Estimate convexity adjustment to translate forward bond price to futures price.
Pricing Parameters
Estimate the convexity bias between Forward and Futures prices.
Formula Used
Adjustment = 0.5 × σ² × T × Duration² × Price
Futures Price ≈ Forward Price - Adjustment
This Hull approximation estimates how much lower the futures price should be relative to the forward price due to the negative convexity bias in bond futures settlement.
The Convexity Adjustment is a correction factor applied to the Forward Price of an asset to derive its theoretical Futures Price. In the context of Bond Futures (prices) or Eurodollar Futures (rates), it accounts for the impact of daily Mark-to-Market (MTM) settlement.
While Forwards and Futures are similar, they are not identical. A Forward contract is settled only at maturity. A Futures contract is settled daily. When interest rates are volatile, this difference creates a pricing gap.
Forward vs. Futures: The Crucial Difference
Forward Contract: No cash changes hands until the very end. You are not exposed to interest rate risk on your profit/loss during the life of the trade.
Futures Contract: You effectively "realize" your profit or loss every single day. If you make a profit, you can reinvest that cash. If you lose, you must finance that loss.
Because interest rates (financing costs) are correlated with bond prices, this creates a bias.
The Mechanics of the Bias
For Bonds, prices fall when financing rates rise. This correlation is negative.
Scenario A (Rates Rise): Bond prices fall. The Short position makes a profit. The Short receives cash *immediately* and can reinvest it at the *new, higher* interest rate. This is excellent for the Short.
Scenario B (Rates Fall): Bond prices rise. The Short position loses money. The Short must pay cash, but borrows at the *new, lower* interest rate. This reduces the pain of the loss.
Since the Short position wins 'double' when they win (reinvesting at high rates) and loses 'less' when they lose (borrowing at low rates), the Short position is more valuable. Paradoxically, in a competitive market, this advantage forces the Futures Price down relative to the Forward Price.
The Impact of Volatility
The adjustment is proportional to the square of volatility (σ²). This means:
Doubling volatility quadruples the convexity adjustment.
In calm markets, the adjustment is tiny and often ignored. In volatile markets (like 2008 or 2022), the adjustment becomes massive, creating significant arbitrage opportunities for those who can price it correctly.
Implications for Hedging
If you use Futures to hedge a Forward exposure (or vice versa) without accounting for convexity, you will be under-hedged. The "tail risk" (extreme convexity events) will leave you exposed exactly when the market moves the most.
Frequently Asked Questions
Expert answers on Pricing Adjustments
Is the futures price usually higher or lower?
For Bond Futures, it is typically lower than the Forward Price. For Interest Rate Futures (like Eurodollars, where the underlying is the Rate itself), the rate is typically higher (meaning the implied price/index is lower).
What happens if correlation is zero?
If the asset price and interest rates are uncorrelated, the convexity adjustment is zero. Forward Price = Futures Price. This relies on the theorem by Cox, Ingersoll, and Ross (CIR).
Does this apply to commodities?
Generally not as much. Commodity prices (Gold, Oil) are not strongly correlated with short-term financing rates in the same mechanical way bonds are.
How accurate is the Hull approximation?
It is a Taylor Series expansion approximation. It works well for short maturities (less than 5 years). For very long-dated options or futures, more complex stochastic models (like BGM) are required.
Why is Duration squared in the formula?
Because convexity is the second derivative of price with respect to yield. The formula is deriving the price impact of the volatility of yields, which relates to price via duration.
Can the adjustment be positive?
Yes, if the asset is positively correlated with interest rates. In that case, the Long position has the financing advantage, pushing the Futures Price above the Forward Price.
How do I estimate yield volatility?
You can use the implied volatility from Swaptions (volatility of swap rates) or calculate historical standard deviation of daily yield changes.
What is "Cheapest to Deliver" (CTD)?
Bond futures allow the seller to deliver any of several eligible bonds. They will choose the one that is cheapest to buy relative to the futures conversion factor. This option value also depresses the futures price.
Is this related to "Convexity" of a bond?
Yes, they are cousins. Bond Convexity makes the bond price curve "smile." Futures Convexity Adjustment accounts for how that smile interacts with daily cash settlement.
Why don't simple calculators include this?
Because it requires estimating volatility, which is not observable directly in price. It is an advanced pricing metric for derivatives traders.
Summary
The Convexity Adjustment Calculator quantifies the pricing gap between Forwards and Futures.
It highlights the hidden value of daily liquidity and financing in volatile markets.
Use this tool to avoid overpaying for futures contracts in high-volatility environments.
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Estimate convexity adjustment to translate forward bond price to futures price.
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