Question

The spot price for gold is $1,200 and the the 6-month risk-free rate is 3% continuously compounded.

(1) Calculate the 6-month Futures price of gold.

(2) Describe a trading strategy to make arbitrage profits if the quoted futures price is $10 lower than the fair value you calculate in Part (1). Show the cash flows to each element of your trading strategy.

(3) Describe a trading strategy to make arbitrage profits if the quoted futures price is $10 more than the fair value you calculate in Part(1). Show the cash flows to each element of your trading strategy.

Answer 1

1.) 6 month future price of gold = spot price * e^rt (in case of continuous compounding)

where, e = 2.71828 (mathematical constant used for continuous compounding rates)

r = rate of interest

t = time period for which it was compounded

6 month future price of gold = $1200*(2.71828)^(0.06*1)

6 month future price of gold = $1200*1.06(using calculator)

6 month future price of gold = $1272

2.) trading strategy to make arbitrage profits if the quoted futures price is $10 lower than the fair value you calculate in Part (1)

Quoted price will be $1272-$10 = $1262, since quoted price is lower than fair value, arbitrage can be achieved with following steps:

Step 1: Sale gold today in cash market(assuming trader has stock of gold, otherwise he can not be benefitted) at the spot price, deposit the proceeds for 6 months @3% componded continuously

   Cash inflow at 6 month end= $1200(sale proceeds received)*(2.71828)^(0.06*1) as continous compounding

Cash inflow at 6 month end = $1272(withrawal of deposited amount)

Step 2: Purchase gold future at the quoted price i.e $1262 at the 6 month end

Cash outflow at 6 month end = $1262(contract will be settled at quoted price irrespective of actual price on that date)

Net cash flow = Inflow in step 1 - outflow in step 2

  Net cash flow = $1272-$1262 = $10(gain from arbitrage)

3.) trading strategy to make arbitrage profits if the quoted futures price is $10 more than the fair value you calculate in Part(1)

Quoted price will be $1272+$10 = $1282, since quoted price is higher than fair value, arbitrage can be achieved with following steps:

Step 1: Buy gold today in cash marketat the spot price, by borrowing $1200 for 6 months @3% componded continuously

Cash outflow at 6 month end= $1200(borrowed amount)*(2.71828)^(0.06*1) as continous compounding

Cash outflow at 6 month end = $1272(payment of borrowing)

Step 2: Sale gold future at the quoted price i.e $1282 at the 6 month end

Cash inflow at 6 month end = $1282(contract will be settled at quoted price irrespective of actual price on that date)

Net cash flow = Inflow in step 2 - Outflow in step 1

  Net cash flow = $1282-$1272 = $10(gain from arbitrage)