Shocking Treasury Futures with Conversion Factors and CTD Selection
Summary
The note addresses how to estimate Treasury futures prices under interest-rate shocks when the cheapest-to-deliver bond might change. It explains that the futures price need not be known first to identify the cheapest-to-deliver security. A simplified bond-based calculation compares each eligible bond’s financed dirty price, net of coupon value, after dividing by its conversion factor; the minimum is the corresponding theoretical futures price under the stated assumptions.
That calculation omits the delivery option, which can be incorporated with stochastic interest-rate models of varying complexity. For small shocks, the response suggests holding the base-case cheapest-to-deliver bond fixed as a practical approximation. The document gives no model details, calibration, numerical example, or guidance for larger shocks, where delivery optionality and a change in the cheapest-to-deliver bond may matter.
Key ideas
- Theoretical Treasury futures pricing should account for each eligible bond’s conversion factor.
- A bond-based comparison can identify the cheapest-to-deliver security without first using the futures price.
- The simplified financed-price calculation omits the delivery option embedded in the contract.
- Holding the base-case cheapest-to-deliver bond fixed may be reasonable for small shocks, while larger shocks need more care.
Tags
Full text
# How to calculate rate shocked bond future prices (effective duration)
# How to calculate rate shocked bond future prices (effective duration)
I am looking to build out parallel interest rate shocks for a treasury future. I know how to calculate the cheapest to deliver in the base case and how to calculate the theoretical future price using cost of carry formula: F0 = (S0 - I) * e^(rT) Where:
- F0 = Theoretical Futures Price
- S0 = Spot Price
- I = Present Value of Coupons
- r = Financing Rate
- T = Time to Future's Delivery Date in years
However, I'm not sure what the convention is on interest rate shocks. In a shock scenario the cheapest to deliver could theoretically change, but you'd need to know the cheapest to deliver in order to calculate the theoretical future price for a given rate shock. You need the future price to calculate the cheapest to deliver. Questions:
- Given the circular reference, is it best practice to assume that the cheapest to deliver security remains the same, and simply shock that security's price, then input that to the theoretical future price formula? Or is there some sort of iterative solution?
- Is using the theoretical future price to calculate the price for the shocks the correct approach?
- For the delivery options' value, is it usually taken into account or handwaved away? The value could be determined in the base case, is it good enough to apply that value to all the shocked prices, or is a more quantitative approach usually used?
## Answer by Andrea (score 0, accepted)
https://quant.stackexchange.com/a/81264
You don't need the bond future price to compute the CTD.
Many people use the IRR to select the CTD, but this is the circular argument: in reality a futures contract is an option on a basket of bonds, and it should be priceable from the bonds. Since the futures is a lot more liquid than bonds, people look at it the wrong way round and we have circular dependency.
Your formula above misses the Conversion Factor and should read as
$F0=\min_i \frac{(S0_i - I_i) * e^{r_i T}}{CF_i}$
(assuming by Spot Price you mean Dirty Price)
This does not take into account the option, which can be incorporated with more or less complex stochastic rate models.
Then, if you rather not make it too complicated, you can fix the CTD, which is valid for small shocks.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.