<p>I heard in a lecture at my university, that measure theory is applied in finance. This field also operates with negative probabilities a lot.</p>

<p>Anyways, I don't see the practical intuition behind using negative probabilities in a bond-pricing model. You can check this article by <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1773077" rel="nofollow">Burgin and Meissner: Negative Probabilities in Financial Modeling (Wilmott Magazine March 2012)</a>.</p>

<p>I think you should migrate your question to <a href="http://quant.stackexchange.com">Quant SE</a>.</p>

CC BY-SA 3.0
2016-07-17T14:04:49.893Z
2016-07-17T14:04:49.893Z
Übel Yildmar
<p>In <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.592.2043&amp;rep=rep1&amp;type=pdf" rel="nofollow">Half of a Coin: Negative Probabilities</a>, the author mentions bond duration.</p>

<p>Suppose we have payments at times $t = 1,2,...,n$ denoted respectively by $R_1, R_2, ..., R_n$ and the discount factor is $v = \frac{1}{1+i}$ where $i$ is effective interest rate. Then the bond value today is given by</p>

<p>$$B = \sum_{t=1}^{n} R_tv^t$$</p>

<p>The bond duration is</p>

<p>$$D = \frac{\sum_{t=1}^{n} tR_tv^t}{\sum_{t=1}^{n} R_tv^t}$$</p>

<p>It can be seen that $$D = E[T]$$</p>

<p>where</p>

<p>$T$ is a random variable with range $t = 1,2,...,n$ each having probability $\frac{R_t v^t}{B}$</p>

<p>The author says something like we can have negative probabilities if we have negative $R_t$'s. So this is a kind of bond where instead of making a payment we get a certain amount of money? Is there such a thing? Or is that only in theory?</p>

CC BY-SA 3.0
2016-07-14T16:15:05.087Z
Also, in a traditional bond, you make a lump-sum payment and receive smaller payments over time. Not sure what you mean by making a payment.
Kontorus
CC BY-SA 3.0
2016-07-15T03:34:11.417Z
You cannot have a negative probability.
123
CC BY-SA 3.0
2016-07-16T02:00:03.657Z
@123 You can if it's in a quasiprobability space. Edited
BCLC
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2016-07-16T03:56:40.567Z
No. One cannot reasonably discuss negative probabilities. The negative probability of an event is ridiculous. Perhaps you mean negative expected values? That is certainly possible. Negative values. Sure.
123
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2016-07-16T04:11:45.943Z
A quasiprobability distribution is something entirely different than a probability. This is definitely true since the former violates the axioms of the later. If you want to discuss quasiprobability distributions as used in quantum mechanics or mathematical finance then that's fine but perhaps be more clear.
123
CC BY-SA 3.0
2016-07-16T08:08:28.717Z
@123 So can one reasonably discuss negative probabilities?
BCLC
CC BY-SA 3.0
2016-07-16T08:09:45.623Z
@123 why is the negative probability of an event ridiculous?
BCLC
CC BY-SA 3.0
2016-07-16T20:56:49.210Z
No, one cannot. One can reasonably discuss quasiprobability spaces. These violate the axioms of probability. One must clearly make that distinction. I find the notion ridiculous because there is no meaningful,operational interpretation of a 'negative probability'.
123
CC BY-SA 3.0
2016-07-17T16:00:45.170Z
@123 negative quasiprobability? Or maybe negative probability in a quasiprobability space?
BCLC
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2016-07-18T14:06:57.443Z
@BCLC negative probabilities are similar to negative weight or negative distances; they are impossible.
Kontorus
CC BY-SA 3.0
2016-07-18T17:03:55.323Z
@Kontorus well depends on the space? In a regular metric or probability space I guess so. What about in a quasiprobability space?
BCLC
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2016-07-18T17:06:18.683Z
@Kontorus last I checked we are in Eco se not physics se. Is negative money impossible? Did you check the link?
BCLC
<mathematical-economics><finance><financial-economics><probability><bonds>
Negative probabilities - Can we have negative payments in bonds?
BCLC
# Negative probabilities - Can we have negative payments in bonds?

(Asked by: username_0 on 2016-07-09)

In [Half of a Coin: Negative Probabilities](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.592.2043&rep=rep1&type=pdf), the author mentions bond duration.

Suppose we have payments at times $t = 1,2,...,n$ denoted respectively by $R\_1, R\_2, ..., R\_n$ and the discount factor is $v = \frac{1}{1+i}$ where $i$ is effective interest rate. Then the bond value today is given by

$$B = \sum\_{t=1}^{n} R\_tv^t$$

The bond duration is

$$D = \frac{\sum\_{t=1}^{n} tR\_tv^t}{\sum\_{t=1}^{n} R\_tv^t}$$

It can be seen that $$D = E[T]$$

where

$T$ is a random variable with range $t = 1,2,...,n$ each having probability $\frac{R\_t v^t}{B}$

The author says something like we can have negative probabilities if we have negative $R\_t$'s. So this is a kind of bond where instead of making a payment we get a certain amount of money? Is there such a thing? Or is that only in theory?

**Tags:** mathematical-economics, finance, financial-economics, probability, bonds

> **Comments:**
> **username_1**: Also, in a traditional bond, you make a lump-sum payment and receive smaller payments over time. Not sure what you mean by making a payment.  
> **username_2**: You cannot have a negative probability.  
> **username_0**: @username_2 You can if it's in a quasiprobability space. Edited  
> **username_2**: No. One cannot reasonably discuss negative probabilities. The negative probability of an event is ridiculous. Perhaps you mean negative expected values? That is certainly possible. Negative values. Sure.  
> **username_2**: A quasiprobability distribution is something entirely different than a probability. This is definitely true since the former violates the axioms of the later. If you want to discuss quasiprobability distributions as used in quantum mechanics or mathematical finance then that's fine but perhaps be more clear.  
> **username_0**: @username_2 So can one reasonably discuss negative probabilities?  
> **username_0**: @username_2 why is the negative probability of an event ridiculous?  
> **username_2**: No, one cannot. One can reasonably discuss quasiprobability spaces. These violate the axioms of probability. One must clearly make that distinction. I find the notion ridiculous because there is no meaningful,operational interpretation of a 'negative probability'.  
> **username_0**: @username_2 negative quasiprobability? Or maybe negative probability in a quasiprobability space?  
> **username_1**: @username_0 negative probabilities are similar to negative weight or negative distances; they are impossible.  
> **username_0**: @username_1 well depends on the space? In a regular metric or probability space I guess so. What about in a quasiprobability space?  
> **username_0**: @username_1 last I checked we are in Eco se not physics se. Is negative money impossible? Did you check the link?  

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## 1 answers

### Answer by username_3 on 2016-07-17. Score: 2

I heard in a lecture at my university, that measure theory is applied in finance. This field also operates with negative probabilities a lot.

Anyways, I don't see the practical intuition behind using negative probabilities in a bond-pricing model. You can check this article by [Burgin and Meissner: Negative Probabilities in Financial Modeling (Wilmott Magazine March 2012)](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1773077).

I think you should migrate your question to [Quant SE](http://quant.stackexchange.com).


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