3 Determination of Forward and Futures Prices

References

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 determines the fair market rate at contract initiation, ensuring neutrality for both parties.
Valuation assesses the contract’s value after initiation, fluctuating based on market conditions.

Example: Forward Price vs. Value

On November 2, an investor takes a long position in a December Gold futures contract at $1,250/oz. At initiation, the contract’s value is $0 (fair market agreement).
On November 3, if the market price drops to $1,225/oz, the contract’s value turns negative, reflecting a loss for the investor.

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.
Law of One Price: asserts that identical cash flows must have the same price, irrespective of future outcomes.
Arbitrage Principles:
1. Do not use your own money.
2. Do not take any price risk.
Short Selling: Short selling involves selling borrowed securities, aiming to repurchase them at a lower price. Sellers must cover dividends and borrowing fees.

Example: Short Selling

Short 100 shares at $100 each and close the position at $90, while a $3/share dividend is issued:

Profit Calculation:
\[ 100 \times (100 - 90 - 3) = 700 \]
If instead you buy 100 shares, a price drop results in a loss:
\[ 100 \times (90 - 100 + 3) = -700 \]

Investment Assets: Held for potential value appreciation over time (e.g., gold, stocks, bonds).
Consumption Assets: Used in production or consumption (e.g., oil, copper).

3.2 Forward Price for an Investment Asset

Assumptions:

No transaction costs in trading.
All net trading profits are taxed equally.
Borrowing and lending occur at the same risk-free interest rate.
Arbitrage opportunities are immediately exploited.

Notation:

$S_0$: Current spot price of the asset.
$F_0$: Today’s futures or forward price of the asset.
$T$: Time to contract maturity (in years).
$r$: Annualized risk-free interest rate over $T$.

Arbitrage Example

Consider an asset that provides with no income, priced at $40, with a 5% annual interest rate and a 3-month forward contract.

Case 1: Forward Price = $43

Now: Borrow $40, buy the asset, and enter a forward contract to sell at $43.
In 3 months: Sell at $43, repay the loan ($40.50 with interest).
Profit: $2.50.

Case 2: Forward Price = $39

Now: Short sell the asset for $40, invest the proceeds at 5% interest, and enter a forward contract to buy at $39.
In 3 months: Buy back at $39, close the short position, and receive $40.50 from the investment.
Profit: $1.50.

Forward Price Formula

For an investment asset with no income, the no-arbitrage forward price is:

\[ F_0 = S_0 e^{rT} \]

Example: Eliminate arbitrage

Using $S_0 = 40$, $T = 0.25$ (3 months), and $r = 0.05$ (5%):

\[ F_0 = 40 e^{0.05 \times 0.25} = 40.50 \]

This aligns with the theoretical fair price, ensuring no arbitrage.

If the asset generates a known income $I$ (expressed as the present value of the income), the formula adjusts to:

\[ F_0 = (S_0 - I)e^{rT} \]

If the asset provides a yield $q$ (expressed as a continuously compounded annualized rate):

\[ F_0 = S_0 e^{(r - q)T} \]

3.3 Valuing a Forward Contract

At initiation, a forward contract has zero value, ensuring fairness for both parties (excluding bid-offer spreads). Over time, its value can become positive or negative as market conditions change.

Let:

$K$ = Agreed delivery price in the contract.
$F_0$ = Forward price for a contract negotiated today.

The value of a forward contract depends on the difference between the agreed delivery price and the current forward price, discounted at the risk-free rate:

Long position (buying the asset):
\[ V_{\text{long}} = (F_0 - K) e^{-rT} \]
Short position (selling the asset):
\[ V_{\text{short}} = - (F_0 - K) e^{-rT} \]

These formulas show how the risk-free rate $r$ and time to maturity $T$ impact the present value of expected gains or losses.

3.4 Forward vs. Futures Prices

Although forward and futures contracts both involve agreeing to buy/sell an asset at a future date, their prices may diverge due to interest rate volatility.

If interest rates and asset prices are positively correlated $\rightarrow$ Futures price > Forward price
- Daily settlement allows gains to be reinvested at higher rates.
If interest rates and asset prices are negatively correlated $\rightarrow$ Futures price < Forward price
- No daily settlement avoids reinvesting at lower rates.

One notable example is Eurodollar futures, where pricing anomalies arise due to unique market characteristics.

3.5 Forward Price for Specific Assets

Stock Index

A stock index is an investment asset that effectively pays a dividend yield, similar to the income generated by holding its underlying stocks.

The relationship between the futures price ($F_0$) and the spot price ($S_0$) is:

\[ F_0 = S_0 e^{(r-q)T} \]

where, $q$ represents the average dividend yield of the portfolio reflected by the index over the contract’s life.

Warning

This formula assumes the index represents a tradable investment asset. However, a purely numerical index (e.g., Nikkei 225 without its underlying assets) does not qualify as a true investment asset.

Index Arbitrage

Arbitrageurs exploit price discrepancies between futures and spot prices adjusted for dividend yield and the risk-free rate:

If $F_0 > S_0 e^{(r-q)T} \rightarrow$ Buy the underlying stocks, sell futures.
If $F_0 < S_0 e^{(r-q)T} \rightarrow$ Buy futures, short-sell the index’s underlying stocks.

High-frequency trading algorithms often execute these strategies. However, real-world frictions (e.g., execution delays, transaction costs) may prevent perfect arbitrage.

Exchange Rates

A foreign currency behaves like a security yielding interest, where the yield is the foreign risk-free rate ($r_f$). The forward exchange rate is given by:

\[ F_0 = S_0 e^{(r - r_f)T} \]

Exchange Rate Arbitrage

Consider two equivalent strategies for converting 1,000 units of a foreign currency into dollars by time $T$:

Invest in the foreign currency at rate $r_f$, then convert at the forward rate $F_0$.
Convert immediately at spot rate $S_0$, then invest in dollars at rate $r$.

No-arbitrage implies both must yield the same result, leading to:

\[ 1,000 \times e^{r_f T} \times F_0 = 1,000 \times S_0 \times e^{rT} \]

Solving for $F_0$ confirms the forward rate formula:

\[ F_0 = S_0 e^{(r - r_f)T} \]

Commodities

For consumption assets (e.g., oil, wheat, copper), storage costs act as negative income, affecting forward pricing:

\[ F_0 \leq S_0 e^{(r + u)T} \]

where:

$u$ = Storage cost per unit time as a percentage of the asset’s value.

Alternatively, if $U$ represents the present value of storage costs:

\[ F_0 \leq (S_0 + U) e^{rT} \]

These formulas accommodate the costs associated with holding and storing physical commodities, from agricultural products to metals, affecting their forward pricing.

The Cost of Carry

The cost of carry ($c$) represents the total cost of holding an asset, including:

Storage costs
Financing costs
Income earned (e.g., dividends, yields)

For investment assets, the forward price follows:

\[ F_0 = S_0 e^{cT} \]

For consumption assets, storage and financing costs may create lower price bounds:

\[ F_0 \leq S_0 e^{cT} \]

A convenience yield ($y$) represents the benefit of physically holding a consumption asset rather than a derivative. The forward price adjusts to:

\[ F_0 = S_0 e^{(c - y)T} \]

where $y$ reflects scarcity, supply-chain security, and non-financial benefits of physical possession.

3.6 Futures Prices and Expected Spot Prices

Contango ($F_t > S_t$): Futures prices are higher than the current or expected future spot price.
- Occurs when carrying costs (storage, interest) exceed convenience yield.
- Common in gold, silver, natural gas, and agricultural commodities.
- Contango is more common across most futures markets due to carrying costs.
Backwardation ($F_t < S_t$): Futures prices are lower than the expected future spot price.
- Happens when convenience yield is high (scarcity or strong demand for immediate delivery).
- Common in crude oil, perishable goods, and equity index futures.
Futures prices are poor predictors of future spot prices. Empirical evidence shows biases due to risk premia and market inefficiencies. Speculative demand, liquidity constraints, and irrational expectations can further distort pricing.
While futures prices provide useful signals, they should not be relied upon as perfect forecasts of future spot prices.

Tip

What is Contango and Backwardation - CME Institute

3.7 Forward Rate Agreement (FRA)

An FRA is a financial contract where parties exchange a fixed interest rate for a floating rate (e.g., LIBOR, SOFR, SONIA) at a future date, based on a specified notional principal.

No principal exchange occurs—only the difference in interest payments is settled.
The initial value of an FRA is zero since the fixed rate equals the forward rate at inception.
As the forward rate changes over time, the FRA’s value fluctuates.

Example: FRA

Party A and Party B agree to exchange a fixed rate of 3% for the three-month SOFR on a $100 million notional in two years (compounded quarterly).

Party A pays floating SOFR and receives a fixed 3%.
Party B takes the opposite position.

If in two years the SOFR rate is 3.5%, Party A receives a payment from Party B:

\[ 100,000,000 \times (0.035 - 0.030) \times 0.25 = 125,000 \]

This amount is discounted for three months at 3.5%, since payments are made at the two-year mark.

A zero rate is the interest rate for an investment lasting $n$ years, with interest and principal paid at maturity (no intermediate payments).
A forward rate represents the future interest rate implied by current zero rates for future periods. It reflects the cost of borrowing/lending at a future time.

For a forward rate between $T_1$ and $T_2$, given zero rate $R_1$ and $R_2$:

\[ R_F = \frac{R_2 T_2 - R_1 T_1}{T_2 - T_1} \]

Example: Forward Rate Calculation

Year	Zero Rate (% 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

For year 4, given $T_1 = 3$, $T_2 = 4$, $R_1 = 4.6\%$, and $R_2 = 5.0\%$:

\[ R_F = \frac{0.05 \times 4 - 0.046 \times 3}{4 - 3} = 6.2\% \]

This forward rate (6.2%) is higher than the zero rate (5.0%), reflecting an upward-sloping yield curve.

The value of an FRA is determined by:

The difference between the agreed fixed rate ($R_K$) and the current forward rate ($R_F$).
The contract period ($t$).
Discounting the difference to present value.

3.8 Practice Questions and Problems

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?
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?
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?

Solution

$F_0 = 109.085$

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.
- 1. What are the forward price and the initial value of the forward contract?
- 1. 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?

Solution

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?

Solution

$F_0 = 2440.725$

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?

Solution

$F_0 = 735.589$

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.

Solution

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.

Solution

$F_0 = 11.91$

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?

Solution

$F_0 = 43.70$

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?

Solution

$F_0 = 3719.654$

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?

Solution

$F_0 = 0.911$

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
- 1. The value of the foreign currency falls rapidly during the life of the contract
- 1. The value of the foreign currency rises rapidly during the life of the contract
- 1. The value of the foreign currency first rises and then falls back to its initial value
- 1. The value of the foreign currency first falls and then rises back to its initial value