Serving the Quantitative Finance Community

pentagram
Topic Author
Posts: 32
Joined: November 9th, 2010, 11:40 am

### Bond SDE

Hi,

I was trying to replicate the convexity adjustment between FRAs and futures under the Ho-Lee model : convexity adjustment Ho-Lee

As part of doing this the first step was to replicate the SDE for the bond, however, I end up with a different formula to the one Hull calculates in the above pdf.

The Bond SDE formula which Hull uses is,
dP(t,T) = r(t)P(t,T)dt - (T-t)σP(t,T)dz

where P(t,T) = A(t,T)e^[-r(T-t)]
and the Ho-Lee diffusion eq for the short term rate is dr = θ(t)dt + σ dz
By using applying Ito on P, what I get is a different SDE
dP = [r * P(t,T) + (dA/dt)e^[-r(T-t)]]dt - (T-t)P(t,T)dr + [0.5(T-t)^2 P(t,T)σ^2 ]dz
by replacing dr with Ho-Lee SDE, this boils down to ()dt+()dz but the ()dt term of the Bond SDE ends up being very different to Hull's formula,
[rP + (dA/dt)e^(-r(T-t)) - (T-t)Pθ + 0.5P (T-t)^2σ^2]dt
What am I missing/how can I bridge the gap I get by applying Ito and Hull's Bond SDE ?

bearish
Posts: 6419
Joined: February 3rd, 2011, 2:19 pm

### Re: Bond SDE

If you multiply and divide the dA/dt term by A, you can scale out P. Once you remove the r(t)dt term you are left with an expression that has to be equal to zero to satisfy the risk neutral drift requirement, and this is one way to solve for the functional form of A given $\theta$. I personally like to start from the bond price and derive the rate dynamics — on account that it’s easier to differentiate than to integrate — but as long as the model you work with is internally consistent, you should end up in the same place.

pentagram
Topic Author
Posts: 32
Joined: November 9th, 2010, 11:40 am

### Re: Bond SDE

If you multiply and divide the dA/dt term by A, you can scale out P. Once you remove the r(t)dt term you are left with an expression that has to be equal to zero to satisfy the risk neutral drift requirement, and this is one way to solve for the functional form of A given $\theta$. I personally like to start from the bond price and derive the rate dynamics — on account that it’s easier to differentiate than to integrate — but as long as the model you work with is internally consistent, you should end up in the same place.
Thank you for your response. How do we know P has a risk neutral drift ? e.g.
ln[P(t,T_x)]

does not have a risk neutral drift.

bearish
Posts: 6419
Joined: February 3rd, 2011, 2:19 pm

### Re: Bond SDE

Ah - good question! Because P is the price of a non-dividend paying asset. $ln (P)$ is a non-linear (concave) function of a price, and thus not investable.

pentagram
Topic Author
Posts: 32
Joined: November 9th, 2010, 11:40 am

### Re: Bond SDE

Ah - good question! Because P is the price of a non-dividend paying asset. $ln (P)$ is a non-linear (concave) function of a price, and thus not investable.
Thanks, how would you go about "the other way around" deriving the rate dynamics from the bond price?