In Ho Lee model, assuming risk neutral probability is not exactly 0.5, would a change in the volatility of short-term rate affect the price of an interest rate swap? My intuition tells me no as interest rate swap price should only depend on prices of zero at t=0 but my model is throwing a different answer.

Would you it be possible to have some help on both mathematical and intuitive explanations, please?

Many thanks.

  • $\begingroup$ I don't understand your question. Please clarify it. $\endgroup$
    – user16651
    Nov 13, 2016 at 20:31

1 Answer 1


Under the Ho-lee model, \begin{align*} dr_t = \theta_t dt + \sigma dW_t. \end{align*} Then, the price at time $t$ of a zero-coupon bond with maturity $T$ and unit notional is given by \begin{align*} P(t, T) = E\left(e^{-\int_t^T r_s ds} \mid \mathcal{F}_t \right), \end{align*} where $\mathcal{F}_t$ is the information set at time $t$. Note that, for any $s\ge t \ge 0$, \begin{align*} r_s = r_t + \int_t^s \theta_u du + \sigma\int_t^s dW_u. \end{align*} Therfore, \begin{align*} \int_t^T r_s ds &=r_t(T-t) + \int_t^T\left(\int_t^s \theta_u du \right)ds+ \sigma \int_t^T\left(\int_t^s dW_u\right)ds\\ &=r_t(T-t) + \int_t^T\left(\int_u^T \theta_u ds \right)du+ \sigma \int_t^T\left(\int_u^T ds\right) dW_u\\ &=r_t(T-t) + \int_t^T (T-u)\theta_u du + \sigma \int_t^T (T-u) dW_u. \end{align*} Note that $\int_t^T (T-u) dW_u$ is independent of $\mathcal{F}_t$ and normal with zero mean and variance \begin{align*} \int_t^T (T-u)^2 du &= \frac{1}{3}(T-t)^3. \end{align*} Consequently, \begin{align*} P(t, T) &= E\left(e^{-\int_t^T r_s ds} \mid \mathcal{F}_t \right)\\ &=e^{-r_t(T-t) -\int_t^T (T-u)\theta_u du + \frac{\sigma^2}{6}(T-t)^3}. \end{align*} For $t=0$, then \begin{align*} P(0, T)=e^{-r_0 T -\int_0^T (T-u)\theta_u du + \frac{\sigma^2}{6}T^3}.\tag{1} \end{align*} From $(1)$, we see clearly that the bond price depends on the volatility $\sigma$.

  • $\begingroup$ The intuition is that the bond price is a convex function of rates, hence long rate volatility. In practice you would have to adjust the theta parameters to offset this effect , since zero coupon bond prices are known. $\endgroup$
    – dm63
    Dec 14, 2016 at 10:52
  • $\begingroup$ Thanks @dm63. The idea is of course similar to that of the Hull-White model. The volatility parameter can be calibrated using certain option prices, and then the theta parameter can be calibrated to the zero coupon bond prices. If instead we simulate the zero coupon bond prices based on the model, these prices will also depend on the volatility parameter. $\endgroup$
    – Gordon
    Dec 14, 2016 at 15:00

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