Deriving the Zero-Coupon Bond Pricing PDE from Rate Dynamics
Summary
The question asks how a zero-coupon bond pricing equation with a short-rate process is derived and whether it is related to the Black-Scholes equation. The response gives an intuitive account based on expanding the bond price as a function of time and the interest rate. It assumes the price is differentiable in time and twice differentiable in the short rate, then uses a second-order Taylor expansion to represent changes as time and rate move.
This explains why the PDE has a familiar diffusion-equation structure, but it does not provide a complete derivation. The answer notes that some coefficients in the displayed equation are not defined in the supplied material, so their role cannot be confirmed. It therefore offers intuition about the calculus behind the equation rather than a proof of the bond-pricing relation or a full mapping to Black-Scholes. A complete derivation would require the specified short-rate dynamics and pricing assumptions.
Key ideas
- A bond price depending on time and the short rate can be expanded in those variables to describe small changes.
- The derivation requires time differentiability and at least second-order differentiability with respect to the short rate.
- The PDE resembles Black-Scholes because both involve time evolution and a second-order diffusion term.
- The response is only intuitive because the source material does not define all coefficients or provide full derivation details.
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# Zero-coupon bond pricing equation derivation
# 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
## Answer by Fr1 (score 1)
https://quant.stackexchange.com/a/46994
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.
For other examples of the use of Taylor expansion Taylor series expansion (Volatility Trading book) explanation sought and http://kfoster.ccny.cuny.edu/classes/spring2010/eco275/lecturenotes7.htmlShown 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.