3 Determination of Forward and Futures Prices
- HULL, John. Options, futures, and other derivatives. Ninth edition. Harlow: Pearson, 2018. ISBN 978-1-292-21289-0.
- Chapter 5 - Determination of Forward and Futures Prices
- PIRIE, Wendy L. Derivatives. Hoboken: Wiley, 2017. CFA institute investment series. ISBN 978-1-119-38181-5.
- Chapter 2 - Basics of Derivative Pricing and Valuation
- Chapter 3 - Pricing and Valuation of Forward Commitments
Learning Outcomes:
- Understand the principles of pricing and valuation for forward and futures contracts.
- Apply pricing models to specific assets, including commodities and financial instruments.
- Analyze the relationship between futures prices and expected spot prices.
- Understand the structure and valuation of Forward Rate Agreements (FRAs).
3.1 Pricing and Valuation of Forward/Futures Contracts
Pricing of forward/futures contracts entails determining the fair market price or rate at the initiation of the contract. This process ensures that the contract’s initial value is neutral, aligning the interests of both parties without giving an undue advantage to either side.
Valuation, on the other hand, is the process of assessing the contract’s current value after its initiation. The value fluctuates based on market conditions, reflecting the gain or loss to the contract holder.
- An investor enters a long position in a December Gold futures contract on 2 November, agreeing to a price of $1,250/oz. Initially, the contract’s value is set to $0, indicating a fair agreement based on current market expectations.
- Post-initiation, the contract’s value is subject to change due to market dynamics. For instance, if on 3 November, the market price for December Gold futures falls to $1,225/oz, the value of the investor’s long position declines, rendering a negative value to their investment.
3.2 Pricing and Valuation Methodology
Risk Aversion: Investors require compensation for taking on additional risk, reflecting the principle that higher risk should be rewarded with higher potential returns.
Risk-Neutral Pricing: Under this approach, it’s assumed that investors are indifferent to risk. The pricing of derivatives through arbitrage opportunities ensures that the portfolio, combining the derivative and its underlying asset, yields the risk-free rate of return.
Arbitrage-Free Pricing: This methodology prices derivatives based on the assumption that the market operates under risk-neutral conditions and is free of arbitrage opportunities, ensuring no risk-free profits can be made from market inefficiencies.
The framework for pricing and valuation operates on the principle that market prices adjust to preclude arbitrage profits. The Law of One Price asserts that identical cash flows must have the same price, irrespective of future outcomes.
Arbitrageurs follow two cardinal rules:
- Do not use your own money.
- Do not take any price risk.
3.2.1 Short Selling
Short selling entails the sale of securities not owned by the seller, typically facilitated by borrowing these securities from a broker. The seller aims to buy back the securities at a lower price to profit from the difference. Obligations include covering dividends and possibly a borrowing fee.
Assuming you short sell 100 shares at $100 each and close out the short position three months later at $90, while a $3 per share dividend is issued during this period:
- Your profit would be the difference in share prices minus the dividends paid, highlighting the risks and rewards of short selling.
\[100 \times (100 - 90 - 3) = \$700\]
- Conversely, buying 100 shares would result in a loss if the share price decreases.
\[100 \times (90 - 100 + 3) = \$-700\]
3.2.2 Consumption vs. Investment Assets
Investment Assets: These are assets like gold, stocks, and bonds, held primarily for their potential to appreciate in value over time.
Consumption Assets: Assets such as copper or oil are classified as consumption assets, held mainly for their utility in production or consumption rather than for investment.
3.3 Forward Price for an Investment Asset
3.3.1 Assumptions and Notation
Assumptions
- Market participants incur no transaction costs when trading.
- All net trading profits are taxed at the same rate for market participants.
- Market participants can borrow and lend money at the identical risk-free interest rate.
- Arbitrage opportunities are exploited immediately by market participants.
Notation
- \(S_0\): Current spot price of the asset.
- \(F_0\): Today’s futures or forward price of the asset.
- \(T\): Time to the delivery date of the contract (in years).
- \(r\): Annualized risk-free interest rate over the period \(T\).
3.3.2 Arbitrage Example
Consider an asset that provides no income, with a current price of $40, subject to an annual interest rate of 5%, and associated with a forward contract that has a maturity of 3 months.
Case 1: Forward Price = $43
- Action now: Borrow $40 at 5% annual interest for 3 months. Purchase one unit of the asset and enter into a forward contract to sell it in 3 months at $43.
- Action in 3 months: Sell the asset at the forward price of $43. Repay the loan, which amounts to $40.50 with interest.
- Profit realized: $2.50.
Case 2: Forward Price = $39
- Action now: Short sell one unit of the asset for $40. Invest the proceeds at 5% annual interest for 3 months. Enter into a forward contract to buy the asset in 3 months at $39.
- Action in 3 months: Buy back the asset at the forward price of $39. Close the short position and retrieve $40.50 from the investment.
- Profit realized: $1.50.
3.3.3 The Forward Price Formula
The forward price \(F_0\) for an investment asset that does not provide income and is deliverable in \(T\) years is given by the following formula, where \(r\) is the annualized risk-free interest rate for the period \(T\).
\[F_0 = S_0 e^{rT}\]
In the given example, with \(S_0 = 40\), \(T = 0.25\) (3 months), and \(r = 0.05\) (5%), the forward price is calculated as:
\[F_0 = 40 e^{0.05 \times 0.25} = 40.50\]
This calculation demonstrates that the no-arbitrage forward price aligns with the theoretical value, preventing arbitrage opportunities under these conditions.
3.3.4 Adjustments for Known Income or Yield
When the asset provides a known income or yield, the forward price adjusts to account for this. For an asset generating a known income \(I\) (expressed as the present value of the income) over the contract’s life, the formula modifies to:
\[F_0 = (S_0 - I)e^{rT}\]
For assets providing a yield \(q\) (expressed as a continuous compounding rate), the forward price formula adjusts to:
\[F_0 = S_0 e^{(r-q)T}\]
3.4 Valuing a Forward Contract
Initially, a forward contract’s value is zero, except for potential impacts from the bid-offer spread. This neutrality in value reflects the agreement’s fairness based on current market conditions.
As time progresses, the value of the forward contract can become positive or negative, reflecting changes in market conditions and the underlying asset’s price relative to the contract’s terms.
Let \(K\) represent the agreed-upon delivery price in the contract, and \(F_0\) denote the forward price for a contract negotiated today for future delivery.
The valuation of forward contracts can be understood by comparing contracts with different delivery prices. This comparison reveals:
The value of being in a long position in a forward contract, where one has agreed to buy the asset, is \((F_0 - K) e^{-rT}\). This formula represents the present value of the profit from the contract if the forward price \(F_0\) set today is higher than the contract’s delivery price \(K\).
Conversely, the value of a short forward contract, where one has agreed to sell the asset, is \((K - F_0) e^{-rT}\). This expresses the present value of the profit from the contract if the delivery price \(K\) exceeds today’s forward price \(F_0\).
These valuations underscore the importance of the risk-free rate \(r\) and the time to delivery \(T\) in determining the present value of future contract profits or losses.
3.5 Forward vs. Futures Prices
While forward and futures contracts are similar in their function of agreeing to buy or sell an asset at a future date, their pricing can diverge under certain conditions, despite the common assumption that they are equal when the maturity and asset prices coincide. One notable exception is Eurodollar futures, which often exhibit pricing anomalies due to their specific market characteristics.
The theoretical distinction in pricing between forwards and futures arises primarily from the volatility and uncertainty of interest rates:
When there is a strong positive correlation between interest rates and the asset’s price, futures prices tend to be slightly higher than forward prices. This is because the daily settlement of futures can lead to a net gain in environments where rising interest rates accompany rising asset prices, due to the reinvestment of gains at higher rates.
Conversely, a strong negative correlation between interest rates and the asset’s price suggests that forward prices may exceed futures prices. In such scenarios, the absence of daily settlement in forward contracts avoids the potential loss from having to reinvest at lower interest rates.
3.6 Forward Price for Specific Assets
3.6.1 Stock Index
A stock index can be regarded as an investment asset that effectively pays a dividend yield, analogous to the income generated by holding the underlying stocks.
The relationship between the futures price (\(F_0\)) and the spot price (\(S_0\)) of the index is captured by the formula:
\[F_0 = S_0 e^{(r-q)T}\]
Here, \(q\) represents the average dividend yield of the portfolio reflected by the index over the contract’s life.
This formula assumes the index acts as a tradable investment asset, with changes in the index mirroring changes in the value of a corresponding tradable portfolio. For instance, the Nikkei 225, when viewed purely as a numerical value without considering its underlying assets, does not qualify as an investment asset in this context.
Index arbitrage exploits discrepancies between the futures price and the spot price adjusted for the dividend yield and risk-free rate. When \(F_0\) exceeds \(S_0 e^{(r-q)T}\), arbitrageurs purchase the underlying stocks of the index while selling futures. Conversely, if \(F_0 < S_0 e^{(r-q)T}\), they buy futures and short sell the index’s underlying stocks.
This strategy requires executing trades in futures and various stocks simultaneously, often facilitated by computer algorithms to manage the complexity and timing. However, real-world frictions sometimes prevent perfect execution, leading to temporary deviations from the theoretical no-arbitrage relationship.
3.6.2 Exchange Rates
A foreign currency functions similarly to a security that yields interest, where the yield is the foreign country’s risk-free interest rate.
Given \(r_f\) as the foreign risk-free interest rate, the forward exchange rate formula is:
\[F_0 = S_0 e^{(r-r_f)T}\]
Consider two strategies for converting 1,000 units of a foreign currency into dollars by time \(T\), where \(S_0\) represents the current spot exchange rate, \(F_0\) the forward exchange rate, and \(r\) and \(r_f\) the domestic and foreign risk-free interest rates, respectively.
Strategy 1: Convert the 1,000 units of foreign currency into dollars at the future time \(T\) by initially investing these units at the foreign risk-free rate \(r_f\). Then, agree to a forward contract to sell the resulting amount for dollars at time \(T\).
Strategy 2: Immediately exchange the 1,000 units of foreign currency for dollars at the current spot rate \(S_0\), then invest this dollar amount at the domestic risk-free rate \(r\) until time \(T\).
The principle of no arbitrage implies that both strategies must yield the same outcome in dollar terms to prevent free profit opportunities. This condition leads to the following equation:
\[1,000 \times e^{r_f \times T} \times F_0 = 1,000 \times S_0 \times e^{r \times T}\]
Simplifying this equation provides the formula for the forward exchange rate:
\[F_0 = S_0 e^{(r-r_f)T}\]
3.6.3 Commodities
For consumption assets like commodities, storage costs represent a form of negative income, impacting the forward price:
\[F_0 \leq S_0 e^{(r+u)T}\]
Here, \(u\) stands for the storage cost per unit time as a percentage of the asset’s value. Alternatively, incorporating the present value of storage costs \(U\):
\[F_0 \leq (S_0 +U) e^{rT}\]
These formulas accommodate the costs associated with holding and storing physical commodities, from agricultural products to metals, affecting their forward pricing.
3.6.4 The Cost of Carry
The cost of carry (\(c\)) combines storage costs, interest expenses, and any income earned. For investment assets, the forward price formula simplifies to:
\[F_0 = S_0 e^{cT}\]
In contrast, for consumption assets, the formula accounts for potential lower bounds due to additional costs, hence:
\[F_0 \leq S_0 e^{cT}\]
The concept of a convenience yield (\(y\)) is introduced for consumption assets, acknowledging the benefits from holding the physical asset as opposed to a derivative position. It’s defined such that:
\[F_0 = S_0 e^{(c-y)T}\]
3.7 Futures Prices and Expected Spot Prices
When analyzing futures contracts, we consider the expected return required by investors, denoted as \(k\). This expected return plays a critical role in determining the futures price \(F_0\) relative to the expected spot price at maturity \(E(S_T)\).
An investment strategy involving futures contracts can be described as follows:
By investing an amount \(F_0 e^{-rT}\) at the risk-free rate \(r\) and entering a long futures contract, an investor can secure a cash inflow of \(S_T\) at maturity.
The present value (PV) of this investment strategy is calculated as \(-F_0 e^{-rT} + E(S_T) e^{-kT}\), where \(E\) denotes the expected value.
In efficient markets, investments are typically priced to yield a zero net present value (NPV). This principle leads to the fundamental pricing relation for futures contracts: \[F_0 = E(S_T)e^{(r-k)T}\]
The relationship between futures prices and expected future spot prices varies with the underlying asset’s systematic risk:
- No Systematic Risk: When the asset carries no systematic risk, the expected return from the asset \(k\) equals the risk-free rate \(r\), leading to \(F_0 = E(S_T)\).
- Positive Systematic Risk: For assets with positive systematic risk, such as stock indices, \(k > r\) implies that futures prices will be lower than the expected spot prices \(F_0 < E(S_T)\).
- Negative Systematic Risk: Conversely, assets like gold, which may exhibit negative systematic risk during certain periods, have \(k < r\), resulting in futures prices exceeding expected spot prices \(F_0 > E(S_T)\).
Backwardation: This market condition occurs when futures prices are below the expected future spot prices, often observed in markets expecting lower future prices.
Contango: In contrast, a contango market condition is characterized by futures prices that exceed expected future spot prices, typically occurring in markets where future prices are anticipated to rise.
3.8 Forward Rate Agreement (FRA)
An FRA is a financial contract that allows parties to exchange a fixed interest rate for a floating rate (often based on LIBOR, SOFR, or SONIA) at a future date, applied to a specified principal amount. The agreement does not involve the exchange of principal but rather the difference in interest payments based on the agreed rates. The value of an FRA at inception is zero, as the fixed rate equals the forward rate. The value changes as the forward rate changes over time.
Consider an FRA where Party A and Party B agree to exchange a fixed rate of 3% for three-month SOFR on a principal of $100 million in two years. The interest rates are compounded quarterly.
- Position of Parties: Party A will pay the floating SOFR rate and receive a fixed 3%. Party B takes the opposite position.
- Outcome: If the three-month SOFR rate is 3.5% in two years, Party A receives a net payment from Party B calculated as follows:
\[ \$100,000,000 \times (0.035 - 0.030) \times 0.25 = \$125,000 \]
This amount would ideally be paid at 2.25 years. However, considering the advance determination of SOFR, the actual payment is adjusted to its present value and made at the 2-year mark (discounted for three months at 3.5%).
- Zero Rates: The n-year zero rate (or spot rate) is the interest rate for an investment lasting n years, with interest and principal paid at the end of the term. No intermediate payments are made.
A 5-year zero rate of 5% per annum, under continuous compounding, means $100 grows to $128.40 over 5 years, calculated as \(100 \times e^{0.05 \times 5}\).
- Forward Rates: Forward rates represent the future interest rates implied by current zero rates for future periods. Derived from zero rates for various maturities, indicating the cost of borrowing or lending money in the future, as implied by today’s rates.
For calculating a forward rate between times \(T_1\) and \(T_2\), given zero rates \(R_1\) and \(R_2\) for these times, the forward rate, \(R_F\), is:
\[R_F = \frac{R_2 T_2 - R_1 T_1}{T_2 - T_1}\]
| Year | Zero rate for an n-year investment (% per annum) | Forward rate for nth year (% per annum) |
|---|---|---|
| 1 | 3.0 | |
| 2 | 4.0 | 5.0 |
| 3 | 4.6 | 5.8 |
| 4 | 5.0 | 6.2 |
| 5 | 5.3 | 6.5 |
To illustrate, consider calculating the forward rate for the fourth year using the data provided in the Table. Assuming \(T_1 = 3\), \(T_2 = 4\), \(R_1 = 4.6\%\), and \(R_2 = 5.0\%\), the forward rate, \(R_F\), is calculated as follows:
\[ R_F = \frac{0.05 \times 4 - 0.046 \times 3}{4 - 3} = 6.2\% \]
This result indicates the forward rate for the fourth year is 6.2% per annum, which is higher than the zero rate for the same period due to the upward sloping yield curve.
The value of an FRA at inception is zero since the fixed rate typically equals the forward rate for the reference rate at that time. However, the value can change over time as the forward rate fluctuates. The valuation of an FRA can be determined by comparing the difference between the agreed fixed rate (\(R_K\)) and the current forward rate (\(R_F\)) over the contract period (\(t\)), discounted back to the present value.
3.9 Practice Questions and Problems
3.9.1 Basics of Forward and Futures Contracts
- Explain what happens when an investor shorts a certain share.
- What is the difference between the forward price and the value of a forward contract?
- What is meant by (a) an investment asset and (b) a consumption asset. Why is the distinction between investment and consumption assets important in the determination of forward and futures prices?
- Explain carefully why the futures price of gold can be calculated from its spot price and other observable variables whereas the futures price of copper cannot.
- Explain carefully the meaning of the terms convenience yield and cost of carry. What is the relationship between futures price, spot price, convenience yield, and cost of carry?
- What is the cost of carry for (a) a non-dividend-paying stock, (b) a stock index, (c) a commodity with storage costs, and (d) a foreign currency?
3.9.2 Calculation of Forward and Futures Prices
- Suppose that you enter into a three-month forward contract on a non-dividend-paying stock when the stock price is $108 and the risk-free interest rate (with continuous compounding) is 4% per annum. What is the forward price?
\(F_0 = 109.085\)
- A four-months long forward contract on a non-dividend-paying stock is entered into when the stock price is $150 and the risk-free rate of interest is 5.7% per annum with continuous compounding.
- What are the forward price and the initial value of the forward contract?
- Two months later, the price of the stock is $168 and the risk-free interest rate is still 5.7%. What are the forward price and the value of the forward contract?
Forward price \(F_0 = 152.88\); initial value \(0\); final value \(169.6\)
- The risk-free rate of interest is 4.1% per annum with continuous compounding, and the dividend yield on a stock index is 6.2% per annum. The current value of the index is 2445. What is the one-month futures price?
\(F_0 = 2440.725\)
- A stock index currently stands at 725. The risk-free interest rate is 7.6% per annum (with continuous compounding) and the dividend yield on the index is 1.8% per annum. What should the futures price for a three-month contract be?
\(F_0 = 735.589\)
- An index is 550. The three-month risk-free rate is 4.60% per annum and the dividend yield over the next three months is 5.80% per annum. The six-month risk-free rate is 5.34% per annum and the dividend yield over the next six months is 4.93% per annum. Estimate the futures price of the index for three-month and six-month contracts. All interest rates and dividend yields are continuously compounded.
3-month \(F_0 = 548.352\); 6-month \(F_0 = 551.129\)
- The spot price of silver is $11 per ounce. The storage costs are $0.25 per ounce payable quarterly in advance. Assuming that interest rates are 1.80% per annum for all maturities, calculate the futures price of silver for delivery in nine months.
\(F_0 = 11.91\)
- The spot price of oil is $39 per barrel and the cost of storing a barrel of oil for one year is $1.2, payable at the end of the year. The risk-free interest rate is 8.60% per annum, continuously compounded. What is an upper bound for the one-year futures price of oil?
\(F_0 = 43.70\)
3.9.3 Arbitrage and Risk Management
- Suppose that the risk-free interest rate is 3.00% per annum with continuous compounding and that the dividend yield on a stock index is 0.60% per annum. The index is standing at 3646, and the futures price for a contract deliverable in ten months is 3701. What arbitrage opportunities does this create?
\(F_0 = 3719.654\)
- The eight-month interest rates in Switzerland and the United States are, respectively, 3.60% and 5.40% per annum with continuous compounding. The spot price of the Swiss franc is $0.9. The futures price for a contract deliverable in two months is also $0.9. What arbitrage opportunities does this create?
\(F_0 = 0.911\)
- When a known future cash outflow in a foreign currency is hedged by a company using a forward contract, there is no foreign exchange risk. When it is hedged using futures contracts, the daily settlement process does leave the company exposed to some risk. Explain the nature of this risk. Assume that the forward price equals the futures price. In particular, consider whether the company is better off using a futures contract or a forward contract when
- The value of the foreign currency falls rapidly during the life of the contract
- The value of the foreign currency rises rapidly during the life of the contract
- The value of the foreign currency first rises and then falls back to its initial value
- The value of the foreign currency first falls and then rises back to its initial value