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Chapter 5: Problem 16

Suppose that \(F_{1}\) and \(F_{2}\) are two futures contracts on the same commodity with times to maturity, \(t_{1}\) and \(t_{2}\), where \(t_{2}>t_{1} .\) Prove that $$ F_{2} \leqslant F_{1} e^{n\left(t_{2}-t_{1}\right)} $$ where \(r\) is the interest rate (assumed constant) and there are no storage costs. For the purposes of this problem, assume that a futures contract is the same as a forward contract.

Short Answer

Expert verified
\( F_2 \leq F_1 e^{r(z_2-t_1)} \) follows from no-arbitrage, with constant interest rates and zero storage costs.

Step by step solution

01

Understanding the Problem

We have two futures contracts, \( F_1 \) with maturity \( t_1 \) and \( F_2 \) with maturity \( t_2 \). We need to show that \( F_2 \leq F_1 e^{r(z_2-t_1)} \) where the interest rate \( r \) is constant and there are no storage costs.
02

Consider Arbitrage Opportunities

Let's consider the arbitrage opportunity if \( F_2 > F_1 e^{r(z_2-t_1)} \). An arbitrageur could sell the more expensive futures contract \( F_2 \) and invest the proceeds into the cheaper futures contract \( F_1 \). This should yield a risk-free profit, which suggests \( F_2 \) cannot be greater than \( F_1 e^{r(z_2-t_1)} \) in an efficient market.
03

Analyzing the Futures Contract Relationship

Since futures and forward contracts are equivalent under the given conditions, and because storing costs are zero, the price at time \( t_1 \) for the futures contract expiring at \( t_2 \) must incorporate the rate of interest compounded, which ensures that \( F_2 \) should adjust accordingly to bond with the adjusted present value of \( F_1 \).
04

Mathematical Proof

Under no arbitrage conditions, if \( F_2 > F_1 e^{r(z_2-t_1)} \), an arbitrager would exploit the inefficiency, driving \( F_2 \) down to \( F_1 e^{r(z_2-t_1)} \) or below. Thus, proving \( F_2 \leq F_1 e^{r(z_2-t_1)} \).
05

Conclusion

In a no-arbitrage condition with no storage costs, the only possible situation is \( F_2 \leq F_1 e^{r(z_2-t_1)} \), aligning future prices with interest rate considerations over time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arbitrage Opportunities
In finance, an arbitrage opportunity arises when an investor can earn a risk-free profit by simultaneously buying and selling an asset or related assets in different markets. Arbitrage is the practice of exploiting these price differences to earn a profit without any risk. For instance, if you could buy a commodity for a lower price in one market and sell it instantly at a higher price in another, you'd be engaging in arbitrage.
  • Arbitrage relies on the concept of market efficiency. This means markets quickly adjust to eliminate price differences.
  • Arbitrageurs are traders who actively look for these opportunities to earn consistent returns.
  • In the context of futures contracts, arbitrage ensures that prices remain aligned with market conditions and interest rates.
Without arbitrage, markets would not be able to maintain orderly pricing of financial instruments, thus potentially leading to inefficiencies. Utilizing arbitrage opportunities leads to a balance in pricing across related markets.
No-Arbitrage Condition
The no-arbitrage condition is a fundamental concept in finance. It states that in a well-functioning market, it should not be possible to generate risk-free profits. If someone could make a risk-free profit, this would encourage others to exploit the same opportunity, eventually eliminating it as prices adjust.
  • This principle is closely related to arbitrage, as it represents the equilibrium that arises once all arbitrage opportunities have been removed.
  • In the pricing of futures contracts, the no-arbitrage condition dictates that prices should adjust in such a way that no risk-free profit can be achieved through buying and selling these contracts.
  • For it to hold true, the price difference between futures contracts must reflect the compounding interest rate over the time between contracts.
The no-arbitrage condition essentially ensures fair pricing and reflects the consistent application of interest rates in the valuation of financial instruments.
Interest Rate Compounding
Interest rate compounding is the process where the interest earned on an investment is reinvested, so that in subsequent periods, interest is then earned on the initial principal and on the accumulated interest from previous periods. When it comes to futures and forward contracts, compounded interest rates are crucial for understanding how future prices are determined.
  • In financial markets, compounding enables investors to grow their investments more quickly over time.
  • The nature of compounding is exponential, meaning the value of investments grows at an increasing rate.
  • The mathematical expression for continuous compounding is given by \( e^{rt} \), where \( r \) is the interest rate and \( t \) is time.
Understanding compounding is essential for deriving the correct futures price which incorporates the time value of money, aligning with the efficient market hypothesis. This ensures future contracts reflect future value as adjusted by current interest rates.
Forward Contracts
Forward contracts are agreements to buy or sell an asset at a predetermined price and date in the future. Unlike futures, which are standardized and traded on exchanges, forward contracts are customizable agreements typically traded over-the-counter (OTC) between parties.
  • They allow parties to lock in prices for commodities, currencies, or other financial instruments, offering a hedge against future price fluctuations.
  • Because they are customizable, forward contracts can be tailored to fit the specific needs of the contract parties, with specifics such as quantity, commodity, and expiration date all adjustable.
  • While forward contracts offer flexibility, they carry counterparty risk. This means there's a risk that one party may default on the contract obligations.
Understanding the nature of forward contracts helps in appreciating how future prices can be managed and how these contracts provide more precise risk management tools in financial portfolios.

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Most popular questions from this chapter

A trader owns gold as part of a long-term investment portfolio. The trader can buy gold for $$\$ 550$$ per ounce and sell it for $$\$ 549$$ per ounce. The trader can borrow funds at \(6 \%\) per year and invest funds at \(5.5 \%\) per year (both interest rates are expressed with annual compounding). For what range of 1 -year forward prices of gold does the trader have no arbitrage opportunities? Assume there is no bid-offer spread for forward prices.

A bank offers a corporate client a choice between borrowing cash at \(11 \%\) per annum and borrowing gold at \(2 \%\) per annum. (If gold is borrowed, interest must be repaid in gold. Thus, 100 ounces borrowed today would require 102 ounces to be repaid in 1 year.) The risk-free interest rate is \(9.25 \%\) per annum, and storage costs are \(0.5 \%\) per annum. Discuss whether the rate of interest on the gold loan is too high or too low in relation to the rate of interest on the cash loan. The interest rates on the two loans are expressed with annual compounding. The risk-free interest rate and storage costs are expressed with continuous compounding.

Explain what happens when an investor shorts a certain share.

Explain carefully what is meant by the expected price of a commodity on a particular future date. Suppose that the futures price for crude oil declines with the maturity of the contract at the rate of \(2 \%\) per year. Assume that speculators tend to be short crude oil futures and hedgers tend to be long. What does the Keynes and Hicks argument imply about the expected future price of oil?

Suppose that the risk-free interest rate is \(10 \%\) per annum with continuous compounding and that the dividend yield on a stock index is \(4 \%\) per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405 . What arbitrage opportunities does this create?
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