Option Valuation Nomograms

Abstract

HE May/June 1977 issue of Financial Analysts Journal presented a novel method for valuing options by using nomograms.' Nomograms allow one to obtain the value of any option, the position in the underlying stock that will neutralize the option's risk and the probability that it will pay to exercise the option at its maturityall without undertaking any mathematical calculations. This article provides full-sized blank nomograms that the reader can use to apply the technique to both call and put options.2 With only a pencil and ruler, anyone can obtain accurate values for any traded option. The nomogram user requires only four items of information-the maturity of the option, the share price expressed as a percentage of the exercise price, the volatility of the share (i.e., the standard deviation of the continuously compounded annual returns on the share) and the annual rate of interest. (The Probability of Exercise Nomogram requires knowing the annual rate of return expected on the stock, rather than the rate of interest.) By applying this information to the appropriate nomogram, the user can determine the value of the put or call option. The Call Option Value Nomogram provides the value of a call option, expressed as a percentage of the current share price. The Put Option Value Nomogram yields the value of the corresponding put option, expressed as a percentage of the current share price. (This estimate, for reasons explained in the earlier article, slightly understates the value of an American put.) The Hedge Ratio Nomogram will tell the user how many shares to hold against one hundred-share call option contract in order to neutralize the risk of the option position. To find out the number of shares to hold against the corresponding put option contract, the user simply subtracts this number from 100. The Probability of Exercise Nomogram gives the percentage probability that it will pay to exercise a call option at maturity; subtracting this figure from 100 per cent yields the probability for a put option exercisable only at maturity. Each nomogram consists of four quadrants separated by four axes. The investor must draw four lines, one for each item of data, parallel to the axes. Table 1 summarizes the general approach, which is the same regardless of the nomogram being used. Figure 1 illustrates the valuation of an eightmonth call option on a stock that has an annual standard deviation of 60 per cent. Since the exercise price is $40.00, the current share price of $52.00 is 130 per cent of the exercise price.3 The annual interest rate is 10 per cent. First, a vertical line has been drawn through the eight-month maturity, then a horizontal line through the 10 per cent interest rate. Next a vertical line was drawn through the share price as a percentage of the exercise price ( 130 per cent). Finally, a horizontal line was passed through the 60 per cent standard deviation. The intersection of this line with the previous vertical line indicates a call option value of 34 per cent of the share price, or $17.68. The same set of information applied to the Put Option Value Nomogram would indicate a put option value of 6.2 per cent of the share price, or $3.22. The Hedge Ratio Nomogram, given the same data, would specify that 82 shares will neutralize the risk of one hundred-share call option contract. Finally, with the additional information that the expected return on the stock is 16 per cent per annum, the Probability of Exercise Nomogram would tell the user that there is a 77 per cent probability that it will pay to exercise the option.4 The nomograms have two characteristics that greatly enhance their value. First, they allow the reader to examine the sensitivity of the results to changes in the input variables. For example, it is very easy to see the effect on the option value or Elroy Dimson teaches at the Institute of Finance and Accounting, London Graduate School of Business Studies.