<p>In general, if you own an option, holding fixed the expected value of the underlying asset and the strike price of the option, the greater the variance in the distribution of the prices of the option the (weakly) more valuable the option is. Here is a graphical example of a call option at maturity. They grey line indicates the value of the option for each value of the underlying asset where the strike price is \$5. The blue diamonds indicate the payoffs for values of the underlying at 4 and 6. If each are equally likely (p=0.5) then the underlying has an expected value of 5 and the option of \$0.5 (the lower green square). This distribution as a standard deviation of 0.5. Increase the standard deviation to 1.0 but hold the expected value constant at 5 by  changing the two underlying outcomes to \$3 and \$7 (orange squares). The expected value of this option is 1 (higher green square), so the increase in variance of the distribution of the underlying asset has increased the value of the option even though the expected value of the underlying is unchanged.
<a href="https://i.sstatic.net/6Ou9z.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6Ou9z.png" alt="enter image description here"></a></p>

<p>The reason for this is essentially <a href="https://en.wikipedia.org/wiki/Jensen&#39;s_inequality" rel="nofollow noreferrer">Jensen's inequality</a>. We can model the value of an option as a convex function $f(x)$ where $x$ is the underlying. The value of the option is $g(x) = E[f(x)]$ This implies $g(x)$ is increasing in the variance of $x$. We can see this because when we do the Taylor expansion of the function f : $$g(x) \approx E[f(x_0) +  (x - x_0)\cdot f'(x_0) +  (x - x_0)^2 \cdot f''(x_0)]$$</p>

<p>Because of the convexity, the second order term $f''(x_0)$ will be positive. The additive part $E[(x - x_0)^2 \cdot f''(x_0)] = E[(x - x_0)^2] \cdot f''(x_0)$ is increasing in the variance $E[(x - \bar{x})^2]$ so the value of the option is increasing in the variance. </p>

CC BY-SA 3.0
2015-12-14T12:25:18.960Z
Can you please tell me the source of the image you included in your answer?
Karup
CC BY-SA 3.0
2015-12-14T12:26:54.643Z
I made it in Excel.
BKay
CC BY-SA 3.0
2015-12-10T15:01:44.647Z
2015-12-10T15:01:44.647Z
BKay
<p>I was reading about the concept of <a href="https://en.wikipedia.org/wiki/Real_options_valuation" rel="nofollow">real options valuation</a>. It says that ROV has shown that <em>uncertainty in market pay-off enhances the project value</em> (V). How is that? Will it be possible to explain that with an example?</p>

<p>Here by 'uncertainty' we mean <em>stochastic variability</em> of parameter distributions. </p>

<p>{<code>V = f(performance,cost,time,market requirement,market payoff)</code> }</p>

CC BY-SA 3.0
2015-12-10T09:58:23.897Z
Could you please identify where in the Wikipedia article this claim is made?
Adam Bailey
CC BY-SA 3.0
2015-12-10T10:59:29.530Z
Doesn't it simply mean that a right to exert an option later is valuable only insofar as you expect uncertainty to be resolved in the meantime? For instance, keeping the possibility to make an investment later instead of now is valuable if you expect to receive some information about its return prior to the decision.
Oliv
CC BY-SA 3.0
2015-12-10T15:23:35.543Z
The intuition is simple- it is because a mean-preserving spread increases the frequency with which you get higher payoffs..this matters for prices of options because the probability with which you get high values is higher..also it does not matter that the frequency with which you get lower values is higher as well- you just do not exercise the option
ChinG
<risk>
How is project value enhanced by uncertainty in market payoff?
Karup
# How is project value enhanced by uncertainty in market payoff?

(Asked by: username_0 on 2015-12-10)

I was reading about the concept of [real options valuation](https://en.wikipedia.org/wiki/Real_options_valuation). It says that ROV has shown that *uncertainty in market pay-off enhances the project value* (V). How is that? Will it be possible to explain that with an example?

Here by 'uncertainty' we mean *stochastic variability* of parameter distributions.

{`V = f(performance,cost,time,market requirement,market payoff)` }

**Tags:** risk

> **Comments:**
> **username_1**: Could you please identify where in the Wikipedia article this claim is made?  
> **username_2**: Doesn't it simply mean that a right to exert an option later is valuable only insofar as you expect uncertainty to be resolved in the meantime? For instance, keeping the possibility to make an investment later instead of now is valuable if you expect to receive some information about its return prior to the decision.  
> **username_3**: The intuition is simple- it is because a mean-preserving spread increases the frequency with which you get higher payoffs..this matters for prices of options because the probability with which you get high values is higher..also it does not matter that the frequency with which you get lower values is higher as well- you just do not exercise the option  

---

## 1 answers

### Answer by username_4 on 2015-12-10. Score: 1 | Accepted answer

In general, if you own an option, holding fixed the expected value of the underlying asset and the strike price of the option, the greater the variance in the distribution of the prices of the option the (weakly) more valuable the option is. Here is a graphical example of a call option at maturity. They grey line indicates the value of the option for each value of the underlying asset where the strike price is \$5. The blue diamonds indicate the payoffs for values of the underlying at 4 and 6. If each are equally likely (p=0.5) then the underlying has an expected value of 5 and the option of \$0.5 (the lower green square). This distribution as a standard deviation of 0.5. Increase the standard deviation to 1.0 but hold the expected value constant at 5 by changing the two underlying outcomes to \$3 and \$7 (orange squares). The expected value of this option is 1 (higher green square), so the increase in variance of the distribution of the underlying asset has increased the value of the option even though the expected value of the underlying is unchanged.
[![enter image description here](https://i.sstatic.net/6Ou9z.png)](https://i.sstatic.net/6Ou9z.png)

The reason for this is essentially [Jensen's inequality](https://en.wikipedia.org/wiki/Jensen's_inequality). We can model the value of an option as a convex function $f(x)$ where $x$ is the underlying. The value of the option is $g(x) = E[f(x)]$ This implies $g(x)$ is increasing in the variance of $x$. We can see this because when we do the Taylor expansion of the function f : $$g(x) \approx E[f(x\_0) + (x - x\_0)\cdot f'(x\_0) + (x - x\_0)^2 \cdot f''(x\_0)]$$

Because of the convexity, the second order term $f''(x\_0)$ will be positive. The additive part $E[(x - x\_0)^2 \cdot f''(x\_0)] = E[(x - x\_0)^2] \cdot f''(x\_0)$ is increasing in the variance $E[(x - \bar{x})^2]$ so the value of the option is increasing in the variance.

> **Comments:**
> **username_0**: Can you please tell me the source of the image you included in your answer?  
> **username_4**: I made it in Excel.  

---

