Question

6.8. A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. a. What is the value of a six-month European put option with a strike price of $42? b. What is the value of a six-month American put option with a strike price of $42?

Answer 1

		$44 * 1.1 = 48.4
	$40 * 1.1 = $44
$40		$44 * 0.9 = $39.6
	$40 * 0.9 = $36
		$36 * 0.9 = $32.4

risk free rate = 12%

quarterly rate = 12%/4 = 3%

a. Value of European put option at $42 = $3.39 (see computation below)

value of put option at expiry = max(0, strike price - stock price)

		$44 * 1.1 = 48.4	max(0, 42 - 48.4) = $0
	$40 * 1.1 = $44
$40		$44 * 0.9 = $39.6	max(0, 42 - 39.6) = $2.4
	$40 * 0.9 = $36
		$36 * 0.9 = $32.4	max(0, 42 - 32.4) = $9.6

from the table,

value(u,u) = 0

value(u,d) = 2.4

value(d,d) = 9.6

Therefore,

value(u) = ((0.5 * value(u,u) + (0.5 * value(u,d))) * e^-0.03

= ((0.5 * 0) + (0.5 * 2.4)) * 0.9704

= (0 + 1.2) * 0.9704 = $1.16448

value(d) = ((0.5 * value(u,d) + (0.5 * value(d,d))) * e^-0.03

= ((0.5 * 2.4) + (0.5 * 9.6)) * 0.9704

= (1.2 + 4.8) * 0.9704 = $5.8224

value(today) = ((0.5 * value(u) + (0.5 * value(d))) * e^-0.03

= ((0.5 * 1.16448) + (0.5 * 5.8224)) * 0.9704

= (0.58224 + 2.9112) * 0.9704 = $3.39

b. Value of American put option at $42 = $4 (see computation below)

value of put option at expiry = max(0, strike price - stock price)

since an American option can be used at any point of time before contract expiration as well,

value of put option at middle nodes = max(strike price - stock price, value at that node)

		$44 * 1.1 = 48.4	max(0, 42 - 48.4) = $0
	$40 * 1.1 = $44
$40		$44 * 0.9 = $39.6	max(0, 42 - 39.6) = $2.4
	$40 * 0.9 = $36
		$36 * 0.9 = $32.4	max(0, 42 - 32.4) = $9.6

from the table,

value(u,u) = 0

value(u,d) = 2.4

value(d,d) = 9.6

Therefore,

value(u) = max(42 - 44, (((0.5 * value(u,u) + (0.5 * value(u,d))) * e^-0.03))

= max(-4, (((0.5 * 0) + (0.5 * 2.4)) * 0.9704))

= max(-4, (0 + 1.2) * 0.9704

= max(-4, 1.16448) = $1.16448

value(d) = max(42 - 36, (((0.5 * value(u,d) + (0.5 * value(d,d))) * e^-0.03))

= max(6, (((0.5 * 2.4) + (0.5 * 9.6)) * 0.9704))

= max(6, ((1.2 + 4.8) * 0.9704))

= max(6, 5.8224) = $6

value(today) = max(40 - 36, (((0.5 * value(u) + (0.5 * value(d))) * e^-0.03))

= max(4, (((0.5 * 1.16448) + (0.5 * 6)) * 0.9704))

= max(4, ((0.58224  + 3) * 0.9704))

= max(4, 3.4762) = $4