# Zero-coupon bond pricing equation derivation

I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation?

The equation is:

$$\frac{\partial B}{\partial t} + \frac{1}{2} (\alpha r - \beta) \frac{\partial^{2} B}{\partial r^{2}} + (\eta - \gamma r) \frac{\partial B}{\partial r} - rB = 0$$

Where the bond price is:

$$B(t;T) = B(T;T) e^{- \int_{t}^{T} r(s) ds}$$

I assume hes using the Ito Lemma to get this, but how is he applying this?

The equation also looks like the Black-Scholes equation, is there any link there?

Chawla, 2010

• Your bond price formula is wrong. – Gordon Aug 7 at 15:08
• I forgot to mention that this is in a short rate model – Tara Aug 7 at 15:42
• What is your short rate dynamics? The equation $B(t;T) = B(T;T) e^{- \int_{t}^{T} r(s) ds}$ is wrong, unless the short rate $r$ is deterministic. – Gordon Aug 7 at 18:18

Some terms are not explained in the restricted screenshot provided like $$\beta$$ and $$\gamma$$ however, from What I see documented, my suspect is that they are using a Taylor expansion (2nd order) to proxy the generic variation of B after a change in r and t (hint: they indeed assume that B is differentiable at least one time with respect to t). It is also clear that they are assuming that B is also at least twice differentiable with respect to r. So they are representing the new generic price after a change in r and t as the sum of the old price and the change, where the change is proxied by the use of Taylor expansion (that roughly speaking uses the derivatives of the function times the change in the respective variables). In that it is similar to Taylor used in BS. It is pretty clear from the formulas even if, as I said at the beginning, I do not see some terms explained, so I can just give you an intuition.
• Ahh, I forgot to mention, the author mentions "the four functions $\alpha, \beta, \gamma$ and $\eta$ are at our disposal to fit the data so that the random walk for $r$ has desired economic feasability properties" – Tara Aug 7 at 15:45