Fin55. Week 8:Chapter 20: Problems 3(a-c), 5(a-c), 8(a-c), 9(a-d), and 10(a-d)

On this problem, it is one of the hardest ones for me to get my hands around and also difficult for me to try to explain. But let me give it a shot.

The concept of ‘synthetic’ is a financial term that refers to financial instruments that are ‘created’ in order to simulate the cash flows of other financial instruments with different cash flows. This can be done by buying a put option on a stock and simultaneously selling a call option on the same stock. As a result, the value will be the same at the maturity of the options because the values move in different directions. For 8a, to simulate this, the graphs would be a parallel graph as follows:

8(a). Sum of T-bill, call & put

Payoff

60 T-bill

Call

0 60 Stock Price

Put

-60

For 8b, the no-arbitrage price would be as follows:

8(b). With a price of $97 for six-month T-bills, the 6-month risk-free rate is 100/97 -1 = 3.09%

Using put-call parity the “no arbitrage price” is

S= PV(exercise price)+C-P

S = $60/1.0309 + 3.18 - 3.38

S = $58.00

8(c). Since put-call parity indicates the “no arbitrage” price of the stock is $58 and the stock selling at $60, the arbitrage would be to sell the overvalued portfolio (the stock) and use the proceeds ($60) to buy the undervalued portfolio (.6 tbills, long 1 call, short 1 put). This set of trades yields $2 in arbitrage profits. Since by putcall parity we know that the two portfolios will have exactly offsetting terminal payoffs, the trade is riskless.

For problem 10, this will help.

10(a). The transactions needed to construct the synthetic T-Bill would be to long the stock, long the put and short the call.

10(b). Assuming the T-Bill yield was quoted on a bond equivalent basis, the synthetic Treasury bill’s annualized yield can be calculated on the same basis:

First, find the present value of the exercise price of $21:

$21.50 + $4.50 - $5.50 = $20.50. The percentage difference between this value and the exercise price is the risk-free rate, which we multiply by 4 to estimate the annual rate:

[$21.00 - $20.50)/$20.50] x 4 = 9.76%

10(c). The strategy would be to short 21 actual T-bills and to long 100 synthetic T-bills.

Assuming the actual T-bill was quoted on a bond equivalent basis, the actual T-bill

gives a 0.75% quarterly return.

Immediately, the short actual T-bill position pays:

$210,000/1.0075 = $208,437

At the time of creation, the long position in the synthetic T-bill would be:

Long stock $215,000

Long put $ 45,000

Short call -$ 55,000

$205,000

Therefore the net cash flow is:

$208,437- $205,000= $3,437

10(d). The approach to calculating net cash flow gives the same result whether the calculation is done for three months or six months. At the three-month expiration, the value of the long synthetic position is:

X = P + S – C

where X = exercise price, P = put price, C = call price, and S = stock price

At expiration X = P + S – C

= $0 + $23 - $2

= $21 per share or $2,100 per contract

Total cash flow of the long synthetic position = 100 x $2,100

= $210,000

Total cash flow of the short Treasury bill position = 21 actual Treasury bills x $10,000

= $210,000

Net cash flow = $210,000 - $210,000 = $0

$0 cash flow at three months would be worth $0 at six months. Alternatively, if the stock price at expiration is $23:

Long position = $23

Short call position = $21 - $23 = $ -2

Long put position = $ 0

Short Treasury bill position = -$21

Net position $ 0