Bond Futures Option Delta and Interest-Rate Risk
Summary
The note distinguishes price delta and gamma for options on bond futures from interest-rate delta and gamma. Price delta measures how an option’s value responds to a change in the futures price; the response explains that interest-rate delta generally refers to sensitivity to rate risk factors in the underlying instruments or valuation framework. It therefore should not automatically be read as rho, which concerns sensitivity to the risk-free funding rate used in theoretical pricing.
The explanation frames portfolio value as a function of multiple risk factors and uses partial derivatives to describe first-order sensitivities. Risk systems may group and subdivide factors, such as different rate curves, and account for correlations between them. It mentions parallel yield-curve shifts as one common simplification. The answer is general rather than specific to a particular CME contract, model, or regulatory framework; precise meanings depend on how a firm defines and reports its risk measures.
Key ideas
- Price delta measures option value sensitivity to the underlying futures price.
- Interest-rate delta usually captures exposure to rate risk factors and is not necessarily rho.
- Rho concerns sensitivity to the risk-free rate used in theoretical funding assumptions.
- Risk models can divide rates into separate curves and combine exposures using correlations.
- Parallel yield-curve shifts are a common simplification in risk measurement.
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Full text
# Delta on Bond Future Options
# Delta on Bond Future Options
When talking about Options on Bond Future on CME (American options), we have 2 definitions of Delta and Gamma. One is 'Price Delta/Gamma' and one is 'Interest Rate Delta/Gamma'.
My understanding is that Price Delta/Gamma is the classic definition of Option delta/gamma (sensitivity and its sensitivity of option's price to changes in underlying's price).
At face value, 'Interest Rate Delta/Gamma' implies sensitivity and its sensitivity of option's price to changes to interest rates.
Is that the right understanding? If so, isn't "Interest Rate Delta" same as Rho?
## Answer by Alexander McFarlane (score 1)
https://quant.stackexchange.com/a/36004
Interest rate delta depends on how you define it but generally (according to banking regulation anyway) is not the same as $\rho$
- $\rho$ is related to the risk free rate to do with the theoretical funding of a position assuming no arbitrage.
- Interest rate delta is related to the sensitivities present in the basket of underlyings w.r.t. different rates curves
The simplest way to understand this is the treat the PV as a function $f(\mathbf{x})$ of a vector of risk factors (e.g. rates, credit spread etc.), $\mathbf{x}$, that you represent with a Taylor expansion.
If the $i^{\text{th}}$ sensitivity is a single rate curve there exists a first order term in the expansion
$$ f(\mathbf{x}) \propto \frac{\partial f (\mathbf{x})}{\partial x_i} $$ which you could also denote $\Delta_i$
As touched on above, what happens in risk management is that $\mathbf{x}$ is decomposed into subsets that are treated as independent (with associated correlations that can be applied later) $$\mathbf{x} = \text{Rates} + \text{Credit} + \dots$$
and this can go further in rates as
$$\text{Rates} = 3m Libor + 6mLibor + 6mEuribor + \dots$$
A lot of risk models will currently look at a parallel shifts in the yield curve and then apply some kind of correlation factor at the various stages of decomposition aboveShown 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.