Question

Go to the Online Learning Center at www.mhhe.com/bkm , link to Chapter 20, and find there a spreadsheet containing monthly values of the S&P 500 Index. Suppose that in each month you had written an out-of-the-money put option on one unit of the index with an exercise price 5% lower than the current value of the index.

a. What would have been the average value of your gross monthly payouts on the puts over the 10-year period October 1977–September 1987? The standard deviation?

b. Now extend your sample by one month to include October 1987, and recalculate the average payout and standard deviation of the put-writing strategy. What do you conclude about tail risk in naked put writing?

Short answer

a. Average gross monthly payout = 0.2437; standard deviation = 1.0951

b. Average gross monthly payout = 0.6875; standard deviation = 5.0026

Step-by-step solution

Calculation of average value of gross monthly payouts ‘a’

As per the spreadsheet from the link, end of month value for S&P 500 in September 1977 was $96.53

Hence exercise price of the put at the beginning of October 1977:

$\begin{array}{rcl}\text{Exercise price of put}& =& \text{Exercise price}\times \text{Value of index}\\ & =& 95\%\times \$96.53\\ & =& \$91.7035\\ & & \end{array}$

At the end of the October, the value of index is 92.34, hence put would have expired out of money and writer’s payout was zero.

The first month with positive payout = January 1978

The exercise price for its put for beginning January:

$\begin{array}{rcl}\text{Exercise price of put}& =& \text{Exercise price}\times \text{Value of index}\\ & =& 95\%\times \$950.10\\ & =& \$90.3450\\ & & \end{array}$

At the end of January, the value of index was 89.25 hence the option writer’s payout

$\begin{array}{rcl}\text{Writer's payout}& =& \text{Value of index in the beginning}-\text{Value of index in end}\\ & =& \$90.3450-\$89.25\\ & =& \$1.0950\\ & & \end{array}$

Hence the average gross monthly payout = 0.2437 and the standard deviation = 1.0951

 Step 3: Calculation of average payout and standard deviation ‘b’

In October 1987, the S&P decreased by more than 21% from 321. 83 to 251.79

The exercise price of the put written at the beginning of October 1987:

$\begin{array}{rcl}\text{Exercise price of put}& =& \text{Exercise price}\times \text{Value of index}\\ & =& 95\%\times \$321.83\\ & =& \$305.7385\\ & & \end{array}$

The payout of the option’s writer at the end of October:

$\begin{array}{rcl}\text{Writer's payout}& =& \text{Value of index in the beginning}-\text{Value of index in end}\\ & =& \$305.7385-\$251.79\\ & =& \$53.9485\\ & & \end{array}$

The average gross monthly payout from October 1977 to October 1987 = 0.6875 and the standard deviation = 5.0026