Question

1. the spot price on the S&P 500 is $20,000 and the 1-year futures price is $20,609.09. The 1-year risk-free rate is 3% per annum with continuous compounding.

   1. Draw the profit and payoff function for the long future. (Provide labels for the axes and label a point on the functions above, below and at the futures price)

   2. Note an S&P 500 futures contract is for 250 times the S&P 500. Calculate what the payoff and profit at expiration is if the spot price is _______.

      i. $19,500 ii. $20,000 iii. $21,100

Answer 1

Sol 1: The profit and payoff function for the long future. Spot Price if being compounded at risk free rate of 3% p.a continuously then after one year the spot price could be = Spot Price* e^rt = Spot Price = $20000*e^0.03 = $ 20609.09. In this case Spot Price and Future Price are same then there is no Profit / Loss and it is at the Futures Price.

Solution (i) Profit and Payoff is spot price is $19500 and 1 year future price is $20,609.09. Spot price after 1 year @ Risk Free Rate @ 3% being continuously compounding =19500*e^0.03*1 = $20,093.86. In this case we should short the Futures thus our payoff should be Future Price - Spot Price after Year 1 i.e. $20,609.09-$20,093.86 = $ 515.23 the profit.

(ii) Profit and Payoff is spot price is $20000 and 1 year future price is $20,609.09. Spot price after 1 year @ Risk Free Rate @ 3% being continuously compounding =20000*e^0.03*1 = $20,609.09. In this case we should our payoff should be Future Price = Spot Price after Year 1 i.e. $20,609.09-$20,609.09 = $ 0.00 profit/loss.This is at the future price.

(iii) Profit and Payoff is spot price is $21,100 and 1 year future price is $20,609.09. Spot price after 1 year @ Risk Free Rate @ 3% being continuously compounding =21000*e^0.03*1 = $21,742.59. In this case we should go Long on Futures thus our payoff should be Spot Price - Future Price after Year 1 i.e. $21,742.59-$20,609.09 = $ 1,133.50 the profit.