Assume a money market has interest rate process $R(t)$. In Shreve's Stochastic Calculus for Finance II, formula (5.2.17) on page 215 defines the discounted process as $$ D(t) = e^{-\int_0^t R(s) ds}. $$

Why is $D(t)$ called "discounted" process?

Does it mean that any value at time t times $D(t)$ will give its present value at time $0$?

  • $\begingroup$ Note that this question is really barely on topic: I guess you're not a professional quant are you? Anyway, a quick answer was enough, but please look at the faq in the future. $\endgroup$ – SRKX Mar 27 '13 at 13:09
  • $\begingroup$ @SRKX: I have taken a course for the first four chapters of the book by Shreve, and I am learning the rest of the book now. $\endgroup$ – Ethan Mar 27 '13 at 13:17
  • $\begingroup$ I get that. But the question is trivial; you even guessed the answer inside. So I guess you can see why I say that if you were a professional quant, you wouldn't have asked it. You can accept the answer if you're OK with it. $\endgroup$ – SRKX Mar 27 '13 at 13:23

Indeed, $D(t)$ is the discount factor used to compute the present value of a cash flow at time $t$:

$$PV = D(t) \cdot CF_t$$

It is more convenient to write it that way when you assume stochastic interest rate because you don't have to write the integral all the time.


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