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Unformatted text preview: STAT 2820 Chapter 6 Futures/Forward Price by K.C. Cheung 6.1 No-Arbitrage Principle An arbitrage opportunity allows one to get something from nothing, i.e. to make a sure profit with no risk → free lunch ! A fundamental idea in modern financial economics is that there is no free lunch → no arbitrage. Hence if two investments have the same payoffs in the future, they must sell at the same price (i.e. ‘law of one price’). However, in reality, the prices can be different due to the existence of transaction cost, such as commissions, information cost. 6.2 Model for forward price 6.2.1 Assume the market is perfect , i.e. the following are true: 1. no transaction cost in trading (e.g. brokage fee), no tax; 2. borrowing rate is the same as the lending rate; 3. no credit risk in lending money; 4. short selling is always possible. Then the no-arbitrage forward price today is given by F = S e rT , where T is the time until delivery date for the forward contract (in years), S is the price of the underlying asset today, r is the risk-free interest rate compounded continuously. This formula means that the forward price that could exclude arbitrage opportunity is given by the accumulated value at maturity of the spot price of the underlying asset. 6.2.2 We are now going to prove the above result using the no-arbitrage principle. Recall that it costs nothing to long or short a forward contract at time 0. Let F be the unknown forward price. We want to express F in terms of S ,r,T . Consider the following trading strategy: Time Transactions Cash Flow ($) t = 0 Borrow F e- rT + F e- rT Buy 1 unit of underlying asset- S Short one forward contract Total CF F e- rT- S t = T Repay loan- F Sell 1 unit of underlying asset + F Total CF STAT 2820 Chapter 6 2 Observe that the total cashflow at the maturity date is 0. By the no-arbitrage principle, if a trading strategy has zero cashflow at maturity, the cost of constructing such strategy at time 0 must also be zero. The reason is that if the cost of constructing such strategy is negative, such strategy will bring a sure profit, and hence is an arbitrage; if the cost of constructing such strategy is positive, such strategy will bring a sure loss, and hence an opposite strategy will be an arbitrage. Therefore, we have F e- rT- S = 0 , which is the stated result. 6.2.3 The proof above also shows that the no-arbitrage forward price is unique : in a perfect market, there is one and only one forward price that could exclude arbitrage opportunity. 6.2.4 The following example shows how to construct arbitrage when the actual forward price is different from the no arbitrage forward price. Example (No arbitrage forward price): On January 1, the price of a certain non-dividend-paying stock is $40 per share. A futures contract will mature in one year. Risk-free interest rate is 10% p.a., com- pounded annually. Each futures contract is on 10 shares of the stock. Assuming a perfect market, the futures price (per share of stock) must be...
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