13.1 Definitions
An option is a contract that gives you a right, but not an obligation, to buy or sell an asset at a predetermined price (the exercise price) at or before some future time (the maturity date). The asset is usually a number of stocks, and we will concentrate on such options. We also distinguish between European and American options. A European option can only be exercised at the maturity date, while an American option can be exercised at any time before the maturity date. We will restrict our discussion to European options.
An option that gives you the right to buy a stock is named a call option, while an option that gives you the right to sell a stock is named a put option. In each case, there is one buyer and one seller of the option. We say that the buyer has a long position and that the seller has a short position in the option. This gives four different positions:
Long Call = You have bought an option that gives you the right to buy a stock from your counterpart at the exercise price on the maturity date.
Short Call = You have sold an option that gives your counterpart the right to buy a stock from you at the exercise price on the maturity date.
Long Put = You have bought an option that gives you the right to sell a stock to your counterpart at the exercise price on the maturity date.
Short Put = You have sold an option that gives your counterpart the right to sell a stock to you at the exercise price on the maturity date.
Call options
A call option is a right to buy a stock at a predetermined price on a future date. Suppose you buy a call option today for an option premium of $5. This option gives you the right to buy a specified stock in six months at the exercise price X = $80. The pay-off diagram at the top of figure 13.1 illustrates how the profit or loss on the option depends on the stock price on the maturity date. If the stock price S is less than $80 on the maturity date, you will not buy the stock for $80 because you can buy it cheaper in the market. The option expires out of the money, and you choose not to exercise the option. Anyway, if the stock price S is more than $80 on the maturity date, you can buy the stock for $80 and immediately sell it in the market making a profit. In this case the option expires in the money, and you choose to exercise the option.
The upper part of figure 13.1 shows your pay-off diagram when you have bought the call option (long call). The pay-off diagram does not include the option premium, because the option premium is paid six months before the maturity date. Remember that the pay-off diagram describes the situation on the maturity date.
Your counterpart that has sold the call option will not be forced to sell the stock on the maturity date if the stock price is less than $80. Anyway, if the stock price is more than $80, he will be forced to sell the stock for $80. This is less than the market price, and he loses money. This is illustrated at the lower part of figure 13.1. We see that the potential profit and loss on call options are unlimited for the two participants.
To draw the diagrams in figure 13.1, we have written the different stock prices in column A and calculated profits in columns B and C by the IF function. When calculating the buyer’s profit in cell B9 we first check if the stock price in cell A9 is less than the exercise price in cell B5. If this is the case, cell B9 is set to the option premium with a negative sign (–B4) which is 0 in figure 13.1. If the stock price is larger than the exercise price, the profit is calculated as stock price – exercise price – option premium. If we want to include the option premium in the diagrams, this can be done by writing the option premium in cell B4.
Figure 13.1 Pay-off diagrams for a call option.
With the symbols S and X representing the stock price and the exercise price respectively on the maturity date, we have the following possible outcomes. (The option premium is not included in the profit.)
Put options
A put option is a right to sell a stock at a predetermined price on a future date. Suppose you buy a put option today with the exercise price X = $60 (in six months). See figure 13.2. If the stock price S is more than $60 on the maturity date, the option is of no value. You will not sell the stock for $60 because you can sell it at a higher price in the market. If the stock price S is less than $60, you can buy it at a lower price in the market, and sell it for $60 with a profit.
The seller of the put option will not have to buy the stock if S > $60 on the maturity date. If S < $60, he will be forced to buy the stock for $60. This represents a loss for him, since he can buy the stock at a lower price in the market. In figure 13.2 we see that the potential profit and loss on a put option are limited to the exercise price for the two participants (when we ignore the option premium).
Figure 13.2 Pay-off diagrams for a put option.
For a put option we have the following possible outcomes. (The option premium is not included.)
13.2 Combinations
Combinations of options (sometimes including stocks) can be made to reduce risk or to increase profit. You can for instance reduce the risk on a stock investment by buying a put option on the stock. This is illustrated in figure 13.3. (The option premium is not included in the figure.) In this case you buy a stock for $50, and at the same time you buy a put option on the stock with the exercise price X = $50. This means that you are guaranteed a minimum price of $50 on the maturity date. The price you pay for this guarantee is the option premium. The total position, which is the sum of the two dashed curves and illustrated by the solid curve in figure 13.3, is called a protected put.
Figure 13.3 Protective put.
Assume an OPEC meeting is coming up, and that this meeting probably will lead to changed oil prices. You expect that the stock price for a specific oil company will be more than $60 or less than $40 in six months. If you have the position illustrated by the solid curve in figure 13.4, you will gain a profit if the stock price rises or falls significantly. This position is called a strangle. The figure also illustrates that this position can be obtained by the combination of buying a put option with an exercise price of $40 and a call option with an exercise price of $60.
Figure 13.4 Bought strangle.
It is possible to obtain any position and any form on the pay-off diagram by combining options, stocks and/or risk-free investments.
The current price on a stock is $50. You expect that this price will rise moderately in the following months. Thus, you want to combine options and invest in a bull spread as illustrated in figure 13.5.
Figure 13.5 Bull spread.
A combination of a bought call option with an exercise price of $50 and a sold call option with an exercise price of $75 results in the bull spread illustrated in figure 13.6. The profits on the options are calculated in the cells E4:F24. The content of cell E4 shows how this profit is calculated for the bought call option using the IF function.
Figure 13.6 A bull spread can be obtained by a combination of a bought call option and a sold call option.
13.3 Binomial model for valuing options
In the binomial model we assume that the underlying stock price for an option increases with the factor u or decreases with the factor d during the next period. More precisely, if the stock price today is S, then the next period’s stock price will either move up to Su or down to Sd. If the risk-free interest rate is r, we have the following restrictions:
The value of the option is given by the expected discounted pay-offs which are calculated by risk-neutral probabilities. The risk-neutral probability q of an up-move in the stock price (from S to Su) is given by:
The risk-neutral probability of a down-move in the stock price (from S to Sd) is then 1 – q.
The next example demonstrates how values for a call and a put option can be calculated by the binomial model. The relation between the value of a call option and the value of a put option is given by the put–call parity:
C0 is the value of a call option, P0 is the value of a put option, and PV (X) is the present value of the exercise price X.
A stock’s present price is $85. Assume that this price will increase by 20 per cent or drop by 10 per cent during the next year. The risk-free interest rate is 7.0 per cent per year. Find today’s value of a call and a put option with exercise prices of $80.
The risk-neutral probabilities of an up-move and a down-move in the stock price are given by:
The possible cash flows for the stock are illustrated in figure 13.7.
Figure 13.7 Possible cash flows for a stock.
The possible cash flows for a call option on the stock are as follows. If the stock price increases, the value of a call option on the stock becomes Su – X = $102 − $80 = $22 on the maturity date. If the stock price decreases, the same call option will be worthless on the maturity date since Sd < X (i.e. $76.50 < $80).
Figure 13.8 Possible cash flows for a call option.
In the spreadsheet in figure 13.9 the value $22 is calculated in cell D14 by the function MAX(D10-$B$4;0). This function returns the largest value of the difference $102 – $80 and 0. All formulas and functions in this spreadsheet are shown in figure 13.10.
Figure 13.9 Binomial model for valuing options.
Today’s value of the call option can be calculated to:
This calculation is done in cell B15 in the spreadsheet.
Figure 13.10 Binomial model for valuing options.
The possible cash flows for a put option on the stock will be as follows. If the stock price increases, the option will be worthless on the maturity date since Su > X. If the stock price decreases, the value of the option becomes X – Sd = $80 − $76.50 = $3.5 on the maturity date.
Figure 13.11 Possible cash flows for a put option.
Today’s value of the put option can be calculated to:
This value can also be calculated by the put–call parity:
Example 13.2 demonstrated a one-period binomial model. Let’s go a step further and study such models with several periods. Examples 13.3 and 13.4 demonstrate binomial models with 4 and 12 periods respectively (i.e. 4 and 12 periods during a year). The trading of stocks and options happens continuously during a year. For this reason the model will improve if the number of periods in a year approaches infinity. Such a model will be discussed in section 13.4.
A stock’s present price is $80. Assume that this price will increase by 10 per cent or drop by 5 per cent during a quarter. The risk-free interest rate is 3.0 per cent per quarter. Find today’s value of a call option on the stock with an exercise price of $70. The option expires after four quarters (i.e. one year).
The risk-neutral probabilities of an up-move and a down-move in the stock price during a quarter are given by:
In the spreadsheet in figure 13.12, possible stock prices are calculated quarter by quarter. At the end of the first quarter the stock price will be Su = $80 * 1.1 = $88 or Sd = $80 * 0.95 = $76. During the second quarter the stock price can go from $88 to $88 * 1.1 = $96.80 or to $88 * 0.95 = $83.60. Alternatively the stock price can go from $76 to $76 * 1.1 = $83.60 or to $76 * 0.95 = $72.20.
If the stock price goes up and up during the first and second periods, the price will be $96.80 at the end of the second period. If the price goes up and down, or down and up, the price will be $83.60. Finally, if the price goes down and down, the price will be $72.20 at the end of the second period. We see that one development results in the price $96.80, two developments in the price $86.30, and one development in the price $72.20 after two quarters. Thus the probability is 1/4 for the price $96.80, 2/4 = 1/2 for the price $83.60, and 1/4 for the price $72.20. This corresponds to a binomial distribution with p = 0.5 and n = 2. In this distribution P(X = 0) = 1/4, P(X = 1) = 1/2, and P(X = 2) = 1/4.
The spreadsheet in figure 13.12 shows that this development continues to the end of the fourth quarter. We see that one path results in the stock price $117.13, four paths in the stock price $101.16, and so on. The probabilities for the different final stock prices are calculated both from the number of paths and from a binomial distribution by the function BINOMDIST.
The corresponding values for a call option on the maturity date are calculated further down in column J. In cell J20 we find the function MAX(J8 – $B$4;0) which returns the difference between the stock price and the exercise price if the stock price exceeds the exercise price. Otherwise it returns the value 0. In column H we use the risk-neutral probabilities and the values in column J to calculate all possible values for the option at the end of the third quarter. In cell H21 we find the following calculation:
Similar calculations are performed in columns F, D and B. In cell B24 we find the present value of the call option, which is $18.01.
Figure 13.12 Binomial model for valuing options.
A stock’s present price is $90. Assume that this price will increase by 5 per cent or drop by 2 per cent during a month. The risk-free interest rate is 1.0 per cent per month. Find today’s value of a call option on the stock with an exercise price of $75. The option expires after 12 months (i.e. one year).
Given: S = 90, X = 75, r = 0.01, u = 1.05 and d = 0.98 (12 months).
The risk-neutral probabilities of an up-move and a down-move in the stock price during a month are:
In the spreadsheet in figure 13.13 the present value for the call option is calculated to $23.45 by a similar procedure as shown in example 13.3.
The probabilities for the different values of the call option on the maturity date are calculated by the function BINOMDIST. The binomial distribution represented by these probabilities is illustrated by dots in the diagram in figure 13.13. We have also drawn a curve through the dots. This curve represents the distribution we get if the number of periods (during a year) approaches infinity. The central limit theorem says that a symmetrical binomial distribution approaches a normal distribution when n approaches infinity (in the binomial distribution). This leads us to section 13.4 and the Black–Scholes model for valuing options.
Figure 13.13 Binomial model for valuing options.
13.4 Black–Scholes model for valuing options
The curve in figure 13.13 represents the distribution of possible values for a call option on the maturity date. The option’s value is calculated as S – X. Thus a curve showing the corresponding values for the stock must have a similar shape. The curve in figure 13.13 is skewed with a tail to the right. The explanation for this is that we have a lognormal distribution.
Figure 13.14 illustrates the lognormal distribution for a stochastic variable X. In the upper part of the figure the distribution is drawn with a linear scale on the horizontal axis. This gives a skewed distribution. In the lower part we have a logarithmic scale on the horizontal axis. This gives the shape of a normal distribution.
Trading stocks and options is a continuous activity. Thus we need a model where the number of periods per year approaches infinity. This leads us to the Black–Scholes model.
The Black–Scholes model assumes a lognormal distribution for the stock prices at the maturity date, and is widely used for valuing options. In this model the present value C0 of a call option is calculated as follows:
Figure 13.14 Lognormal distribution.
The symbols have the following meanings:
S0 |
= |
Present stock price |
N (d) |
= |
P(Z < d) in a standard normal distribution |
X |
= |
Exercise price |
r |
= |
Return per year on a risk-free investment (at continuous compounding) |
T |
= |
Time in years to the maturity date |
σ |
= |
Standard deviation for the yearly return on the stock |
The model assumes a constant risk-free rate and volatility (expressed by σ) during the time to maturity. Another assumption is that no dividends are paid on the stock in the same period.
The present value P0 of a put option can be calculated from the present value of a call option and the put–call parity:
Due to the continuous compounding, the present value of the exercise price must be calculated as:
A combination of the last three equations gives the Black–Scholes formula for valuing a put option:
The Black-Scholes formula is valid for European options. American options can be exercised before the maturity date. This possibility represents a value. Thus the Black-Scholes formula gives the minimum value of an American option.
A stock’s present price is $120. The standard deviation for the yearly return on the stock is 50 per cent (σ = 0.5). A call option on the stock has an exercise price of $100 and a maturity date in three months. The return per year on a risk-free investment is 8 per cent. Calculate the present value of the call option by the Black–Scholes model. Then calculate the present value of a put option under similar conditions.
First we calculate:
In the spreadsheet in figure 13.15 d1 is calculated in cell C11 by the formula “=(LN(C5/C6)+ (C7+0,5*C9^2) *C8)/(C9*SQRT(C8))” and d2 in the cell C12 by the formula “=C11 – C9*SQRT(C8)”. The value for S0 is written in cell C5, the value for X in cell C6, and so on.
N(d1) and N(d2) are calculated by the function NORMSDIST in cells F11 and F12 respectively:
N (0.934) = 0.825 N (0.684) = 0.753
The present value of the call option is then calculated in cell C14 to:
The present value of the put option is calculated in cell C15 by the put–call parity:
Figure 13.15 Valuation of options by the Black–Scholes formula model.
13.5 Time value and intrinsic value for options
At the maturity date, the value of an option will depend only on the underlying stock price and the exercise price. At an earlier point in time, an option’s value will also depend on other factors such as time to maturity, the stock’s volatility, and the risk-free interest rate during the time to maturity.
Take a look at figure 13.16 and assume that you buy a call option today with a maturity date in six months and an exercise price of $100. First assume the present stock price is $90. If the maturity date had been today, the option would have been of no value. Anyway, since the maturity date is in six months the stock price may increase so that the option ends up with a positive value on the maturity date. Thus, the present value of the call option is positive although the stock price is only $90. In this case we say that the option has only a time value and that it is out of the money. The stock price may also decrease in the future. Anyway, the value of the option can never be negative, since you will not exercise the option if the stock price is less than the exercise price on the maturity date.
Figure 13.16 indicates that the time value of the option decreases if the stock price decreases. If the present stock price is $120 and above the exercise price the option has an intrinsic value in addition to the time value. In this case we say that the option is in the money.
Assume now that the maturity date is in three months. In figure 13.16 the curve for three months to maturity is drawn below the curve for six months to maturity. This is due to the fact that the probability of an increase in the stock price is lower when there is only three months left to maturity. Thus the time value decreases when the maturity approaches. If the stock price is $120, the exercise price is $100, and the option’s market value is $30, the option’s intrinsic value will be $120 – $100 = $20. Thus the time value must be $30 – $20 = $10. If the stock price is $120 on the maturity date, the time value is 0 and the intrinsic value $20.
Figure 13.16 Value of a call option as a function of the underlying stock price 6 months and 3 months before maturity.
A call option on a stock has an exercise price of $100. The standard deviation for the yearly return on the stock is 35 per cent (σ = 0.35). The return per year on a risk-free investment is 5 per cent. Calculate the option’s time value and intrinsic value when the maturity date is in six months and the stock price is $110.
Figure 13.17 Time value and intrinsic value for a call option.
In the spreadsheet in figure 13.17 the present value of the call option is calculated to C0 = 17.63 in cell C14 by the Black–Scholes formula. The intrinsic value of the option is $110 – $100 = $10. Thus the time value is $17.63 – $10 = $7.63. The intrinsic value and the time value are calculated in cell C16 and C17 respectively.
The diagram in figure 13.17 shows the option’s total value and intrinsic value as functions of the stock price. The total values (C0) are calculated by the Black-Scholes formula in a data table in the cells C21:C37. The intrinsic values are calculated by the function MAX which returns the largest value of (stock price – exercise price) and 0.
Problems
13-1. The position illustrated in figure 13.18 is a bought bear spread, which can be valuable if the underlying stock price decreases in the future. (The option premiums are included in the figure.) Explain how this position can be obtained by a combination of options.
Figure 13.18 Bear spread.
13-2. The two positions in figure 13.19 are a sold straddle and a bought straddle. (The option premiums are included in the figure.) You might want to sell a straddle if you expect the underlying stock price to be approximately constant in the next period. If you expect that the stock price will increase or decrease significantly, you should rather buy a straddle. Explain how these positions can be obtained by combining options.
Figure 13.19 Straddle.
13-3. Figure 13.20 illustrates the pay-off diagram for a stock you have bought for $100. Show how you can combine options and obtain a similar position. Such a combination is called a synthetic stock.
Figure 13.20 Synthetic stock.
13-4. Explain how you can obtain the position in figure 13.21 (a butterfly) by combining options.
Figure 13.21 Butterfly.
13-5. You buy a stock for $120. At the same time you sell a call option on the stock with an exercise price of $120. Draw the pay-off diagram for this position, which is called a covered call. What might the purpose be for this position?
13-6. A stock’s present price is $120. Assume that this price will increase by 15 per cent or drop by 10 per cent during the next year. The risk-free interest rate is 5.0 per cent per year. Use a binomial model and calculate the present values of a call option and a put option on the stock with $130 as exercise prices. The options expire after one year.
13-7. A call option on an underlying stock expires after six months with an exercise price of $250. Assume that the risk-free interest rate will be 8.0 per cent per year in the future. The present stock price is $220. Assume that the stock price will be $265 or $190 in six months, and calculate the present value of the option by a binomial model.
13-8. A stock’s present price is $150. Assume that this price will increase by 7 per cent or drop by 5 per cent during the next three months. The risk-free interest rate is 1.5 per cent per quarter. Calculate the present value of a call option on the stock that expires after one year with an exercise price of $155.
13-9. A put option on an underlying stock expires after six months with an exercise price of $185. The present stock price is $170. The risk-free interest rate will be 5.0 per cent for the next six months. Assume that the stock price will be $210 or $150 in six months, and calculate the present value of the option by a binomial model.
13-10. A stock’s present price is $120. The standard deviation for the yearly return on the stock is 50 per cent. The return per year on a risk-free investment is 7 per cent. Use the Black–Scholes model and calculate the present value of a call option on the stock that expires in one year with an exercise price of $130.
13-11. A stock’s present price is $170. The standard deviation for the yearly return on the stock is 30 per cent. The return per year on a risk-free investment is 5 per cent. Use the Black–Scholes model to calculate the present values of call and put options on the stock that expires in six months.
a) Calculate present values of a call and a put option with exercise prices of $150.
b) Calculate present values of a call and a put option with exercise prices of $180.
c) Assume that the standard deviation for the yearly return on the stock is 50 per cent. Calculate present values of a call and a put option with exercise prices of $150.
d) Assume that the options expire in one year and that the standard deviation for the yearly return on the stock is 30 per cent. Calculate present values of a call and a put option with exercise prices of $150.
e) Explain why the present values of the options change from a to b, from a to c, and from a to d.
13-12. A call option on a stock has an exercise price of $80. The standard deviation for the return on the stock is 50 per cent and the return on a risk-free investment is 7 per cent during the time to maturity. Calculate the option’s time value and intrinsic value when the maturity date is in three months and the present stock price is $95.
13-13. A call option on a stock has an exercise price of $120. The standard deviation for the return on the stock is 30 per cent and the return on a risk-free investment is 5 per cent during the time to maturity. Calculate the option’s time value and intrinsic value when the maturity date is in six months and the present stock price is $110.