Why a Bond Price Is Not Generally Lognormal When Its Yield Is

Summary

The document considers whether a bond price is lognormally distributed when a yield-related process is lognormal. The answer explains that the usual condition for a lognormal price process—a constant proportional diffusion coefficient—does not generally hold here. In the cited price dynamics, both a duration-like term and the yield factor depend on the yield and therefore on the bond price, making the proportional volatility state-dependent. Consequently, lognormality of the yield alone does not imply lognormality of a finite-maturity bond price.

A special case is given for a perpetual bond with level coupons. Its price is coupon income divided by yield, so its proportional price change is the negative proportional yield change; under the stated setup, a lognormal yield then implies a lognormal price. This is a model-specific exception rather than a general bond-pricing result. The document provides a qualitative argument but does not spell out all model assumptions or establish distributional properties for other coupon structures.

Key ideas

• A price process is not generally lognormal when its proportional volatility depends on the state.
• A yield being lognormal does not by itself make a finite-maturity bond price lognormal.
• For a level-coupon perpetuity, price is inversely proportional to yield.
• The perpetuity relationship makes the proportional price change the negative proportional yield change under the stated setup.
• The special case does not establish lognormality for other bond structures.

Tags

# Is this process log normally distributed?


# Is this process log normally distributed?












I came across a question that I guess $P$ is lognormally distributed.

where $y_n$ is log-normally distributed.

Am I right on the guessing?

Here is the full solution if interested.( my guessing comes from the last two equations)

## Answer by dm63 (score 4, accepted)

https://quant.stackexchange.com/a/68513

I’m sorry to say you’re not correct in your conclusion. The basic problem is that in the second to last equation $$dP/P=D^*\sigma_y y_n dW_t$$ the $D^* $ is not constant but is a function of $y_n$ and therefore of $P$. Also you have a $y_n$ present which also is a function of $P$. Hence $dP/P$ is not constant and therefore P is not in general lognormal.

Interestingly there is a case where $P$ is indeed lognormal if $y_n$ is. Consider the case of a perpetual bond with level coupons C where n is infinite. Then you can show from the first equation that the price $ P=C/y$. Hence $dP=-Cdy/y^2$ and $dP/P= -dy/y$. Then lognormality of $y$ implies lognormality of $P$. But for all other cases it is not true.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.