Question

6. Suppose you invest $10,000 in a 1–year–equity–linked–CD. At maturity, the CD is guar- anteed to pay the invested amount, plus 50% of the percentage gain (if any) during the year on the stock index to which it is linked. At the time you invest, the stock index is prices at $1,500. Your payoff in one year is $10,000×(1+c×C(S1,K,1))), where C(S1,K,1) is a one year call option on the index with strike price K, and c is a constant. Determine c and K. What would the value of the stock index have to be at the end of the year (at maturity of the CD) in order for the CD to pay you $12,000?

Answer 1

Hello,

If you invest 10,000 in 1 CD linked with stock index priced at 1500, you buy 10,000/1500 units of that stock index which is equal to 25/3 units.

From the information provided in the question, we can write equations as -

Payoff = 10,000 + 50% of 25/3 X (Price of stock index at Year end - 1500). --- eq 1
Given Price at Year end is >= 1500
as gain during the year = No. of units of stock X Increase in price of stock.

Second equation provided in the question is

Payoff = 10000 X (1 + cXC(S1,K,1))) --- eq 2

where, c - constant (unknown)

C - Value of Call Option

S1 - Spot Price of the underlying stock

K - Strike Price

1 - Time to expiry is 1 years.

At any point, both these equations will give the same answer as they denote the same variable (payoff).

Let us assume, we are at the point of expiry of the CD (ie. one year from now) -

Let us also assume that at expiry, the value of underlying stock index went up to 1800 and Strike Price, K is less than 1800

From equation 1,

Payoff = 10,000 + 50% of 25/3 X (Price of stock index at Year end - 1500)

Payoff = 10,000 + 50% of 25/3 X (1800-1500)

Payoff = 10,000 + 50% of 25/3 X 300

= 10,000 + 1250 = 11,250

From equation 2,

Payoff = 10000 X (1 + cXC(S1,K,1)))

= 10000 X (1 + c X (1800 - K)) as at expiry of call option, its value is Price of underlying - Strike Price (if Price of Underlying > Strike Price and we have already assumed that K is less than 1800)

= 10000 + 10000 X c X (1800 - K)

Combining eq 1 and eq 2

11250 = 10000 + 10000 X c X (1800 - K)

.125 = 1800 c - cK ---- eq A

Now, Let us assume that at expiry, the value of underlying stock index went up to 2100 and Strike Price, K is less than 1800

From equation 1,

Payoff = 10,000 + 50% of 25/3 X (Price of stock index at Year end - 1500)

Payoff = 10,000 + 50% of 25/3 X (2100-1500)

Payoff = 10,000 + 50% of 25/3 X 600

= 10,000 + 2500 = 12,500

From equation 2,

Payoff = 10000 X (1 + cXC(S1,K,1)))

= 10000 X (1 + c X (2100 - K)) as at expiry of call option, its value is Price of underlying - Strike Price (if Price of Underlying > Strike Price and we have already assumed that K is less than 1800)

= 10000 + 10000 X c X (2100 - K)

Combining eq 1 and eq 2

12500 = 10000 + 10000 X c X (2100 - K)

.250 = 2100 c - cK ---- eq B

Subtracting eq A from eq B

.125 = 300 c

c = .125/300 = 1/2400

Substituting value of c in eq B

.250 = 2100/2400 - K/2400

600 = 2100 - K

K = 1500

These values of c and K also satisfy the condition, when the price of stock index at the end of year is less than 1500.

To solve the second part of the question,

From equation 1,

Payoff = 10,000 + 50% of 25/3 X (Price of stock index at Year end - 1500)

12,000 = 10,000 + 50% of 25/3 X (Price of stock index at Year end - 1500)

4,000 = 25/3 X (Price of stock index at Year end - 1500)

480 = Price of stock index at Year end - 1500

Price of stock index at Year end = 1980