Forwards futures and swaps valuation - Cost-of-carry pricing...

Learning Outcomes

After reading this article, you will be able to calculate forward, futures, and swap contract prices using the cost-of-carry model, interpret the role of cash flows, storage costs, and yield effects, and identify how convexity impacts derivative valuations. You will recognize the use of arbitrage in enforcing fair pricing and apply these concepts to practical CFA Level 2 exam scenarios.

CFA Level 2 Syllabus

For CFA Level 2, you are required to understand how forwards, futures, and swaps are priced using the cost-of-carry approach and how convexity impacts their values. Specifically, you should be able to:

• Explain and apply the cost-of-carry pricing model for forwards, futures, and swaps.
• Calculate forward and futures prices for assets with or without income, storage, or carrying costs.
• Identify arbitrage opportunities when traded prices deviate from theoretical values.
• Describe convexity effects in derivative pricing.
• Understand the impact of non-linear payoffs (convexity) on futures and swap valuations.

Test Your Knowledge

Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision.

1. What is the formula for pricing a forward contract on a dividend-paying stock using the cost-of-carry method?
2. How does convexity affect the valuation of a futures contract compared to a forward contract?
3. True or False: Storage costs can cause the futures price of a commodity to be higher than its spot price.
4. Which arbitrage strategy is appropriate when a forward contract is trading above its fair value?

Introduction

Derivative contracts such as forwards, futures, and swaps are priced to prevent arbitrage by linking their values to the cost of holding (carrying) the reference asset. The cost-of-carry model incorporates factors such as financing rates, income (like dividends or coupon payments), and storage costs. Convexity arises from the non-linear payoffs in derivatives, most notably affecting futures because of daily marking-to-market. A solid understanding of cost-of-carry and convexity effects is essential for accurate contract valuation and exam success.

Key Term: cost-of-carry
The total cost incurred for holding an asset until a future date. Includes financing, storage, insurance, and less any income received from the asset.

Key Term: convexity
The measure of how a derivative’s value changes with changes in the reference asset price, reflecting non-linear price behavior, especially for futures contracts.

Key Term: arbitrage
The risk-free profit achieved by exploiting differences between actual and theoretical prices of related assets or contracts.

COST-OF-CARRY PRICING OF FORWARDS, FUTURES, AND SWAPS

The cost-of-carry model provides the no-arbitrage price for forward and futures contracts. It assumes an investor can borrow at the risk-free rate to purchase the asset, or use the asset as collateral or generate income from the asset.

The general cost-of-carry formula for a forward contract is:

Where:

• $FP$ = Forward price
• $S_0$ = Spot price of the asset
• $PV_{income}$ = Present value of any income from asset during the contract
• $r$ = Financing rate (typically risk-free rate)
• $T$ = Time to settlement in years

Key Term: forward price
The agreed price for future delivery, calculated based on the spot price adjusted for carry costs and income effects.

Applying Cost-of-Carry to Different Assets

• Stocks (with Dividends): The present value of expected dividends is subtracted from $S_0$.
• Bonds (with Coupons): The present value of coupons during the contract is subtracted.
• Commodities: Storage and insurance costs (carry costs) are added; convenience yield is subtracted if applicable.
• Currencies: Both currencies have interest rates; cost-of-carry model adjusts for interest differential.

Worked Example 1.1

A 180-day forward contract is written on a share trading at $50. The annual risk-free rate is 6%, and an $0.80 dividend is expected in 90 days. Calculate the no-arbitrage forward price.

Answer:

• $PV_{dividend} = 0.80 / (1 + 0.06)^{0.25} \approx 0.786$
• $FP = (50 - 0.786) \times (1 + 0.06)^{0.5} \approx 49.214 \times 1.02956 = 50.68$
• The no-arbitrage price is $50.68.

ARBITRAGE AND CASH-AND-CARRY STRATEGIES

When actual forward or futures prices deviate from the no-arbitrage price, arbitrage becomes possible:

• If the forward is overpriced: Borrow money, buy the asset, sell the forward (cash-and-carry arbitrage).
• If the forward is underpriced: Sell the asset short, invest the proceeds, buy the forward (reverse cash-and-carry).

Worked Example 1.2

An oil futures contract is trading at $74, while the fair price is $72 using cost-of-carry. What arbitrage strategy can lock in a profit?

Answer:

• Borrow to buy oil for $70, store it (storage: $1), finance: 6% annual.
• Sell the futures and deliver oil at $74 in 1 year.
• Net profit = $74 (futures settlement) – $70 (purchase) – $1 (storage) – $3 (interest) = $0.

CONVEXITY EFFECTS IN FORWARD AND FUTURES PRICING

Convexity reflects the non-linear payoff of derivatives. For futures, daily marking-to-market results in cash flows that impact the contract’s value compared to an equivalent forward.

When interest rates and the value of the reference asset are correlated, the value of a futures contract will deviate from an equivalent forward contract:

• Positive convexity (asset price and rates positively correlated): Futures price will be higher than the forward price.
• Negative convexity (asset price and rates negatively correlated): Futures price will be lower than the forward price.

Convexity arises because gains on futures are reinvested at prevailing rates (which may differ over time), introducing an option-like skew.

Worked Example 1.3

A bond and its associated futures contract are both priced using the same cost-of-carry inputs. Why might the futures price differ from the forward price if interest rates are volatile?

Answer:

• With rate volatility, the frequent marking-to-market of futures means gains are received sooner, and reinvested at different, possibly higher rates. This increases the expected return (positive convexity) and can make the futures price higher than the equivalent forward price.

Exam Warning

It is a common error to ignore storage costs, dividends, or coupons in forward pricing. Always adjust the spot price for any carry costs (positive or negative), or you may misprice the contract and overlook arbitrage.

Summary

Forwards, futures, and swaps are priced to deter arbitrage using the cost-of-carry model. This model adjusts for interest rates, asset income, and storage or convenience yields. Convexity introduces differences between forward and futures prices due to the impact of marking-to-market and reinvestment, especially under volatility. Arbitrage mechanisms ensure that market prices cluster close to theoretical values.

Key Point Checklist

This article has covered the following key knowledge points:

• Cost-of-carry pricing for forwards, futures, and swaps with various asset types
• Adjusting for dividends, coupons, or storage costs in contract pricing
• Identifying arbitrage opportunities and related trading strategies
• Convexity effects and their impact on futures vs. forward values
• The role of interest rate volatility in relative pricing of derivatives

Key Terms and Concepts

• cost-of-carry
• convexity
• arbitrage
• forward price