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Michael Brenndoerfer

Forward and Futures Pricing and Hedging › Futures Contracts: Exchange-Traded Derivatives

Michael BrenndoerferNovember 14, 20256 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 15Forward and Futures Pricing and Hedging › Futures Contracts: Exchange-Traded Derivatives

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While forwards are over-the-counter contracts between two parties, futures are standardized contracts traded on organized exchanges. The CME Group, ICE, and Eurex are major futures exchanges trading contracts on equity indices, interest rates, currencies, and commodities. The move from bilateral to exchange-traded contracts fundamentally changes the risk profile and operational mechanics, though the economic exposure remains similar.

Key Differences from ForwardsLink Copied

Several features distinguish futures from forwards, each addressing specific limitations of the bilateral forward market:

Standardization: Contract sizes, delivery dates, and specifications are fixed by the exchange. This standardization creates fungibility: all March S&P 500 futures are identical, enabling liquid trading.
Central clearing: The exchange clearinghouse becomes counterparty to all trades, eliminating bilateral credit risk. Buyers and sellers need not evaluate each other's creditworthiness; they only need confidence in the exchange.
Daily settlement (marking-to-market): Gains and losses are settled daily through margin accounts. This prevents losses from accumulating to dangerous levels and ensures credit exposure remains manageable.
Margin requirements: Traders must post collateral (initial margin) and maintain minimum balances (maintenance margin). This collateral backs the clearinghouse's guarantee.

Daily Settlement MechanicsLink Copied

The daily settlement process works as follows. At the end of each trading day, the exchange calculates a settlement price for each contract based on trading activity. Your margin account is credited or debited the day's price change multiplied by the contract size. If your account falls below the maintenance margin, you receive a margin call requiring immediate deposit of additional funds.

This process turns a single future payment into a series of daily payments. Rather than waiting until expiration to realize your gains or losses, you experience them incrementally each day. We can simulate this process in Python to track the daily cash flows and cumulative P&L:

In[12]:

Code

import numpy as np
import pandas as pd

# Simulate daily settlement for a long futures position
np.random.seed(42)

contract_size = 50  # e.g., E-mini S&P 500 = $50 × index
initial_futures_price = 5000
days = 10

# Simulate daily price changes
daily_returns = np.random.normal(0.001, 0.015, days)
futures_prices = initial_futures_price * np.cumprod(1 + daily_returns)
futures_prices = np.insert(futures_prices, 0, initial_futures_price)

# Calculate daily settlements
daily_changes = np.diff(futures_prices)
daily_cash_flows = daily_changes * contract_size
cumulative_pnl = np.cumsum(daily_cash_flows)

settlement_df = pd.DataFrame(
    {
        "Day": range(1, days + 1),
        "Futures Price": futures_prices[1:],
        "Price Change": daily_changes,
        "Daily P&L": daily_cash_flows,
        "Cumulative P&L": cumulative_pnl,
    }
)

Out[13]:

Console

Daily Settlement Example: Long 1 E-mini S&P 500 Futures
=================================================================
Initial futures price: 5000
Contract multiplier: $50

 Day  Futures Price  Price Change  Daily P&L  Cumulative P&L
   1       5,042.25         42.25   2,112.68        2,112.68
   2       5,036.84         -5.42    -270.76        1,841.92
   3       5,090.81         53.97   2,698.57        4,540.49
   4       5,212.20        121.39   6,069.63       10,610.12
   5       5,199.11        -13.09    -654.73        9,955.39
   6       5,186.05        -13.06    -653.02        9,302.37
   7       5,314.08        128.03   6,401.71       15,704.07
   8       5,380.57         66.49   3,324.36       19,028.43
   9       5,348.06        -32.51  -1,625.50       17,402.93
  10       5,396.93         48.87   2,443.64       19,846.57

Total P&L over period: $19,846.57

Out[14]:

Visualization

Line chart of simulated futures prices over 10 days with a dashed line marking the initial price of 5000. — Futures price evolution over 10 days. The price path determines the daily settlement variation margin.

Bar chart of daily P&L gains and losses overlaid with a line showing cumulative P&L over the 10-day settlement period. — Daily and cumulative profit and loss (P&L). The daily settlement process converts the price path into a stream of daily cash flows (bars), which sum to the cumulative P&L (line).

The daily settlement has an important implication: your P&L is realized incrementally each day, not as a lump sum at expiration. This means cash flows occur throughout the contract's life, affecting the time value of money. Daily gains can be withdrawn and invested; daily losses must be funded immediately. This timing difference creates subtle distinctions between futures and forward prices.

Futures vs Forward PricesLink Copied

For constant interest rates, futures and forward prices are identical. The daily settlement merely changes when you receive money, not how much you ultimately receive. With constant rates, the present value of receiving cash daily equals the present value of receiving it all at expiration.

However, when interest rates are stochastic and correlated with the underlying asset, differences emerge. The daily settlement creates a subtle advantage or disadvantage depending on the correlation structure.

Consider a long futures position on an asset positively correlated with interest rates. When the asset price rises, rates tend to rise too. The daily gains are credited to your margin account and can be reinvested at higher rates. Conversely, when prices fall, rates fall, and you fund your losses at lower rates. This asymmetry benefits the futures holder: gains arrive when reinvestment rates are high, and losses occur when borrowing rates are low.

The opposite applies when asset prices are negatively correlated with rates. In this case, gains arrive when reinvestment rates are low, and losses occur when funding costs are high, disadvantaging the futures holder. The relationship is captured by the convexity adjustment:

F^{\text{futures}} = F^{\text{forward}} \cdot e^{\rho \sigma_S \sigma_r T^2}

where:

$F^{\text{futures}}$ : futures price
$F^{\text{forward}}$ : forward price
$\rho$ : correlation between the asset price and interest rates
$\sigma_S$ : volatility of the asset price
$\sigma_r$ : volatility of interest rates
$T$ : time to maturity

For most equity and commodity futures with short maturities (under one year), this adjustment is negligible, often just a few basis points. For interest rate futures like Eurodollar contracts, where the underlying is by definition correlated with rates, the adjustment matters and must be accounted for in pricing and hedging.