Calculating Forward Prices: Income vs. No Income

The No-Arbitrage Principle

At the heart of derivative pricing is a simple but profound idea: two assets or portfolios with the same future payoffs must have the same price today. If they didn't, an arbitrageur could simultaneously buy the cheaper one and sell the more expensive one, locking in a guaranteed, risk-free profit. This activity, repeated by many market participants, would quickly force the prices to converge.

This "law of one price" is our primary tool. To price a forward contract, we will construct a simple arbitrage argument.

Pricing a Forward on a Non-Income-Paying Asset

Let's start with the simplest case: a forward contract on an investment asset that provides no income and has no storage costs during the life of the contract, such as a non-dividend-paying stock.

Let's define our notation:

• $S_0$: Spot price of the asset today.
• $F_0$: Forward price agreed upon today for delivery at time $T$.
• $T$: Time to maturity of the forward contract (in years).
• $r$: The continuously compounded risk-free interest rate.

Consider two portfolios created at time 0:

• Portfolio A: Buy one share of the stock. The cost today is $S_0$. At time $T$, its value will be $S_T$.
• Portfolio B: Enter a long forward contract to buy one share at price $F_0$ at time $T$. Also, invest an amount of cash equal to $F_0{e}^{-rT}$ at the risk-free rate. The cost today is $F_0{e}^{-rT}$. At time $T$, your investment will have grown to $F_0$. You use this cash to settle the forward contract and receive the share, which is worth $S_T$.

Both portfolios result in owning one share of the stock at time $T$. Their payoffs are identical. Therefore, by the no-arbitrage principle, their costs today must be equal:

Rearranging this gives us the fundamental no-arbitrage forward pricing formula:

This equation tells us that the forward price is simply the spot price grown at the risk-free rate of interest over the term of the contract. The term $e^{rT}$ is the future value factor for continuous compounding.

The following resource provides a more detailed explanation with numerical examples of what happens when this relationship does not hold.

Adjusting for Income and Costs: The Cost of Carry

The simple formula $F_0=S_0{e}^{rT}$ assumes that holding the underlying asset generates no cash flows. In reality, many assets do. Stocks pay dividends, bonds pay coupons, and commodities incur storage costs. These cash flows are known as the cost of carry.

The general principle is:

• Benefits of holding the asset (like income) reduce the forward price.
• Costs of holding the asset (like storage) increase the forward price.

The generalized formula adjusts the spot price for the present value of all net costs associated with holding the asset until maturity. Let's explore the most common cases.

1. Known Cash Income (e.g., Stock Dividends)

If an asset is expected to pay a known cash income during the life of the forward contract, holding the physical asset is more attractive than holding the forward. The holder of the physical asset receives the income, while the holder of the forward contract does not.

To prevent arbitrage, the forward price must be lower to compensate. The formula is adjusted by subtracting the present value of the income, $I$, from the spot price before growing it at the risk-free rate.

Here, $I$ is the present value of all dividends (or other cash income) expected to be paid before the contract's expiration at time $T$.

2. Known Yield (e.g., Stock Index)

Some assets, like a broad stock market index, pay dividends so frequently that it is more practical to model the income as a continuous yield, $q$. This yield reduces the effective growth rate of the investment from the arbitrageur's perspective. The net cost of carry becomes $r-q$.

The formula becomes:

This formula is essential for pricing futures on stock indices like the S&P 500.

3. Foreign Currencies

A foreign currency can be treated as an asset that provides a "yield"—the foreign risk-free interest rate, $r_f$. If you hold the foreign currency, you can deposit it in a foreign bank and earn interest at $r_f$. This is a direct analogy to the known yield case.

Therefore, we can adapt the known yield formula, where the domestic risk-free rate is $r$ and the "yield" on the asset (the foreign currency) is $r_f$.

Here, $S_0$ is the spot exchange rate (e.g., dollars per euro), $r$ is the domestic risk-free rate, and $r_f$ is the foreign risk-free rate. This relationship is also known as covered interest rate parity.

The following video provides an excellent intuitive derivation of this formula.

To see this in action with a numerical example, let's turn back to the FIU slides.

4. Commodities with Storage Costs

Finally, consider commodities like gold, silver, or oil. Unlike financial assets, holding a physical commodity often involves storage costs. These can be thought of as a negative yield. If the storage cost is modeled as a continuous rate $u$ (as a percentage of the asset's value), it adds to the cost of carry.

The formula becomes:

If storage costs are a known present value amount, $U$, the formula is analogous to the known cash income case, but with the cost added instead of subtracted:

An important distinction arises with consumption assets like copper or corn. While arbitrageurs can correct an overpriced forward by buying the spot asset and shorting the forward, they may not be able to correct an underpriced forward. This is because the reverse strategy would require short-selling the commodity, which may be difficult or impossible if most of it is held by consumers who need it and are unwilling to lend it out. This can lead to an inequality, $F_0\le S_0{e}^{(r+u)T}$. For investment assets like gold, which are primarily held for investment, this is not an issue.

Conclusion

In this lesson, we established the fundamental principle of no-arbitrage pricing and applied it to forward contracts. We saw that a single, powerful logic—the cost of carry—allows us to derive the fair price for forwards on a wide variety of assets.

Key Takeaways:

• No-Arbitrage Principle: Assets or portfolios with identical future payoffs must have the same price today.
• Cost of Carry: The forward price is the spot price adjusted for the net costs or benefits of holding the asset until the delivery date. The generic formula is $F_0=S_0{e}^{(\text{cost of carry})T}$.
• Key Formulas (Continuous Compounding):
  • No Income: $F_0=S_0{e}^{rT}$
  • Known Cash Income $I$: $F_0=(S_0-I)e^{rT}$
  • Known Yield $q$: $F_0=S_0{e}^{(r-q)T}$
  • Foreign Currency (yield $r_f$): $F_0=S_0{e}^{(r-r_f)T}$
  • Storage Costs $u$: $F_0=S_0{e}^{(r+u)T}$

In our next lesson, we will continue to use this no-arbitrage framework. We will apply it to options to derive put-call parity, a crucial relationship that links the prices of European calls and puts with the underlying stock and a risk-free bond.