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Ho–Lee Bond Pricing and Volatility’s Effect on Swap Value

Article Quant Q&A · Author: Larry
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Summary

The document explains why changing short-rate volatility can affect an interest rate swap’s value in the Ho–Lee model, despite the intuition that swap pricing depends on current zero-coupon bond prices. It presents the short rate as a drift term plus a constant-volatility Brownian component and prices a zero-coupon bond as the risk-neutral expectation of discounted future cash flows.

Integrating the short rate over the bond’s life gives a normally distributed random component. Taking the expectation of the exponential discount factor introduces a variance contribution, so the bond price formula contains a term proportional to volatility squared and the cube of time to maturity. Since swap valuation uses bond prices, volatility can therefore affect the model’s prices. The explanation assumes the stated Ho–Lee dynamics and does not discuss calibration or reconcile the result with a fully calibrated initial yield curve, where the drift may need adjustment to fit observed bond prices.

Key ideas

  • In the Ho–Lee model, the short rate evolves with deterministic drift and a Brownian volatility term.
  • The integrated short rate contains a normally distributed component with variance that grows with the cube of the time horizon.
  • Taking the expected discount factor introduces a volatility-dependent term in the zero-coupon bond price.
  • Because swaps are valued using discount factors or bond prices, volatility can affect their modeled value.

Tags

Fixed incomeDerivatives pricingVolatility
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# Pricing interest rate swap in Ho Lee model


# Pricing interest rate swap in Ho Lee model












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.

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/30997

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 $\sigma$.

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