Fin 441 Homework1

Homework 1 (Call/Put options)

1. Consider an option that expires in 72 days. The bid and ask discounts on the T-bill are $7.44 and $7.20 respectively, Find the appropriate risk free rate. (Assuming1 year, $100 T-bill).

• Bid (purchase) = ($100 - $7.44) = $92.56
• Ask (sell) = ($100 - $7.20) = $92.80
• Average discount = ($7.44 +$7.2)/2 = $7.32
• 72 day T-bill = $100 – ($7.32)(72/360) =  $98.536
• Risk free rate = ($100/$98.536)^(365/72) – 1 = 7.76% 

For questions 2-6 current stock price is 165.13. The expirations are July 17, August 21, and October 16. Risk free rates are .0516, .0550, and .0588 respectively.

1. Compute the intrinsic values, time values, and lower bounds of the following calls.

1. July 160

• Intrinsic value = $165.13 - $160 = $5.13
• Time Value = $6 - $5.13 = $0.87
• Lower bound ≥ Max(0, $165.13 - $160(1+.0516)^(-.038356)) = $5.44 

1. October 155

• Intrinsic value = $165.13 - $155 = $10.13
• Time Value = $14 - $10.13 = $3.87
• Lower bound ≥ Max(0, $165.13 - $155(1+.0588)^(-.282192)) = $12.61 

1. August 170

• Intrinsic value = $165.13 - $170 = -$4.87
• Time Value = $3.20 - $4.87 = -$1.67
• Lower bound ≥ Max(0, $165.13 - $170(1+.0550)^(-.131507)) = $0.00   

1. Compute the intrinsic values, time values, and lower bounds of the following puts.

1. July 165

• Intrinsic value = $165 - $165.13 = -$0.13
• Time Value = $2.35 - $0.13 = -$2.22
• Lower bound ≥ Max(0, $165(1+.0516)^(-.038356) - $165.13) = $0.00   

1. August 160

• Intrinsic value = $160 - $165.13 = -$5.13
• Time Value = $2.75 - $5.13 = -$2.38
• Lower bound ≥ Max(0, $160(1+.0550)^(-.131507) – $165.13) = $0.00   

1. October 170

• Intrinsic value = $170 - $165.13 = $4.87
• Time Value = $9 - $4.87 = $4.13
• Lower bound ≥ Max(0, $170(1+.0588)^(-.282192) - $165.13) = $2.15

1. Check the following combinations of puts and calls, determine whether they conform to the put-call parity, for European options, and identify violations if any.

1. July 155

•  Pe($165.13 + $0.20) = Ce($10.50 + $155(1+.0516)^(-.038356))

                     $165.33 = $165.20

• Difference ≥ $0.10, put-call parity has been violated

1. August 160

• Pe($165.13 + $2.75) = Ce($8.10 + $160(1+.0550)^(-.131507))  

                    $167.88 = $166.98

• Difference ≥ $0.10, put-call parity has been violated

1. October 170

• Pe($165.13 + $9.00) = Ce($6.00 + $170(1+.0588)^(-.282192))  

                    $174.13 = $173.28

• Difference ≥ $0.10, put-call parity has been violated

1. Examine the following pairs of American calls which differ only by exercise price. Determine whether the rules regarding relationship between American calls that differ only by exercise price are violated.

1. August 155 and 160

• Max difference = $160 - $155 = $5
• Actual difference = $11.80 - $8.10 = $3.70
• Actual difference ≤ Max difference, rules not violated

1. October 160 and 165

• Max difference = $165 - $160 = $5
• Actual difference = $11.10 - $8.10 = $3.00
• Actual difference ≤ Max difference, rules not violated

1. Examine the following pairs of American puts which differ only by exercise price. Determine whether the rules regarding relationship between American puts that differ only by exercise price are violated.

1. August 155 and 160

• Max difference = $160 - $155 = $5
• Actual difference = $2.75 - $1.25 = $1.50
• Actual difference ≤ Max difference, rules not violated

1. October 160 and 170

• Max difference = $170 - $160 = $10
• Actual difference = $9.00 - $4.50 = $3.70
• Actual difference ≤ Max difference, rules not violated