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Deriving Risk-Neutral Bond Price Dynamics in an Affine Rate Model

Article Quant Q&A · Author: Mr Frog
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Summary

The note asks for the risk-neutral dynamics of a zero-coupon bond whose price is exponential-affine in a short rate following a linear diffusion. It gives the time equations for the affine coefficients and compares applying Itô’s lemma to the bond price with applying it to the log price.

Applying Itô’s lemma directly to the price and substituting the coefficient equations cancels the volatility-related drift terms. The resulting bond-price process has the short rate as its instantaneous proportional drift and has diffusion proportional to the negative of the affine loading and the short-rate volatility. The log-price calculation provides an intermediate expression that still contains coefficient derivatives. These dynamics follow under the specified risk-neutral model; the note does not discuss calibration, parameter restrictions, or extensions such as a constant rate drift term.

Key ideas

  • Under the stated risk-neutral model, the bond price is exponential-affine in the short rate.
  • Itô’s lemma can be applied directly to the price or to its logarithm.
  • Substituting the affine coefficient equations into the price dynamics leaves a proportional drift equal to the short rate.
  • The bond’s diffusion loading is the negative affine rate loading multiplied by short-rate volatility.

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Fixed incomeDerivatives pricingStatistics
Full text
# What is the Q-dynamics of affine bond prices when r is described by the given model?


# What is the Q-dynamics of affine bond prices when r is described by the given model?












Assuming an Affine term structure model, where bond prices arebe defined as: $$P(t,T)=\exp({A(t,T)-B(t,T)r_t)}$$ and describing the Q-dynamics of the short rate according to the model: $$dr_t=ar_tdt+\sigma dW_t$$hence having: $$ \partial_t{A(t,T)}=-\frac{\sigma^2}{2}B^2(t,T) \\\partial_tB(t,T)=-aB(t,T)-1$$ What is the Q-dynamics of the bond prices $dp(t,T)$?

Would it be correct to start from the P-dyanimics: $$ dp(t,T)=((B(t,T)+1)a+1)r_tp(t,T)dt-\sigma p(t,T)dW_t$$and perform the change of measure by defining the new Brownian motion as $$dW_t^{\mathcal{Q}}=dW_t-\frac{(B(t,T)+1)ar_t}{\sigma}dt$$

## Answer by fes (score 1, accepted)

https://quant.stackexchange.com/a/63327

You can simply use Ito's lemma under the risk neutral measure $Q$.For the log-bond price $p(t,T)$ this gives

$$dp(t,T)=(A_t(t,T)-B_t(t,T)r_t)dt-B(t,T)dr_t$$

$$=[A_t(t,T)-(B_t(t,T)+B(t,T)a)r_t]dt-B(t,T)\sigma dW_t$$

Here $A_t(t,T)$ and $B_t(t,T)$ are partial derivatives wrt $t$ and $W_t$ is Wiener process under $Q$.

## Answer by tommaso1311 (score 1)

https://quant.stackexchange.com/a/63476

Just adding my two cents. Without taking the logarithm of the price, the Ito's Lemma should result in:

$d p(t,T) = \left( \partial_t A(t,T) - \partial_t B(t,T) r + \frac{1}{2}\sigma^2B(t,T)^2 \right)p(t,T) dt - B(t,T) p(t,T) dr_t$

substituting now the partial derivatives and the differential $dr_t$, and simplifying the identical terms:

$d p(t,T) = r_t p(t,T) d t - \sigma B(t,T) p(t,T) d W_t$

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