Lesson 9.1: Pricing and Valuation of Forwards and Futures

Introduction

In this lesson, we will explore the pricing and valuation of forwards and futures contracts. Understanding these financial instruments is crucial for effective risk management and hedging strategies in finance. This lesson aims to build foundational knowledge around the no-arbitrage pricing principle and how to value these contracts during their life cycle. By the end of this lesson, you should be able to:

Understand the no-arbitrage pricing of forward and futures contracts.
Value positions held during the life of the contract.
Price forwards and futures contracts on various underlying assets.
Value existing forward or futures positions.
Explain the key concepts and terminology associated with forwards and futures.

No-Arbitrage Pricing of Forward Contracts

A forward contract is an agreement between two parties to buy or sell an asset at a specific future date for a price agreed upon today. One of the core principles of pricing these contracts is the no-arbitrage condition, which states that there should be no opportunity to earn a risk-free profit through arbitrage.

Pricing Forward Contracts

To price a forward contract, we begin with the fundamental idea that the forward price (F) must be set so that there is no arbitrage opportunity. The price of the underlying asset today (S) will grow at the risk-free rate (r) until the contract's maturity.

The formula for the forward price can be expressed as:

$$F = S e^{rt}$$

Where:
$F$ = forward price
$S$ = spot price (current price of the asset)
$r$ = risk-free interest rate (annualized)
$t$ = time to maturity (in years)

Example 1: Pricing a Forward Contract

Suppose the spot price of a stock today is $100, the risk-free interest rate is 5% (0.05), and the contract matures in 1 year. We can calculate the forward price:

Given:
$S = 100$
$r = 0.05$
$t = 1$

Using the formula:

$$F = 100 e^{0.05 \cdot 1} \approx 100 e^{0.05} \approx 100 \cdot 1.05127 \approx 105.13$$

Thus, the forward price of the stock is approximately $105.13.

Valuing Forward Contracts During Their Life

During the life of a forward contract, its value (V) can change as market conditions change. The value of a forward contract at time $t$ can be expressed as:

$$V_t = S_t - F e^{-rt}$$

Where:

$V_t$ = value of the forward contract at time $t$
$S_t$ = spot price of the underlying asset at time $t$
$F$ = forward price agreed upon at contract initiation
$r$ = risk-free interest rate

Example 2: Valuing a Forward Contract

Continuing from Example 1, if the spot price of the stock at time $t = 0.5$ years is $110, we can calculate the value of the forward contract:

Given:
$S_t = 110$
$F = 105.13$
$r = 0.05$

Using the formula:

$$V_t = 110 - 105.13 e^{-0.05 \cdot 0.5} \approx 110 - 105.13 \cdot 0.97531 \approx 110 - 102.56 \approx 7.44$$

The value of the forward contract at $t = 0.5$ years is approximately $7.44.

No-Arbitrage Pricing of Futures Contracts

Futures contracts are similar to forward contracts but are traded on exchanges and have standardized terms. Like forwards, futures prices are also derived under the no-arbitrage principle. The pricing relationship for futures contracts is primarily dictated by the same factors affecting forward contracts: spot price, risk-free rate, and time to maturity.

Pricing Futures Contracts

The futures price (F_T) can also be calculated using the following formula:

$$F_T = S e^{rt}$$

In this case:

$F_T$ = futures price at maturity
$S$ = spot price of the asset
$r$ = risk-free interest rate
$t$ = time to maturity (in years)

Example 3: Pricing a Futures Contract

Let's consider a contract for the same stock with a spot price of $100, a risk-free rate of 5%, and a maturity of 1 year:

Using the formula:

$$F_T = 100 e^{0.05 \cdot 1} \approx 105.13$$

The futures price at maturity is also approximately $105.13.

Valuing Futures Contracts During Their Life

The value of a futures contract can also change throughout its life. The value (V_T) of a futures contract at any time $T$ can be expressed as:

$$V_T = S_T - F_T e^{-r(T-t)}$$

Where:

$V_T$ = value of the futures contract at time $T$
$S_T$ = spot price at time $T$
$F_T$ = futures price at time of initial contract execution
$r$ = risk-free interest rate
$T$ = time to futures contract maturity
$t$ = current time in futures contract's life

Example 4: Valuing a Futures Contract

Assuming the spot price of the same stock at $T = 0.5$ years is $115, we can calculate the value of the futures contract:

Given:
$S_T = 115$
$F_T = 105.13$
$r = 0.05$
$T - t = 0.5$

Using the formula:

$$V_T = 115 - 105.13 e^{-0.05 \cdot 0.5} \approx 115 - 105.13 \cdot 0.97531 \approx 115 - 102.56 \approx 12.44$$

The value of the futures contract at $T = 0.5$ years is approximately $12.44.

Conclusion

In summary, pricing and valuing forwards and futures contracts is based on the no-arbitrage principle, ensuring that the prices reflect the underlying spot prices, interest rates, and time to maturity. Understanding these concepts is essential for finance professionals in managing risk and making informed investment decisions. The ability to accurately price and value these contracts ensures effective hedging strategies are employed in varying market conditions.

Study Notes

A forward contract is an agreement to buy or sell an asset at a future date for a pre-agreed price.
The pricing of forwards relies on the no-arbitrage principle, calculated using the formula $F = S e^{rt}$.
The value of a forward contract can fluctuate based on changes in the underlying asset's spot price.
Futures contracts are similar to forwards but are standardized and exchanged on futures markets.
The futures price is calculated similarly to forwards using the formula $F_T = S e^{rt}$.
Futures and forwards have different valuation formulas during their life, accounting for time to maturity and changing spot prices.