Pricing Financial Forwards and Futures

FRM Part I · FRM · 6 min read

Welcome to Pricing Financial Forwards and Futures!

Welcome, future FRM charterholders! If you’ve ever wondered how traders decide exactly what a stock or a currency should be worth six months from now, you’re in the right place. In this chapter, we are going to learn the mechanics of pricing.

The "secret sauce" behind everything we do here is a concept called Arbitrage-Free Pricing. Basically, we want to find a price where no one can make "free money" without taking any risk. Don't worry if the math looks a bit intimidating at first—we will break it down step-by-step using logic you can use in everyday life.

1. The Foundation: Investment vs. Consumption Assets

Before we calculate anything, we need to know what we are dealing with. In this section of the FRM curriculum, we focus primarily on Investment Assets.

Investment Assets are assets held by a significant number of people purely for investment purposes (like stocks, bonds, or gold). Consumption Assets are held primarily for use (like oil, copper, or pork bellies).

Why does this matter? Because for investment assets, we can use Arbitrage arguments to find an exact price. If the price is too high or too low, investors will buy or sell the asset until the price returns to "fair value."

Prerequisite Concept: Short Selling

To understand arbitrage, you must understand Short Selling. This involves borrowing an asset you don't own, selling it today, and promising to buy it back later to return it. If the price goes down, you make money!

Quick Review:
• Spot Price ($ S_0 $): The price of the asset right now.
• Forward Price ($ F_0 $): The price agreed upon today for delivery in the future.
• T: Time until the contract matures (usually in years).
• r: The risk-free interest rate (expressed with continuous compounding).

2. Pricing an Investment Asset with No Income

Imagine you want to buy a share of a non-dividend-paying stock in one year. You have two choices:
1. Buy it today at the spot price $ S_0 $ and hold it.
2. Enter a forward contract to buy it in one year at price $ F_0 $.

To make these choices equal, the forward price must account for the time value of money. If you wait a year to pay, you could have kept your money in a bank earning interest!

The Formula:
$ F_0 = S_0 e^{rT} $

Example:
If a stock is $100 today ($ S_0 = 100 $), the risk-free rate is 5% ($ r = 0.05 $), and the contract is for 1 year ($ T = 1 $):
\n$ F_0 = 100 \times e^{0.05 \times 1} \approx \$105.13 $

Common Mistake to Avoid:
Students often forget to convert months into years for $ T $. If the contract is for 6 months, $ T = 0.5 $. Always check your time units!

Key Takeaway: The forward price is just the spot price "pushed forward" by the interest rate. If the forward price is higher than this, you could borrow money, buy the asset, and sell a forward to make a "riskless profit" (Arbitrage!).

3. Adding Income to the Mix

What if the asset pays you money while you hold it? For example, a bond pays a coupon, or a stock pays a dividend. This makes holding the asset more attractive than holding the forward contract (because the forward holder doesn't get the dividend).

A. Known Cash Income ($ I $)

If you receive a specific dollar amount (like a bond coupon), we subtract the Present Value of that income from the spot price before moving it forward in time.

The Formula:
$ F_0 = (S_0 - I) e^{rT} $
Where $ I $ is the present value of all expected income during the life of the contract.

B. Known Continuous Yield ($ q $)

For stock indices (like the S&P 500), dividends are paid so often that we treat them as a continuous "stream" or a yield ($ q $).

The Formula:
$ F_0 = S_0 e^{(r-q)T} $

Analogy: Think of $ r $ as the "cost" of holding the asset (the interest you pay) and $ q $ as the "benefit" (the dividends you get). The forward price only grows by the net cost: $ r - q $.

Key Takeaway: Income reduces the forward price because the person holding the forward contract misses out on that income.

4. Valuing a Forward Contract (Price vs. Value)

This is a major point of confusion for FRM students. Let’s clear it up!

• Forward Price ($ F_0 $): This is the "delivery price" written in the contract. At the start of the contract, its Value is zero because the price is set fairly.
• Value ($ f $): As time passes and the spot price ($ S_t $) moves, the contract becomes "worth" something. If you locked in a buy price of $100 and the price goes to $120, your contract is now valuable!

The Formula for Value (Long Position):
$ f = (F_t - K) e^{-r(T-t)} $
Where $ K $ is the delivery price you agreed to originally, and $ F_t $ is the current forward price for a new contract.

Memory Aid: Value is just the present value of what you would gain if you settled the contract today.

5. Currency Forwards (Interest Rate Parity)

When dealing with currencies, think of the foreign interest rate ($ r_f $) as the "dividend yield" of that currency. If you hold Euros, you can earn interest on them.

The Formula:
$ F_0 = S_0 e^{(r_d - r_f)T} $
• $ r_d $: Domestic risk-free rate (e.g., USD)
• $ r_f $: Foreign risk-free rate (e.g., EUR)

Note: The spot price $ S_0 $ must be quoted as "Domestic units per one Foreign unit" (e.g., 1.10 USD per 1 EUR).

Did you know? This relationship is called Interest Rate Parity. If it didn't hold, banks could move money between countries and make endless risk-free profits!

6. Futures vs. Forwards: Are they the same?

For the FRM exam, we generally assume the Forward Price and the Futures Price are the same. However, there is a tiny theoretical difference.

Because futures are marked-to-market daily, the timing of cash flows matters. If interest rates are strongly correlated with the asset price, futures prices might be slightly different from forward prices.

Quick Rule of Thumb:
• If interest rates are constant or predictable: Futures Price = Forward Price.
• For most exam questions: Treat them as identical unless the question specifically asks about interest rate correlation.

7. The "Cost of Carry" Summary

To wrap everything up, we use a single concept called the Cost of Carry ($ c $) to summarize the formulas we’ve learned.

General Formula:
$ F_0 = S_0 e^{cT} $

• For a non-income stock: $ c = r $
• For a stock index: $ c = r - q $
• For a currency: $ c = r_d - r_f $

Key Takeaway: The cost of carry is simply the cost to store/finance the asset minus the income earned from the asset.

Final Encouragement

Don't worry if you need to read through these formulas a few times. Pricing is all about "Time Travel" for money. Just ask yourself: "If I buy this later instead of now, what interest do I save, and what dividends do I miss?" Once you have that logic down, the math will follow naturally. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the principle of arbitrage-free pricing?

It is the determination of a price for a financial instrument where no market participant can earn a risk-free profit through simultaneous transactions.

What distinguishes an investment asset from a consumption asset in forward pricing?

Investment assets are held for wealth storage by many investors (e.g., stocks, gold), while consumption assets are held primarily for industrial or personal use (e.g., oil, copper).

How does short selling work in the context of arbitrage?

Short selling involves borrowing an asset from a lender, selling it in the market today, and purchasing it back later to return the asset to the lender.

What is the formula for the forward price of an investment asset that provides no income?

$ F_0 = S_0 e^{rT} $

How do you calculate the forward price of an asset with a known cash income?

$ F_0 = (S_0 - I) e^{rT} $, where $ I $ is the present value of the expected income.

What is the formula for the forward price of an asset with a continuous dividend yield?

$ F_0 = S_0 e^{(r-q)T} $, where $ q $ is the continuous yield.

What is the value of a forward contract at the time of its inception?

The value of a forward contract at inception is always zero ($ f = 0 $).

What is the formula for the value of a long forward contract during its life?

$ f = (F_t - K) e^{-r(T-t)} $, where $ K $ is the original delivery price.

What is the formula for the forward price of a currency (Interest Rate Parity)?

$ F_0 = S_0 e^{(r_d - r_f)T} $, where $ r_d $ is the domestic risk-free rate and $ r_f $ is the foreign risk-free rate.

When are futures prices theoretically different from forward prices?

They differ when interest rates are strongly correlated with the asset's price, due to the daily marking-to-market of futures contracts.

How is the 'cost of carry' defined in forward pricing?

The cost of carry ($ c $) is the total cost of financing/storing an asset minus any income earned from it while holding it.

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