Question

• Consider a stock index constructed from two stocks, I(t) = pi(t) + 2p2(t), where pi(t) and p2(t) are the prices of the two stocks at time t, and I(t) is the value of the index at t. • Also consider a futures contract written on the index: If you buy one contract, you receive from the seller the amount I(T), which is the value of the index at time T. If you sell one contract, you pay to the buyer the amount I(T), which is the value of the index at time T. (Here T is the expiry date of the contract.) Let to denote some point in time prior to the expiry date T, and suppose the prices of the two stocks at to are pı(to) = 20, and p2(to) = 30. (a) Suppose that the price of the futures contract at time to is 81. Describe a riskless arbitrage. (b) Suppose instead that the price of the futures contract at time to is 79. Describe a riskless arbitrage

Answer 1

Please refer to the answers as shown below:

Parta) Fair value of Index at time to = 20+2*30 80 Market price of futures contract=81 If we sell the contract at time to, we will receive 81 at time to and will have to pay I(T) at time I The riskless arbitrage is thus to buy 1 share of stock 1 & 2 shares of stock 2 at total cost of 80 Also sell the futures contract & receive 81 at time to. So we made a 81-80 = 1 profit at time to At time T, sell the index portfolio (1 share of s1 & 2 shares of s2) this will yield I(T). Pay this I(T) to the buyer of the contract to close the position. Thus the $1 profit that we made at time to does not change whatever may be the pricis of stock at time T. Thus it is a riskless arbitrage.

Part b) If we sell buy contract at time to, we will have to pay 79 at time to and will receive I(T) at time T The riskless arbitrage is thus to sell (short) 1 share of stock 1 & 2 shares of stock 2 thus receiving 20+60= 80 in total Also buy the futures contract by paying 79 at time to. So we made a 80-79= 1 profit at time to At time T, buy i.e. close the short positions) the index stocks portfolio (1 share of s1 & 2 shares of s2) by paying pl(T) +2p2(T)= I(T). But I(T) will also be received from the seller of the futures contract that we bought thus making it a zero sum transaction at time T. Thus the $1 profit that we made at time to does not change whatever may be the pricis of stock at time T. Thus it is a riskless arbitrage.