Question

Consider a non-dividend paying stock. Recall that we have found the forward price of a stock supposing that the market is perfect. In particular, we assumed that the borrowing rate equals the saving rate and that there is no bid-ask spread in trading the stock. Suppose now that the effective borrowing rate is not equal the effective saving rate, in particular, rb(t,T) >rs(t,T). Suppose in addition that the bid price (when you sell or sell short a stock) is lower than the ask price (when you buy the stock), Sask(t) >Sbid(t).
(a) Use, for example, the standard arguments and derive the upper and lower no-arbitrage boundaries on the stock's forward price. Check that there is a range of forward prices under these conditions such that the arbitrage is possible only when the forward price lies outside of this range.
(b) Suppose that Sbid(t) = 49, Sask(t) = 50, the effective borrowing and saving rates are rb(t,T) = 6% and rs(t,T) = 4%. Find out whether or not there are arbitrage opportunities in the market for the following forward prices: (1) 52; (2) 54. In case there are arbitrage opportunities, describe the strategy that exploits them

Answer 1

a)

The Stock price should not exceed the price required to buy the stock Sask(t) realised by borrowing the amount at rate rb(t,T) along with the Interest on it

 

Therefore, Upper limit = Sask(t) * (1+ rb(t,T)) * (T-t)/365

or Upper limit = Sask(t) * (1+ rb(t,T)) ( assuming T-t = 365 days or one year)

 

Similarly, if One short sells the stock at Sbid(t) , invests the money at rs(t,T) , the Net amount realised after time T-t is Sbid(t)* (1+rs(t,T).If the stock price is below this amount , there is possibility of arbitrage.

 

So, Lower Limit = Sbid(t)* (1+rs(t,T)

Since, , rb(t,T) >rs(t,T) and Sask(t) >Sbid(t).

 

Upper Limit > Lower Limit and hence there is a range of Forward prices F such that

Upper Limit > F > Lower Limit for which there is no Arbitrage

 

Arbitrage is possible only when F> Upper limit or F

 

b)

For the given values , Upper Limit = 50 *(1+0.06)

                                                             = 53

and Lower Limit = 49*(1+0.04) 

                              = 50.96

 

So, (50.96, 53) is the range of Forward prices for which arbitrage is not possible

 

Therefore

1) If the forward price is 52, arbitrage is not possible as the price lies within the no arbitrage zone.

 

2) If the forward price is 54, arbitrage is possible as below:

i) Today, Borrow 50 at 6% and buy the stock with the amount

ii) Today, Sell the forward contract on the stock at 54

iii) At expiry of the contract, sell the stock and get 54

iv) Pay 50*(1 + 0.06) = 53 as the maturity amount of borrowing and take the remaining amount of 1 as arbitrage profit

a. Arbitrage is possible only when F> Upper limit or F

b. So, (50.96, 53) is the range of Forward prices for which arbitrage is not possible