Library/CFA (Chartered Financial Analyst)/JuiceNotes 2024: Derivatives - Chartered Financial Analyst Level I/Pricing and Valuation of Futures Contracts

Pricing and Valuation of Futures Contracts

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Pricing of Futures Contracts at Inception5 min

The pricing of futures contracts is based on the no-arbitrage principle. The specific formula depends on the characteristics of the underlying asset. For an asset with no associated costs or benefits, the futures price is simply the future value of the spot price calculated using the risk-free rate ($S_0 \times (1+r)^T$). When the underlying asset generates income (benefits) or incurs storage costs, these must be accounted for. The formula adjusts the spot price by subtracting the present value of benefits ($PV_0(I)$) and adding the present value of costs ($PV_0(C)$) before applying the compounding factor. For portfolios or foreign exchange assets, a continuous compounding model ($S_0 \times e^{rT}$) is often used.

Key Points

Basic Formula: $f_0(T) = S_0 \times (1+r)^T$ for assets with no cost/benefit.
With Carry Costs: $f_0(T) = [S_0 - PV_0(I) + PV_0(C)] \times (1+r)^T$.
I represents benefits (Income); C represents Costs.
Continuous compounding applies to portfolios/FX: $f_0(T) = S_0 \times e^{rT}$.

Mark-to-Market Valuation and Comparison with Forwards6 min

A defining feature of futures contracts is the daily mark-to-market (MTM) process. While forward contracts have a fixed price until maturity and accumulate value without interim cash flows, futures reset daily. Gains or losses are realized immediately through the variation margin, keeping the value of the futures contract at zero at the start of each day. This mechanism ensures that counterparty risk is minimized, as exposures are settled daily rather than accumulating until contract expiration. Futures also require an initial margin and maintenance margin, whereas forwards typically do not require upfront cash.

Key Points

Futures settle daily (MTM); Forwards settle at maturity.
Futures require initial and variation margins.
Daily settlement reduces counterparty credit risk in futures.
Accumulated MTM gain/loss on futures is similar to a comparable forward.

Interest Rate Futures, FRAs, and Convexity Bias6 min

Interest rate futures are standardized contracts used to manage interest rate exposure. They are distinct from Forward Rate Agreements (FRAs) in their settlement and pricing mechanics. A key distinction involves the price-yield relationship: interest rate futures have a linear relationship, while interest rate forwards (and bonds) are convex (non-linear). This discrepancy leads to a 'convexity bias'. Additionally, the correlation between asset prices and interest rates affects the relative pricing of futures versus forwards. If the correlation is positive, futures are preferred (higher price) because margin cash flows are received when rates are high (allowing reinvestment at high rates).

Key Points

Rising Rates: Short Futures or Long FRA (Pay Fixed).
Falling Rates: Long Futures or Short FRA (Receive Fixed).
Positive Correlation (Asset/Rates): Futures Price > Forward Price.
Futures Price vs. Yield is linear; Forwards are convex.
Basis Point Value (BPV) = Notional Principal * 0.01 percent * Period.

Questions

Question 1

Which of the following formulas correctly represents the pricing of a futures contract for an asset with no costs or benefits?

Pricing and Valuation of Futures Contracts

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