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Bond Pricing When the Short Rate Follows Geometric Brownian Motion

Article Quant Q&A · Author: bcf
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Summary

The document asks whether a pure discount bond has an analytical price when its risk-neutral short rate follows a geometric Brownian motion. It expresses the bond price as the expected exponential of the accumulated short rate and observes that the rate at a given time is lognormally distributed. The main challenge is the distribution of the time integral of that process, rather than the distribution of the rate itself.

The response identifies the short-rate model as the same geometric Brownian motion used for stock prices in Black–Scholes and connects the integral to the pricing of Asian options. It points toward that literature as relevant background and concludes that this setup has no closed-form bond price. The exchange supplies no derivation, numerical approximation, or alternative pricing method, so it identifies a useful connection and limitation without showing how to compute the expectation in practice.

Key ideas

  • The bond price is the risk-neutral expectation of the exponential of the negative integrated short rate.
  • A geometric Brownian motion short rate is lognormally distributed at fixed times.
  • Pricing requires the distribution or expectation of the time integral of that process.
  • The integral of geometric Brownian motion is related to Asian option valuation.
  • The response states that the model has no closed-form bond price but provides no computational alternative.

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Fixed incomeDerivatives pricingStatistics
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# Analytical Bond Price under Rendlemen-Bartter?


# Analytical Bond Price under Rendlemen-Bartter?












Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process $$ dr_t = ar_t dt + \sigma r_t dW_t, $$ does anyone know of an analytical bond price formula? We know that the time $t$ price of a pure discount $T$-bond $P(t,T)$ is $$ P(t,T) = E_Q\left(e^{-\int_t^T r_s \, ds}\right). $$ We also know that $$ r_t \mid r_0 \sim \log \mathcal{N}\left(\log r_0 + \left(a + \frac{\sigma^2}{2}\right)t, \sigma^2 t\right). $$ Now let $R := \int_t^T r_s ds$. The bond pricing equation becomes $$ P(t,T) = E_Q(e^{-R}), $$ so the questions is really, "what is the MGF for $-R$?" I'm having some trouble working this out for myself, in particular, what is the distribution of $R$?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/25492

I edit this answer to give more details.

The process for $r$ above is the geometric Brownian motion (GBM) used to model stock prices in the Black-Scholes framework. Thus the question is about (the expectation of the) exponetial of the integral of GBM. The intergral of GBM is closely connected to Asian options. Thus one can study the literature about this topic.

According to this https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian3_doc.pdf

I would answer No: there is no closed formula for $P$ in your model.

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