Question

Describe actions you could take at time 0 to exploit an arbitrage opportunity (if any). Calculate the resulting profit (per stock unit).If you given the following:

1.The current bid price and ask price of a stock are 30 and 31, respectively.

2.The stock pays dividends continuously at a rate proportional to its price. The dividend yield is 3%

3.The continuously compounded lending and borrowing rates are 6% and 7%, respectively.

4.The transaction costs are:

•A $1 transaction fee, paid at time 0, for buying or selling each unit of the stock.

•A $2 transaction fee, paid at expiration, for settling a forward contract on the stock

5.The 3-year forward price on the stock is 28.

Answer 1

If the stock is bought and hold till maturity and delivered against the forward contract

Amount required to buy the stock = $31

Amount required for transaction cost = $1

So, amount borrowed at 7% = $32

Future value of the amount after 3 years to be paid (net of dividends received)

=31*exp((0.07-0.03)*3) +1 *exp(0.07*3)

= $36.186

Along with this , $2 also needs to be paid

So, total amount payable after 3 years = $36.186+ $2 = $38.186

Whereas the amount received by delivering the stock against forward contract is only $28

So, arbitrage is not possible this way

If the stock is sold short and amount invested till maturity and and then the stock bought against the forward contract

Amount gained by selling the stock = $30

Amount spent in transaction cost = $1

So, Net amount received and invested at 6% = $29

Future value of the amount after 3 years to be received (net of dividends)  

=30*exp((0.06-0.03)*3) -1*exp(0.06*3)

= $31.628

After paying $2 to settle the forward contract

So, Net amount available after 3 years = $31.628 - $2 = $29.628

Whereas the amount to be paid for taking delivery of the stock against forward contract is only $28

So, arbitrage profit = $29.628 - $28 = $1.628