to buy it outright may buy an option at a fraction of the price of outright purchase, people who want to limit their downside just to the price of the option, people who trade options in their own right as assets and people who exploit pricing relations between options and other factors.

Who sells options?

This one is a bit trickier. There is a difference between selling a call option and buying a put option. For a start, the loss on the former is unlimited while the lost on the latter is limited to the price of the option. Sometimes people sell options to make money from the premiums received and hope that the loss from managing the position will not offset the premiums received. A larger group of natural sellers are insurance companies who would like to reduce their exposure to the market and use options as a means of synthetically shorting some of their holdings.

How are these things priced?

Usually with a lot of difficulty! Early work on option pricing was done by a guy called Bachelier for the equivalent of the doctoral's thesis. He assumed that stock prices were normally distributed and priced options according to the expected payoff at the end of the option's life. For his efforts he managed to get an honourable mention and ended the rest of his life in relative obscurity as a rural teacher. A century or so later, options started trading in Chicago and graduate students Fisher Black, Myron Scholes and Robert Merton were attacking the same problem. Building on the work of Paul Samuelson, they were able to create the option pricing formula now commonly known as the Black-Scholes option formula. Their contribution was not so much the derivation of the formula, which had already been done many times before beginnning with Bachelier, but the economic insight into a crucial part of the pricing argument, thus allowing the elimination of a variable which had previously had to be estimated at best, and guessed at worst. This variable was the market price of risk. The option pricer in this form use the original Black-Scholes formulas to price the european option. The formulas are

[Black-Scholes formula]

for the call option and put option respectively where the variables S, X, d, r, sigma and T are the stock price, strike price, continuously compounded dividend yield, continuously compounded riskless rate, volatility and time to expiration. The function N(.) is the normal cumulative probability function used in statistics. Unfortunately, this formula does not work for american options and a numerical approach has to be used instead. The numerical approach used in this option pricer creates a lattice in accordance with the Cox-Ross-Rubinstein methodology to simulate the diffusion process. Thirty time steps are used in the lattice. However, because of inherent problems with numerical methods and issues with rates of convergence, it may somtimes be noted that the american options may trade sightly less than their european counterparts. This problem is most extreme the less the option is in the money as the chances of early exericise are mitigated and the numerical model then becomes an approximation of the Black-Scholes formula.

How do I use the pricer?

This should be fairly easy. Just type in the price of the stock, its strike, the dividend yield to expiration, the cash interest rate until expiration, the volatility for the stock and the time to expiration. The stock price and strike should be entered in dollars and cents. The dividend yield, interest rate and volatility are measured in percentages per annum. The interest rate to use would be the treasury rate of the government debt (bond, note or bill) of the same expiration as the option. Next select the units of time to measure the remaining expiration and then enter the time to expiration. Now choose the call option or put option radio button for the type of option, and whether the option is european or american.

How do I compute volatility?

Volatility as used means the annualised standard deviation of the logarithm of returns. Although this sounds like mouthful, in reality it is not so bad. Say you already have the stock price going back for a year.

  1. The first step is to work out the returns. This done for each day by dividing each day's price by the previous day's price and taking the natural logarithm of the answer:
    [Returns formula]
  2. The next step is to compute the means of the returns. If there are n days of observations, then the mean return is
    [Mean formula]
  3. We are now ready to compute the volatility. This is done by multiplying by 100 to get the percentage, of the annualised standard deviation of the returns:
    [Volatility formula]

Where can I get volatilities from the net?

Volatility is also called standard deviation in the financial industry. At various times this information has been available on the net. Some historical stock volatilities are computed daily on this site at http://www.intrepid.com/~robertl/stock-vols1.html.

What are delta, gamma, and theta?

These are measures of the option's sensitivity to changes in the stock price. Delta is the change in option price for a small change in stock price. Roughly, delta is how much the option price changes for a dollar change in the stock price. Similarly, gamma is the change in delta for small changes in the stock price. Lastly, theta is the change in the option price over the course of a day. Delta, along with gamma give investors an idea of the sensitivity of their option positions as the price of the stock moves. Because options are a ``wasting'' asset whose insurance value decreases with time, theta is a measure of the cost of this insurance on a daily basis.

What else can I use the pricer to value?

The option pricer can be used to price more than just options on stocks.

Shares are call options?

Yes! Think of a company as being composed of its shareholder's equity and debt. Now, the value of a company is always non-negative. If the value of a company is more than its debt, then what remains belongs to the shareholders. On the other hand, if the company is not worth more than its debt, then nothing is left for the shareholders. Thus, shares in a company are like a call option on the value of the company with a strike set to be the amount of debt in the company. This call option has no maturity but can expire when the company is dissolved.

Buying debt means selling put options?

Think of a company as shareholder's equity and debt. If the company is worth more than its debt, the debt holders just receive the value of the debt. On the other hand, if the company is worth less than the debt, the debt holders only get the value of the company back. This is like getting all the debt back but paying back the difference between the company and the value of the original debt. In other words, the holder of the debt has sold a put option with a strike equal to the debt amount as well as being guaranteed to receive the value of the debt.

What next?

This is a very basic option pricer and other improvements account for discrete dividends, stochastic volatilities and a term structure of interest rates. Options are now written on a variety of assets and combinations of asset, currencies, interest rates and commodities with payoffs that can be path dependent or not. All in all, this is just a sampling of the many different and varied types of financial instruments commonly called ``derivatives''.