# Integrating Interest and Dividend Functions

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.

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