# 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?

• The rho is usually associated with the discount factor for the option where as the bond futures's sensitivity to interest rates is a longer term rate. I have heard people refer to delta as rho in this case, so there may be an unclear convention. The option sensitivity to rates is very tricky because you need the entire basket of deliverables, all of their repo rates, and a good model for the delivery option to figure out the sensitivity to a change in rates - along with the shift in rates comes a change in probability of delivery for any bond in the basket. – FinanceGuyThatCantCode May 8 '17 at 17:30
• Interesting thanks. Is this IRDelta a more useful measure for owners of an option than the classic Price Delta ? – options May 9 '17 at 14:01
• your answer should be a comment above really...but the IR delta is much more useful to me - but there is a lot more heavy lifting involved in getting it right - mostly due to all the work described in my comment above. – FinanceGuyThatCantCode May 9 '17 at 14:26
• oh yea - and you probably want a sensitivity to parallel shifts in the yield curve because there are many bonds with different maturities in the basket - and sensitivities to different parts of the yield curve are good to consider. You might even bucket your risks as fixed income people often do, based on your sensitivities to changes in the 3M buckets of the curve (i.e. what is my risk for a 1bp change in the 2y-2y3m forward rate?) - don't need to do that unless you are taking big risk though. – FinanceGuyThatCantCode May 9 '17 at 15:09

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 above