Pricing and Valuation of Futures Contracts

Derivatives

Pricing of Futures Contracts at Inception

Learning Outcome Statement:

compare the value and price of forward and futures contracts

Summary:

The pricing of futures contracts at inception involves setting the initial value of the contract to zero, as no cash is exchanged. The futures price is determined based on the spot price of the underlying asset, compounded at the risk-free rate over the contract's duration. For assets with associated costs or benefits, these are factored into the pricing using present value calculations. The futures price is adjusted to reflect these costs or benefits to maintain a no-arbitrage condition.

Key Concepts:

Initial Value of Futures Contracts

At inception, both forward and futures contracts have an initial value of zero, indicating no immediate profit or loss.

Futures Price Calculation

The futures price at inception is calculated by compounding the spot price of the underlying asset at the risk-free rate over the time to maturity.

Adjustments for Costs and Benefits

For underlying assets that have associated costs (like storage or insurance) or benefits, these are included in the futures price calculation using their present values to ensure the price reflects all relevant economic factors.

No Arbitrage Condition

The futures price is set such that there is no arbitrage opportunity available, meaning the price reflects all known costs, benefits, and the time value of money, ensuring market efficiency.

Formulas:

Initial Futures Contract Value

V_0(T) = 0

At inception, the value of the futures contract is zero.

Variables:

$V_0(T)$ :: Value of the futures contract at time zero

Units: currency units

Futures Price with No Costs or Benefits

f_0(T) = S_0(1 + r)^T

Futures price calculation using discrete compounding when there are no associated costs or benefits.

Variables:

$f_0(T)$ :: Futures price at time zero
$S_0$ :: Spot price of the underlying asset
$r$ :: Risk-free interest rate
$T$ :: Time to maturity

Units: currency units

Futures Price with Continuous Compounding

f_0(T) = S_0 e^{rT}

Futures price calculation using continuous compounding, typically used for indices or foreign exchange.

Variables:

$e$ :: Base of the natural logarithm

Units: currency units

Futures Price with Costs and Benefits

f_0(T) = [S_0 - PV_0(I) + PV_0(C)] (1 + r)^T

Adjustment of futures price to include present values of costs and benefits, ensuring no arbitrage.

Variables:

$PV_0(I)$ :: Present value of benefits at time zero
$PV_0(C)$ :: Present value of costs at time zero

Units: currency units

MTM Valuation: Forwards versus Futures

Learning Outcome Statement:

compare the value and price of forward and futures contracts

Summary:

This LOS explores the differences in valuation and pricing between forward and futures contracts, particularly focusing on how daily settlement impacts the mark-to-market (MTM) values of futures compared to forwards. It uses a practical example involving a gold contract to illustrate these differences over time, highlighting the role of initial and maintenance margins in futures contracts and the fixed nature of forward contract prices.

Key Concepts:

Mark-to-Market (MTM) Valuation

MTM valuation refers to the process of valuing positions and contracts based on current market prices. In futures contracts, MTM occurs daily and any gains or losses are settled immediately, which resets the MTM value to zero daily. For forward contracts, MTM is calculated as the difference between the current spot price and the present value of the forward price, but it is not settled until the contract matures.

Initial and Maintenance Margin

Futures contracts require the posting of an initial margin at the start and a maintenance margin that must be kept throughout the contract duration. If the margin account falls below the maintenance margin due to MTM losses, a margin call is issued, and the margin must be replenished to the initial level.

Forward Contract Pricing

The price of a forward contract, denoted as F0(T), is fixed at the contract inception and does not change throughout the life of the contract. The MTM value of a forward contract is the difference between the current spot price and the present value of the forward price, adjusted for the time value of money.

Futures Contract Pricing

Unlike forward contracts, futures prices can fluctuate daily based on market conditions. The futures price is initially set at the inception but can vary each day. The daily settlement of gains and losses keeps the MTM value of futures contracts reset to zero.

Formulas:

Forward Contract MTM Value

V_t(T) = S_t - F_0(T) \times (1 + r)^{-(T-t)}

This formula calculates the MTM value of a forward contract at any given time t before maturity. It shows the difference between the current spot price and the present value of the forward price, discounted back to time t.

Variables:

$V_t(T)$ :: MTM value of the forward contract at time t
$S_t$ :: Current spot price of the underlying asset
$F_0(T)$ :: Fixed forward price agreed at contract inception
$r$ :: Risk-free rate
$T$ :: Time of contract maturity
$t$ :: Current time

Units: currency units (e.g., USD)

Interest Rate Futures versus Forward Contracts

Learning Outcome Statement:

compare the value and price of forward and futures contracts

Summary:

This LOS explores the differences in valuation and pricing mechanisms between forward and futures contracts, particularly focusing on interest rate futures versus forward rate agreements (FRAs). It highlights how futures contracts offer a standardized, liquid alternative to FRAs, with daily settlement and margin requirements that mitigate counterparty credit risk. The content also explains the pricing formula for interest rate futures and the impact of market reference rate changes on contract value.

Key Concepts:

Forward Contract Pricing and Valuation

Forward contracts have a fixed price established at inception (F0(T)) and are valued based on the difference between the current spot price (St) and the present value of the forward price (PVt[F0(T)]). The MTM value changes are not settled until maturity, leading to potential counterparty credit risk.

Futures Contract Pricing and Valuation

Futures contract prices fluctuate daily and are settled daily through a margin account, which resets the MTM to zero daily. This mechanism reduces counterparty credit risk. The futures price for interest rate futures is calculated using a formula that inversely relates the price to the yield.

Interest Rate Futures Pricing Formula

Interest rate futures are priced based on the market reference rate (MRR) for a given period, using the formula fA,B−A = 100 − (100 × MRRA,B−A), where fA,B−A represents the futures price and MRRA,B−A is the market reference rate for the period from A to B.

Basis Point Value (BPV) Calculation

The BPV for interest rate futures is calculated as the product of the notional principal, a 0.01% change, and the period fraction. This value represents the dollar change in contract value for a one basis point change in the market reference rate.

Formulas:

Interest Rate Futures Price

f_{A,B-A} = 100 - (100 \times MRR_{A,B-A})

This formula calculates the futures price based on the market reference rate for a specific period, showing an inverse relationship between price and yield.

Variables:

$f_{A,B-A}$ :: Futures price for the period from A to B
$MRR_{A,B-A}$ :: Market reference rate for the period from A to B

Units: percentage

Futures Contract Basis Point Value (BPV)

BPV = \text{Notional Principal} \times 0.01\% \times \text{Period}

This formula calculates the change in contract value for a one basis point change in the market reference rate, useful for assessing the impact of rate changes on the contract.

Variables:

$BPV$ :: Basis Point Value
$Notional Principal$ :: The principal amount on which the futures contract is based
$Period$ :: The fraction of the year for the contract period

Units: currency

Forward and Futures Price Differences

Learning Outcome Statement:

explain why forward and futures prices differ

Summary:

Forward and futures contracts, while similar in their symmetric payoff profiles at maturity, exhibit differences in pricing due to their distinct cash flow profiles and settlement mechanisms. Futures contracts involve daily mark-to-market and margin settlements, resetting the MTM value to zero daily. Forward contracts, in contrast, do not require daily settlements and are settled at maturity with a one-time payment reflecting the cumulative value change. These differences, along with the correlation between futures prices and interest rates, and the volatility of interest rates, can lead to price differentials between forwards and futures.

Key Concepts:

Daily Settlement

Futures contracts require daily settlement of gains and losses through a margin account, which resets the MTM value to zero each day.

Forward Contract Settlement

Forward contracts are settled at maturity with a one-time payment that reflects the cumulative change in the contract's value over its term.

Price Correlation and Volatility

The correlation between futures prices and interest rates, along with the volatility of interest rates, can cause differences in the pricing of forward and futures contracts. Positive correlation makes futures more attractive, while negative correlation favors forwards.

Convexity Bias

A specific difference in price changes for interest rate futures versus forward contracts due to the different handling of price movements and interest rate changes.

Formulas:

Interest Rate Futures Contract Value

V = P \times \left[1 + \left(\frac{r}{n}\right)\right]

Calculates the value of an interest rate futures contract based on the principal, the annual interest rate, and the number of periods per year.

Variables:

$V$ :: contract value
$P$ :: principal amount
$r$ :: annual interest rate
$n$ :: number of periods per year

Units: currency

Interest Rate Forward and Futures Price Differences

Learning Outcome Statement:

explain why forward and futures prices differ

Summary:

Forward and futures contracts, while similar in their payoff profiles at maturity, differ in their valuation and pricing due to their distinct cash flow profiles over the contract duration. Futures contracts involve daily settlements and margin requirements, which reset the futures price daily. Forward contracts, on the other hand, do not require daily settlements and are settled at maturity. These differences in cash flow timing and management can lead to price differences under certain conditions, particularly when interest rates are variable or correlated with futures prices.

Key Concepts:

Margin and Settlement Differences

Futures contracts require posting of initial margin and daily mark-to-market settlements, which reset the futures price daily to zero MTM value. Forward contracts are settled at maturity with a one-time cash settlement, without daily requirements.

Price Correlation with Interest Rates

If futures prices and interest rates are positively correlated, futures contracts can be more attractive due to the ability to reinvest profits at higher rates. Conversely, a negative correlation makes forwards more attractive.

Convexity Bias

The non-linear price changes in forwards due to discounting, known as convexity bias, affect the pricing especially over longer periods. This bias is absent in futures, which have a linear payoff profile.

Formulas:

Net Payment for FRA

Net Payment = (MRR_{B-A} - IFR_{A,B-A}) \times Notional Principal \times Period

Calculates the net payment at maturity of a Forward Rate Agreement based on the difference between the market rate at settlement and the implied rate at the start, adjusted for the principal and period.

Variables:

$MRR_{B-A}$ :: Market Reference Rate at settlement
$IFR_{A,B-A}$ :: Implied Forward Rate at the start of the period
$Notional Principal$ :: Principal amount on which the FRA is based
$Period$ :: Duration of the FRA

Units: currency

Cash Settlement for FRA

Cash Settlement (PV) = \frac{Net Payment}{1 + \frac{MRR}{4}}

Calculates the present value of the net payment at maturity of a Forward Rate Agreement, discounted by the market reference rate.

Variables:

$Net Payment$ :: Net payment calculated from the FRA
$MRR$ :: Market Reference Rate at settlement

Units: currency

Effect of Central Clearing of OTC Derivatives

Learning Outcome Statement:

explain why forward and futures prices differ

Summary:

Central clearing of derivatives, by imposing futures-like margining requirements on OTC derivative dealers and their end users, reduces the cash flow impact differences between exchange-traded derivatives (ETD) and over-the-counter (OTC) derivatives. This alignment in margin requirements leads to a reduction in the price differences between futures and forwards for the same underlying asset and maturity period.

Key Concepts:

Margin Requirements

Central clearing introduces margin requirements for OTC derivatives similar to those for futures, requiring dealers and end users to post cash or highly liquid securities, which aligns the financial obligations of parties in both markets.

Cash Flow Impact

The similar margining requirements reduce the variance in cash flow impacts between futures and forwards, making their pricing more comparable and reducing the price disparity for the same underlying and maturity.

Price Convergence

By standardizing the financial obligations (through margin requirements) across ETD and OTC derivatives, central clearing helps in narrowing the price gap that might otherwise exist due to different credit risks and liquidity conditions.

Formulas:

Net Payment for FRA

Net Payment = (MRR_{B-A} - IFR_{A,B-A}) \times Notional Principal \times Period

This formula calculates the cash settlement for a pay-fixed FRA, considering the difference between the market rate at maturity and the implied forward rate, multiplied by the notional principal and the fraction of the year represented by the period.

Variables:

$MRR_{B-A}$ :: Market Reference Rate at maturity minus the rate at the start
$IFR_{A,B-A}$ :: Implied Forward Rate from start to maturity
$Notional Principal$ :: Principal amount on which the FRA is based
$Period$ :: Duration of the FRA

Units: currency

Present Value of FRA Settlement

PV = \frac{Net Payment}{1 + \frac{MRR_{2m,3m}}{4}}

This formula discounts the net payment of the FRA to its present value, considering the market reference rate applicable over the period divided by the number of periods per year (quarterly in this case).

Variables:

$Net Payment$ :: Initial net payment calculated from FRA
$MRR_{2m,3m}$ :: Market Reference Rate for the specific period

Units: currency

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