Interpreting Numerical Delta for Bond Options and Rate Derivatives
Summary
The document examines why a numerically estimated delta for a zero-coupon bond call may differ from the analytical delta under a Vasicek-based option formula. It describes using central differences on option values as the underlying bond price changes, and notes that the numerical result appears to rise more slowly as the option moves in the money. Similar confusion is reported for swaptions.
The response points to a mismatch in what is being measured: the analytical delta is being compared as a percentage of the underlying’s delta, while the numerical calculation may not account for the underlying bond’s DV01. Differences in rate sensitivity can also become more pronounced at longer maturities. The document offers this as a diagnostic explanation, rather than a complete numerical method; a follow-up BPV-based relationship is reported by the questioner as unsuccessful.
Key ideas
- Option delta depends on the definition and units of the underlying sensitivity being measured.
- Finite differences in bond price should be compared consistently with the analytical bond price delta.
- The underlying bond’s DV01 may be needed when estimating option sensitivity through interest rates.
- Longer maturities can magnify distortions when rate sensitivity is not accounted for.
- The proposed explanation is diagnostic and the questioner reports that a BPV-based adjustment did not resolve the issue.
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Full text
# Numerical delta of Bond Options
# Numerical delta of Bond Options
I'm trying to calculate the delta for bond Call options. I'm using the vasicek model which gives the following solution for a Zero-coupon bond call option:
$Z = N P(t,S) \Phi(d_1) - K P(t,T) \Phi(d_2).$
Differentiating the above equation with respect to the underlying $P(t,S)$ gives the classical delta
$\Delta = N \Phi(d_1).$
where
$d_1 = \frac{ln \left(\frac{NP(t,S)}{KP(t,T)}\right) + \frac{\sigma_p^2}{2}}{\sigma_p}$ , $d_2 = d_1 - \sigma_p$.
Comparing the analytical delta with the numerical first derivative (with the call formula as input) using a central difference scheme
$ Z' = \frac{Z(r_{i+1})-Z(r_{i-1})}{P(t,S)(r_{i+1})-P(t,S)(r_{i-1})} $
or any other higher-order stencil gives a very different result. For example, while the former approaches 1 as the price of P(t,S) increases the latter doesn't reach 0.7. See the figure below. The X-axis is the price of the bond $P(t,S)$. Here $t<T<S$.
These discrepancies increase whith the Option maturity.
Is there any trick to find numerical deltas for IR derivatives? I faced the same problem with Swaptions.
I think my confusion lies in the underlying.
Thank you. Allan
## Answer by closedloop (score 1)
https://quant.stackexchange.com/a/17096
It appears that you are plotting your analytical delta as a % of the delta of the underlying. This is why the delta converges to 100%
As for the numerical delta, it could be that you are not adjusting for the DV01 of the underlying. This would explain why the numerical delta still increases as the option gets more in the money and why the distortion is larger for longer maturities (larger DV01).
## Answer by Allan Jonathan (score 0)
https://quant.stackexchange.com/a/16681
I tried to use the BPV/delta relashionship
$\Delta = \frac{ \frac{\partial Z}{\partial r}-\frac{\partial P(t,T)}{\partial r}\frac{Z}{P(t,T)} } {\frac{\partial P(t,S)}{\partial r}}$
but it doesn't work as well.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.