Question

1. The owner of a European put option on a share has the right, but not the obligation, to sell one share for a fixed price known as the option’s strike price, on a fixed date known as the option’s expiry date. If the share price equals S on the option’s expiry date then the option’s payoff function is the larger of 0 and K − S, where K is the option’s strike price. This question illustrates what we can learn about the prices of Arrow-Debreu securities if we can observe the prices of a set of put options on the same stock, but with different strike prices. We assume that the option expires at date 1 and that the state of nature at date 1 is the share price at date 1, where the share price can take the values $0, $1,$2, . . . , $99, $100. Suppose that the Arrow-Debreu security paying out in state “$j” has price θ_j, for j = 0, 1, 2, . . . , 100.

1. Fill in the following table, where Portfolio A comprises one short position in a put option with strike price $51 and one long position in a put option with strike price $52; and Portfolio B comprises one short position in a put option with strike price $52 and one long position in a put option with strike price $53. The missing entries are the date 1 payoffs of the indicated assets (and portfolios) if the share price at date 1 equals the value given at the head of each column.

Share price on option’s expiry date	$47	$48	$49	$50	$51	$52	$53
Payoff of an option with strike $51
Payoff of an option with strike $52
Payoff of an option with strike $53
Portfolio A
Portfolio B

2. Use the current option prices given in the table above to calculate the prices of Arrow-Debreu securities that pay out when the date 1 share price equals $S, for each of S = 48, 49, 50, 51, 52.

Strike price	$47	$48	$49	$50	$51	$52	$53
Price of a put option with the indicated strike price	6.6879	7.0821	7.5017	7.9467	8.4140	8.9044	9.4092

Answer 1

ANSWER 1 Payoff of a put option is calculated by MAX (strike price - stock price, 0)

here we have Strike Price (K) = 51

and stock price as 47, 48 ... , 53

thus by calculating in excel we get the answer for eg if K = 51 and Stock price = 47, thus pay off of Put option will be MAX ( 51-47 , 0 ) i.e. MAX (4,0) therefore Payoff will be $4. Thus by calculating all in the table.

Portfolio A comprises one short position in a put option with strike price $51 and one long position in a put option with strike price $52.

example:- when Stock price is 47 , the Payoff of Portfolio A will be = -4 (short position in a put option with strike price $51) + 5 (one long position in a put option with strike price $52) = 1

doing the same in other stock prices also.

Portfolio B comprises one short position in a put option with strike price $52 and one long position in a put option with strike price $53.

example:- when Stock price is 47 , the Payoff of Portfolio A will be = -5 (short position in a put option with strike price $52) + 6 (one long position in a put option with strike price $53) = 1

doing the same in other stock prices also.

Share price on option’s expiry date	$             47	$             48	$             49	$             50	$             51	$             52	$             53
Payoff of a PUT option with strike $51	4.00	3.00	2.00	1.00	0.00	0.00	0.00
Payoff of a PUT option with strike $52	5.00	4.00	3.00	2.00	1.00	0.00	0.00
Payoff of a PUT option with strike $53	6.00	5.00	4.00	3.00	2.00	1.00	0.00
Portfolio A	1.00	1.00	1.00	1.00	1.00	0.00	0.00
Portfolio B	1.00	1.00	1.00	1.00	1.00	1.00	0.00

ANSWER 2 To calculate the prices of Arrow-Debreu securities we can Simply Subtract Put option price from the Strike Price.

S = K - P₀

Strike price	$         47	$         48	$         49	$         50	$         51	$         52	$         53
Price of a put option with the indicated strike price	6.6879	7.0821	7.5017	7.9467	8.414	8.9044	9.4092
Stock Price	40.3121	40.9179	41.4983	42.0533	42.586	43.0956	43.5908