Discounting Under Stochastic Interest Rates

Summary

The document explains why the value at time t of a unit payment due at time T is expressed using the accumulated short rate from t to T. It defines the growth of a unit investment from time zero to each date, then obtains the discount factor by dividing current accumulated wealth by wealth at maturity. This ratio simplifies to the exponential of the negative rate integral over the remaining period.

The explanation clarifies the distinction between an integral from zero to t, which describes past accumulation, and one from t to T, which discounts a future payment back to t. The result is an algebraic identity for a realized path of rates; it does not derive a particular stochastic interest-rate model or address expectations, risk premia, or option valuation beyond the discount factor. The document is a concise conceptual clarification rather than a broader treatment of pricing under stochastic rates.

Key ideas

• A unit invested at time zero grows according to the accumulated short rate through the current date.
• The value at time t of a unit paid at T is the ratio of accumulated wealth at t to accumulated wealth at T.
• That ratio equals the exponential of the negative integral of rates from t to T.
• The integral from zero to t measures past growth, while the remaining-period integral performs discounting.

Tags

# Stochastic Interest Rates in Option pricing


# Stochastic Interest Rates in Option pricing












My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from `0` to `t`. Surely, it's a negative integral from `t` to `T`? Her notes are full of mistakes so I cannot figure out if I'm not getting something or if she's made a mistake.

## Answer by Kevin (score 1)

https://quant.stackexchange.com/a/53941

$Value(t)$ is the value at time $t$ of receiving \$1 at time $T$. Thus, indeed $$Value(t)=\exp\left(-\int_t^T r_u\mathrm{d}u\right).$$

This expression is also known as (stochastic) discount factor. Let $Wealth(t)=\exp\left(\int_0^t r_u\mathrm{d}u\right)$ be the value at time $t$ of investing \$1 at time $0$. Your \$1 grows at the stochastic rate $r_t$. Then, \begin{align*} Value(t)&=\frac{Wealth(t)}{Wealth(T)} \\ &= \frac{\exp\left(\int_0^t r_u\mathrm{d}u\right)}{\exp\left(\int_0^T r_u\mathrm{d}u\right)} \\ &= \exp\left(\int_0^t r_u\mathrm{d}u-\int_0^T r_u\mathrm{d}u\right) \\ &= \exp\left(-\int_t^T r_u\mathrm{d}u\right) \end{align*} It's nothing else than discounting (computing present values).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.