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How are interest rate and dividend functions integrated over time in practice?

For example, what does it mean in practice to discount a current price by $e^{\int_{t_m}^{T}r_s ds }$ where $r_s$ is the interest rate function or $e^{\int_{t_m}^{T}\delta_s ds }$ where $\delta_s$ is the dividend fuction.

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For a fixed interest rate or dividend payment/yield, integrating the function becomes:

$\large{P_T = } \Large{e^{\int_t^T r_s ds} = \Large{e^{r_s (T - t)}}}$

$P_T$ is usually understood to be the future value of a zero-coupon bond with a face value of \$1. The present value, $P_t$, is simply then:

$\large{P_t = } \Large{e^{\int_t^T -r_s ds} = \Large{e^{r_s (t - T)}}}$

If $r_s$ is a function, then that must be accounted for in the integration. For complex functions, it is common to estimate the quantity through discretization techniques. When discounting a discrete function, a product function may be used:

$\large{P_T=\prod_{t}^T(r_s + 1)}$;

for fixed $\large{r_s}$, $\large{P_T \to (r + 1)^{(T - t) + 1}}$

If $r_s$ is not a deterministic function (i.e., it is stochastic), we must now take the expectation under some risk-neutral measure, $\mathbb{Q}$, where the price of a zero-coupon bond at time, $T$, equals:

$\large{ \mathbb{E}[ P_T] = \mathbb{E^Q} [e^{\int _{t}^{T}r_{s}\,ds}|{\mathcal{F}}_{t}]}$

where $\mathcal{F_t}$ is a natural filtration process.

There exist many methods and models by which to express the expectation of a stochastic interest rate. E.g.: Cox-Ingersol-Ross; Ho-Lee; Hull-White; Merton; etc... The "best" model just depends on what you're trying to do.

When dividends are also present, it is common and mathematically convenient to assume they are a fixed proportion of $P_t$. The effect of proportional dividend is often thought of a exerting a "decay" on future value; it is thus common for pricing models to take the expectation of a "forward" value, $F_t$.

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