<p><span class="math-container">$e^{-rt}$</span> is the <em>continuous</em> discounting factor while <span class="math-container">$(1 + r)^{-t}$</span> is its discrete counterpart.</p>

<p>Identically/equivalently, there is the continuous manner of computing factors of variation, e.g. <span class="math-container">$1 + \ln \frac{x_{0+t}}{x_0}$</span>, and its discrete counterpart <span class="math-container">$\frac{x_{0+t}}{x_0}$</span>.</p>

<p>It all depends on whether you think you are dealing with a continuous quantity or not.</p>

<p>In asset pricing (or finance in general) the underlying notion is that of <a href="https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding" rel="nofollow noreferrer">continuous (interest) compounding</a>.</p>

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2020-05-11T18:34:00.190Z
2020-05-11T19:30:32.947Z
2020-05-11T19:30:32.947Z
keepAlive
<p>So I was reading <a href="https://analystprep.com/study-notes/frm/part-1/binomial-trees/" rel="nofollow noreferrer">here</a> where they calculate the expected value of an option at present given the expected value of the option in a year by calculating</p>

<p><span class="math-container">$$C_0 = C_1 e^{(-r)}$$</span></p>

<p>where <em>r</em> is the interest rate. But where does <em>e</em> come from here? Shouldn't present value just be calculated using</p>

<p><span class="math-container">$$C_0 = C_1 / (1 + r)$$</span></p>

<p>What am I missing?</p>

<interest-rate><asset-pricing>
Calculating present value using Euler's number
Ferus
# Calculating present value using Euler's number

(Asked by: username_0 on 2020-05-11)

So I was reading [here](https://analystprep.com/study-notes/frm/part-1/binomial-trees/) where they calculate the expected value of an option at present given the expected value of the option in a year by calculating

$$C\_0 = C\_1 e^{(-r)}$$

where *r* is the interest rate. But where does *e* come from here? Shouldn't present value just be calculated using

$$C\_0 = C\_1 / (1 + r)$$

What am I missing?

**Tags:** interest-rate, asset-pricing

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## 1 answers

### Answer by username_1 on 2020-05-11. Score: 3 | Accepted answer

$e^{-rt}$ is the *continuous* discounting factor while $(1 + r)^{-t}$ is its discrete counterpart.

Identically/equivalently, there is the continuous manner of computing factors of variation, e.g. $1 + \ln \frac{x\_{0+t}}{x\_0}$, and its discrete counterpart $\frac{x\_{0+t}}{x\_0}$.

It all depends on whether you think you are dealing with a continuous quantity or not.

In asset pricing (or finance in general) the underlying notion is that of [continuous (interest) compounding](https://en.wikipedia.org/wiki/Compound_interest#Continuous_compounding).


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