Differentiating a Conditional Discount-Bond Expectation in the Ho–Lee Model

Summary

The document considers a short rate that follows a Brownian model with deterministic time-dependent drift and asks how to differentiate the conditional expectation of a discount factor from the current time to a future date. The proposed calculation differentiates the exponential inside the expectation and factors out the current short rate. The question notes that this result conflicts with a separate Ho–Lee bond-dynamics derivation, signaling that the time dependence of the integration limits and the conditioning information need careful treatment.

The reply recommends rewriting the integrated short rate using stochastic integration by parts, then substituting the short-rate dynamics to separate known and random terms before taking the conditional expectation and differentiating. It points readers toward a fuller derivation but does not provide it here. The document therefore offers a useful calculation route, not a complete solution; it does not work through the resulting expectation or resolve the original discrepancy.

Key ideas

• A conditional discount-factor expectation depends on both the integration limits and the information available at the current time.
• Differentiating the exponential alone may not account for every time dependence in the expression.
• Stochastic integration by parts can rewrite the integrated short rate in terms of the current rate and its increments.
• After substitution of the rate dynamics, conditional expectation can be evaluated before completing the time derivative.

Tags

# Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$


# Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$












Let $r$ a random process defined by :

$$dr_t=\theta(t)dt + \sigma dW_t$$

$\theta$ is deterministic in $t$ and $W$ a brownian motion.

I don't know where my calculation below is going wrong :

Let $R=\int r(s)ds$

then : $$\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right] = \mathop{\mathbb{E}}\left[ \frac{d}{dt} e^{-(R_T - R_t)}|\mathscr{F}_t \right] = \mathop{\mathbb{E}}\left[ r(t) e^{-(R_T - R_t)}|\mathscr{F}_t \right] = r(t) \mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$$

But regarding this question Bond dynamics in Ho Lee model my computation is not correct

Any help please?

## Answer by FunnyBuzer (score 2)

https://quant.stackexchange.com/a/44054

You can use the stochastic integration by parts to solve the integral inside the expectation: $$\int_t^Tr_sds=Tr_T-tr_t-\int_t^Tsdr_s=(T-t)r_t+\int_t^T(T-s)\underbrace{dr_s}_{:=\theta(s)ds+\sigma dW_s}$$ Solve this integral and then take the expectation and solve the derivative. For a complete answer see Ho and lee derivation for short rates model.

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