### American-Style Options- The Effect of American-Style Options

American-Style Options

The Effect of American-Style Options

The ability to exercise early has no effect on the price of calls in the absence of dividends and only a moderate impact if dividends are taken into account. The principal difference between European- and American-style options occurs for put premiums. For American-style options, there is a significant upward effect on put premiums, whether or not dividends are present. Unfortunately, there is no neat, concise, and exact formula for American-style put premiums. A good analytic approximation for these premiums is given by a formula developed by Giovanni Barone-Adesi and Robert Whaley, as follows:

where the European put premium is the one calculated using the Merton variation, and P* is the critical stock price determined from the procedure outlined below. The early exercise premium is given by:

where:

N(-h*) again denotes the cumulative normal distribution function, where h* is the dividend adjusted parameter specified in the Merton variation, as evaluated at the critical stock price P*:

The critical stock price P* is the iterative solution to the following equation:

where the European put premium is evaluated at the critical stock price P*.

Numerical Example

I illustrate the use of the methodology with the same example used for the Merton variation, in which the strike price and stock price are \$100, the volatility is 0.35, the risk-free interest rate is 6 percent, there are 24 months till expiration, and the annual dividend rate is 2 percent.

We then compute in order:

The American put premium with a 2 percent dividend rate is therefore its European put value of \$14.596 plus \$1.288, or \$15.884. As mentioned earlier, the critical stock price P*= 58.1844 is determined by trial and error so as to satisfy Equation. To verify that this is the case, we calculate the value of the European put premium using the Merton variation at that particular stock price, as follows:

The left-hand side is given by:

while the right-hand side is calculated as:

Because the two values are identical (within computational round-off), the critical stock price of 58.1844 is confirmed.

Table  European vs. American Put Premiums

Dividend Rate                        European Put                       American Put
0%                                           13.314                                   14.880
1                                              13.948                                   15.373
2                                              14.596                                   15.884
3                                              15.259                                   16.417

Comparison Table

Table compares European- and American-style put premiums where the strike price and stock price are \$100, the volatility is 0.35, the risk-free interest rate is 6 percent, there are 24 months till expiration, and the annual dividend rate ranges from percent to 3 percent. As is clearly apparent, the premiums in all cases are greater than the one utilized for the zero-dividend, European-style case. Because of this, the premiums, retention rates, and account values shown in each run throughout this book would have been somewhat higher if actual dividends and early exercise fights were taken into account.

Computer Program for American Put Premiums

A relatively short BASIC program for calculating American put premiums follows. By using a Newton-Raphson iterative search technique, the critical stock price can usually be obtained to within three decimal places in just four or five iterations, as shown in the numerical example that follows the BASIC program.

Numerical Examples

As examples of the kind of results obtainable using the computer program, look at both at-the-money and out-of-the money situations. The first is the same one that was done by hand earlier.

Stock Price: \$100
Strike Price: \$100
Risk-Free Interest Rate (e.g., .06): .06
Volatility (e.g., 0.35): .35
Annual Dividend Rate (e.g., .02): .02
Time to Expiration in Months: 24

No.                          POLD                         PNEW                         F
1                            100.0000                     66.8516                      38.8177
2                              66.8516                     59.0941                        5.8858
3                              59.0941                     58.1949                        0.5523
4                              58.1949                     58.1819                        0.0078
5                              58.1819                     58.1819                        0.0000

Critical Stock Price:                                                                                            \$58.1819
20% of Price:                                   \$2,000.00                  @10%:                    \$1,000.00
Less OTM                                                 0.00                                                                -
Total:                                                \$3,588.40                                                  \$2,588.40

Margin Requirement: \$3,588.40 (per contract)

Stock Price: \$100
Strike Price: \$85
Risk-Free Interest Rate (e.g., .06): .06
Volatility (e.g., 0.35): .35
Annual Dividend Rate (e.g., .02): .02
Time to Expiration in Months: 24

No.                          POLD                          PNEW                          F
1                            100.0000                     59.5921                       51.3915
2                              59.5921                     50.7781                         7.1807
3                              50.7781                     49.4864                         0.8096
4                              49.4864                     49.4546                         0.0190
5                              49.4546                     49.4546                         0.0000

Critical Stock Price:                                                                                              \$49.4546
20% of Price:                                   \$2,000.00                   @10%:                     \$1,000.00
Less OTM:                                         1,500.00                                                                  -
Total:                                                \$1,403.03                                                    \$1,903.03

Margin Requirement: \$1,903.03 (per contract)

Computer Program for American Call Premiums

For completeness, I've also developed a short BASIC program for calculating American call premiums. By using a Newton-Raphson iterative search technique, the critical stock price can usually be obtained to within three decimal places in a half dozen or so iterations, as shown in the example that follows. The reason more iterations are needed to arrive at American call premiums is because in the absence of dividends, there is no early exercise premium for calls. As a result, the smaller the dividend rate, the higher the critical stock price and the longer it takes to converge to it.

Numerical Examples

As examples of the results obtainable with this program, I again consider both at-the-money and out-of-the money situations. The first is the same one that was done by hand, earlier. Note that the computation of call premiums typically takes longer to converge than does the computation of put premiums.

Stock Price: \$100
Strike Price: \$100
Risk-Free Interest Rate (e.g., .06): .06
Volatility (e.g., 0.35): .35
Annual Dividend Rate (e.g., .02): .02
Time to Expiration in Months: 24

No.                          POLD                          PNEW                          F
1                            100.0000                     170.6420                      33.7405
2                            170.6420                     257.8725                      13.3574
3                            257.8725                     372.5958                        5.7306
4                            372.5958                     425.3898                        1.5634
5                            425.3898                     427.2815                        0.0528
6                            427.2815                     427.2836                        0.0001
7                            427.2836                     427.2833                       -0.0000

Critical Stock Price:                                                                                          \$427.2834
20% of Price:                                  \$2,000.00                      @10%:                 \$1,000.00
Less OTM                                                0.00                                                                --
Total:                                               \$4,204.07                                                  \$ 3,204.07

Margin Requirement: \$4,204.07 (per contract)

Stock Price: \$100
Strike Price: \$115
Risk-Free Interest Rate (e.g., .06): .06
Volatility (e.g., 0.35): .35
Annual Dividend Rate (e.g., .02): .02
Time to Expiration in Months: 24

No.                           POLD                          PNEW                           F
1                             100.0000                     182.3433                       46.6304
2                             182.3433                     277.3311                       17.7129
3                             277.3311                     407.3093                         7.6294
4                             407.3093                     486.6183                         2.4358
5                             486.6183                     491.3691                         0.1327
6                             491.3691                     491.3751                         0.0002
7                             491.3751                     491.3751                        -0.0000

Critical Stock Price:                                                                                           \$491.3751
20% of Price:                                  \$2,000.00                   @10%:                     \$1,000.00
Less OTM:                                        1,500.00                                                                 --