<p>I believe there is not a unique price. Say, instead of buying the option  you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order stochastically dominates the option. So, no matter your probability beliefs about the states,  in that setting you'd never pay $0.5$ for the option which pays less in every state. This suggests the right price is less than $0.5$. Similarly, buying $0.25$ units of the $S^0_1$ asset or $0.167$ units of the $S^1_1$ asset would likewise stochastically dominate the option. In fact, because for $0.375$ units of asset $S^1_2$, which costs on $0.375$, you can still have an asset that pays out $[0.3, 0.45, 0.6]$, it seems unlikely that the price could even be as high as $0.375$. Asset 0 implies a price below $0.4$ and asset 1 below $0.45$</p>

<p>Some python code to solve:</p>

<pre><code>import numpy as np
S0 = np.array([[2],[3],[1]])
D = np.array([[1,2,3], [2,2,4], [0.8, 1.2, 1.6]])
X = np.array([[0.0],[0.2],[0.6]])
phi = np.dot(np.linalg.inv(D.transpose()), X)
print('The weights of the portfolio that replicates payoff X are: \n', phi)
P_X = np.dot(S0.transpose(), phi)
print('With a price: ', P_X)
print('Normalizing to pay a fixed price P_X for each of the three assets, what payoffs can you get?')
D_norm = D/(2*S0)
print(D_norm)
print('Notice that all three first order stochastically dominate the option for a price of: ', P_X)
print(D_norm - X.transpose())
print('Using each of the base assets, what\'s the minimum quantity that dominates?')
D_relative = X.transpose() / D
print(D_relative)
Min_dominating_fraction = np.max(D_relative,axis=1)
print('Minimum fraction of each of the assets that dominates X\n', Min_dominating_fraction)
P_Min_dominating_fraction = S0.transpose() * Min_dominating_fraction 
print('At prices of: ', P_Min_dominating_fraction)
print('Therefore the option price should be less than: ', np.min(P_Min_dominating_fraction))
</code></pre>

<p>This code doesn't spell out the price of the option, it just show my calculations for the paragraph above. I believe the real price of this option would actually be zero if shorting is permitted. If you buy three units of asset $S^2_0$  and short one unit of $S^1_0$ you get an asset with payouts $[ 0.4,  1.6,  0.8]$. This position costs nothing to take, has positive payouts for all states, and first order stochastically dominates the option itself. Since it is possible to make a better than replicating portfolio at zero cost the price should be zero. Oh the insanity at work when arbitrages are present!</p>

CC BY-SA 3.0
2015-12-17T00:59:55.170Z
re the last paragraph, what exactly do you mean? the $\psi_i$'s are not the probabilities. the $q_i$'s are. They are not nonnegative but they sum to 1
BCLC
CC BY-SA 3.0
2015-12-17T01:05:51.690Z
I think I misunderstood some of your notation. Let me think on it a bit and rewrite it.
BKay
CC BY-SA 3.0
2015-12-18T12:10:22.400Z
Thanks BKay. Wasn't notified though I think. Will read later
BCLC
CC BY-SA 3.0
2015-12-18T12:19:30.983Z
how to run it w/o downloading python? ['ImportError: No module named numpy  '](http://www.tutorialspoint.com/execute_python_online.php)
BCLC
CC BY-SA 3.0
2015-12-18T12:23:47.757Z
Try pasting the code in here: https://www.pythonanywhere.com/try-ipython/
BKay
CC BY-SA 3.0
2015-12-18T12:57:52.593Z
OKay BKay, this is what I got: [0.37499999999999994](https://www.pythonanywhere.com/try-ipython/). Is that what it's supposed to be? What is that exactly? The least upper bound? What algorithm is this exactly? Not a computational finance person. Also, I don't suppose you know how to find positive lower bounds do you? :P
BCLC
CC BY-SA 3.0
2015-12-18T13:21:04.770Z
Please see my further update/
BKay
CC BY-SA 3.0
2015-12-18T13:36:08.803Z
Post answer in [QF](http://quant.stackexchange.com/questions/22228) too? :P I can give you the bounty there
BCLC
CC BY-SA 3.0
2015-12-17T00:58:16.763Z
2015-12-18T13:20:52.320Z
2015-12-18T13:20:52.320Z
BKay
<p>Assume that we have a general one-period market model consisting of d+1 assets and N states.</p>

<p>Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$,  on the asset $S_1^2$ with strike price $K = 1$ given that</p>

<p>$$S_0 =\begin{bmatrix}
2 \\ 
3\\ 
1
\end{bmatrix}, S_1 = \begin{bmatrix}
S_1^0\\ 
S_1^1\\ 
S_1^2
\end{bmatrix}, D = \begin{bmatrix}
1 &amp; 2 &amp; 3\\ 
2 &amp; 2 &amp; 4\\ 
0.8 &amp; 1.2 &amp; 1.6
\end{bmatrix}$$</p>

<p>where the columns of D represent the states for each asset and the rows of D represent the assets for each state</p>

<hr>

<p>What I tried:</p>

<p>We compute that:</p>

<p>$$X = \begin{bmatrix}
0\\ 
0.2\\ 
0.6
\end{bmatrix}$$</p>

<p>If we solve $D'\phi = X$, we get:</p>

<p>$$\phi = \begin{bmatrix}
0.6\\ 
0.1\\ 
-1
\end{bmatrix}$$</p>

<p>It would seem that the price of the European call option $\Pi(0;X)$ is given by the value of the replicating portfolio</p>

<p>$$S_0'\phi = 0.5$$</p>

<hr>

<p>On one hand, if we were to try to see if there is arbitrage in this market by seeing if a state price vector $\psi$ exists by solving $S_0 = D \psi$, we get</p>

<p>$$\psi = \begin{bmatrix}
0\\ 
-0.5\\ 
1
\end{bmatrix}$$</p>

<p>Hence there is no strictly positive state price vector $\psi$ s.t. $S_0 = D \psi$. By <a href="http://dybfin.wustl.edu/research/papers/arbetc7.pdf" rel="nofollow">'the fundamental theorem of asset pricing'</a> (or <a href="http://www.nuff.ox.ac.uk/users/murphya/Teaching/Arbitrage.pdf" rel="nofollow">'the fundamental theorem of finance'</a> or <a href="http://www.jhqian.org/apt/apbook.pdf" rel="nofollow">'1.3.1' here</a>), there exists arbitrage in this market.</p>

<hr>

<p>On the other hand the price of 0.5 seems to be confirmed by:</p>

<p>$$\Pi(0;X) = \beta E^{\mathbb Q}[X]$$</p>

<p>where $\beta = \sum_{i=1}^{3} \psi_i = 0.5$ (sum of elements of $\psi$) and $\mathbb Q$ is supposed to be the equivalent martingale measure given by $q_i = \frac{\psi_i}{\beta}$.</p>

<p>Thus we have</p>

<p>$$E^{\mathbb Q}[X] = q_1X(\omega_1) + q_2X(\omega_2) + q_3X(\omega_3)$$</p>

<p>$$ = 0 + \color{red}{-1} \times 0.2 + 2 \times 0.6 = 1$$</p>

<p>$$\to \Pi(0;X) = 0.5$$</p>

<hr>

<p>I guess $\therefore$ that we cannot determine the price of the European call using $\Pi(0;X) = \beta E^{Q}[X]$ because there is no equivalent martingale measure $\mathbb Q$</p>

<p>So what's the verdict? Can we say the price is 0.5? How can we price even if there is arbitrage?</p>

<hr>

<p>Edit: I noticed that one of the probabilities, in what was attempted to be the equivalent martingale measure, is negative. I remember reading about <a href="https://en.wikipedia.org/wiki/Negative_probability#Finance" rel="nofollow"><strong>negative probabilities</strong></a>, but these links <a href="http://poseidon01.ssrn.com/delivery.php?ID=649017121009107028094121105090018069032034022058026059113005070074031084099122001112103042027101022023115111118120122066015017046051062093060076111091084020114102061047044088127066112028029002069089095091082087104064117087123080025112103020025082122&amp;EXT=pdf" rel="nofollow">1</a> <a href="http://www.espenhaug.com/NegativeProbabilitiesHaug.pdf" rel="nofollow">2</a> mentioned by Wiki seem to assume absence of arbitrage so I think they are not applicable. Or are they?</p>

<p>Is it perhaps that this market can be considered to be arbitrage-free under some <a href="https://en.wikipedia.org/wiki/Quasiprobability_distribution" rel="nofollow">quasiprobability measure</a> that allows negative probabilities?</p>

<hr>

<p>Edit (to address a deleted answer):</p>

<p>Thanks BKay.</p>

<p>1 So you mean there is no unique price for $X$ but we can find upper bounds? Like in your example the least upper bound so far is 0.3 then we can continue to find lower upper bounds $u_1, u_2, ...$ (or even higher lower bounds $l_1, l_2, ...$) to say the price of $X$ is in $[0,\inf_n u_n]$ (or $[\sup_n l_n,\inf_n u_n]$)?</p>

<p>2 Re stochastic domination, I haven't heard that term in classes, but I think I read about that before. Might that depend on the (quasi)probability measure? Under this probability measure $0.5 S_1^2$ dominates $X$ but what about under some quasiprobability measure?</p>

<p>3 the $q_i$'s, not the $\psi_i$'s are the probabilities</p>

CC BY-SA 3.0
2015-12-15T14:11:25.287Z
This question seems to be more suited for quant.stackexchange.
Wecon
CC BY-SA 3.0
2015-12-15T14:25:17.203Z
@Wecon Thanks. Stochastic calculus questions seem to be asked and answered here though.
BCLC
<finance><probability><asset-pricing><arbitrage>
Pricing a European call option while absence of arbitrage is violated
BCLC
# Pricing a European call option while absence of arbitrage is violated

(Asked by: username_0 on 2015-12-13)

Assume that we have a general one-period market model consisting of d+1 assets and N states.

Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$, on the asset $S\_1^2$ with strike price $K = 1$ given that

$$S\_0 =\begin{bmatrix}
2 \\
3\\
1
\end{bmatrix}, S\_1 = \begin{bmatrix}
S\_1^0\\
S\_1^1\\
S\_1^2
\end{bmatrix}, D = \begin{bmatrix}
1 & 2 & 3\\
2 & 2 & 4\\
0.8 & 1.2 & 1.6
\end{bmatrix}$$

where the columns of D represent the states for each asset and the rows of D represent the assets for each state

---

What I tried:

We compute that:

$$X = \begin{bmatrix}
0\\
0.2\\
0.6
\end{bmatrix}$$

If we solve $D'\phi = X$, we get:

$$\phi = \begin{bmatrix}
0.6\\
0.1\\
-1
\end{bmatrix}$$

It would seem that the price of the European call option $\Pi(0;X)$ is given by the value of the replicating portfolio

$$S\_0'\phi = 0.5$$

---

On one hand, if we were to try to see if there is arbitrage in this market by seeing if a state price vector $\psi$ exists by solving $S\_0 = D \psi$, we get

$$\psi = \begin{bmatrix}
0\\
-0.5\\
1
\end{bmatrix}$$

Hence there is no strictly positive state price vector $\psi$ s.t. $S\_0 = D \psi$. By ['the fundamental theorem of asset pricing'](http://dybfin.wustl.edu/research/papers/arbetc7.pdf) (or ['the fundamental theorem of finance'](http://www.nuff.ox.ac.uk/users/murphya/Teaching/Arbitrage.pdf) or ['1.3.1' here](http://www.jhqian.org/apt/apbook.pdf)), there exists arbitrage in this market.

---

On the other hand the price of 0.5 seems to be confirmed by:

$$\Pi(0;X) = \beta E^{\mathbb Q}[X]$$

where $\beta = \sum\_{i=1}^{3} \psi\_i = 0.5$ (sum of elements of $\psi$) and $\mathbb Q$ is supposed to be the equivalent martingale measure given by $q\_i = \frac{\psi\_i}{\beta}$.

Thus we have

$$E^{\mathbb Q}[X] = q\_1X(\omega\_1) + q\_2X(\omega\_2) + q\_3X(\omega\_3)$$

$$ = 0 + \color{red}{-1} \times 0.2 + 2 \times 0.6 = 1$$

$$\to \Pi(0;X) = 0.5$$

---

I guess $\therefore$ that we cannot determine the price of the European call using $\Pi(0;X) = \beta E^{Q}[X]$ because there is no equivalent martingale measure $\mathbb Q$

So what's the verdict? Can we say the price is 0.5? How can we price even if there is arbitrage?

---

Edit: I noticed that one of the probabilities, in what was attempted to be the equivalent martingale measure, is negative. I remember reading about [**negative probabilities**](https://en.wikipedia.org/wiki/Negative_probability#Finance), but these links [1](http://poseidon01.ssrn.com/delivery.php?ID=649017121009107028094121105090018069032034022058026059113005070074031084099122001112103042027101022023115111118120122066015017046051062093060076111091084020114102061047044088127066112028029002069089095091082087104064117087123080025112103020025082122&EXT=pdf) [2](http://www.espenhaug.com/NegativeProbabilitiesHaug.pdf) mentioned by Wiki seem to assume absence of arbitrage so I think they are not applicable. Or are they?

Is it perhaps that this market can be considered to be arbitrage-free under some [quasiprobability measure](https://en.wikipedia.org/wiki/Quasiprobability_distribution) that allows negative probabilities?

---

Edit (to address a deleted answer):

Thanks BKay.

1 So you mean there is no unique price for $X$ but we can find upper bounds? Like in your example the least upper bound so far is 0.3 then we can continue to find lower upper bounds $u\_1, u\_2, ...$ (or even higher lower bounds $l\_1, l\_2, ...$) to say the price of $X$ is in $[0,\inf\_n u\_n]$ (or $[\sup\_n l\_n,\inf\_n u\_n]$)?

2 Re stochastic domination, I haven't heard that term in classes, but I think I read about that before. Might that depend on the (quasi)probability measure? Under this probability measure $0.5 S\_1^2$ dominates $X$ but what about under some quasiprobability measure?

3 the $q\_i$'s, not the $\psi\_i$'s are the probabilities

**Tags:** finance, probability, asset-pricing, arbitrage
> **Comments:**
> **username_1**: This question seems to be more suited for quant.stackexchange.  
> **username_0**: @Wecon Thanks. Stochastic calculus questions seem to be asked and answered here though.  

---

## 1 answers

### Answer by username_2 on 2015-12-17. Score: 4 | Accepted answer

I believe there is not a unique price. Say, instead of buying the option  you spent 0.5 on a half a unit of the asset $S^2_1$ This asset pays out $[0.4, 0.6, 0.8]$ which first order stochastically dominates the option. So, no matter your probability beliefs about the states,  in that setting you'd never pay $0.5$ for the option which pays less in every state. This suggests the right price is less than $0.5$. Similarly, buying $0.25$ units of the $S^0_1$ asset or $0.167$ units of the $S^1_1$ asset would likewise stochastically dominate the option. In fact, because for $0.375$ units of asset $S^1_2$, which costs on $0.375$, you can still have an asset that pays out $[0.3, 0.45, 0.6]$, it seems unlikely that the price could even be as high as $0.375$. Asset 0 implies a price below $0.4$ and asset 1 below $0.45$
Some python code to solve:
import numpy as np
S0 = np.array([[2],[3],[1]])
D = np.array([[1,2,3], [2,2,4], [0.8, 1.2, 1.6]])
X = np.array([[0.0],[0.2],[0.6]])
phi = np.dot(np.linalg.inv(D.transpose()), X)
print('The weights of the portfolio that replicates payoff X are: \n', phi)
P_X = np.dot(S0.transpose(), phi)
print('With a price: ', P_X)
print('Normalizing to pay a fixed price P_X for each of the three assets, what payoffs can you get?')
D_norm = D/(2*S0)
print(D_norm)
print('Notice that all three first order stochastically dominate the option for a price of: ', P_X)
print(D_norm - X.transpose())
print('Using each of the base assets, what\'s the minimum quantity that dominates?')
D_relative = X.transpose() / D
print(D_relative)
Min_dominating_fraction = np.max(D_relative,axis=1)
print('Minimum fraction of each of the assets that dominates X\n', Min_dominating_fraction)
P_Min_dominating_fraction = S0.transpose() * Min_dominating_fraction 
print('At prices of: ', P_Min_dominating_fraction)
print('Therefore the option price should be less than: ', np.min(P_Min_dominating_fraction))

This code doesn't spell out the price of the option, it just show my calculations for the paragraph above. I believe the real price of this option would actually be zero if shorting is permitted. If you buy three units of asset $S^2_0$  and short one unit of $S^1_0$ you get an asset with payouts $[ 0.4,  1.6,  0.8]$. This position costs nothing to take, has positive payouts for all states, and first order stochastically dominates the option itself. Since it is possible to make a better than replicating portfolio at zero cost the price should be zero. Oh the insanity at work when arbitrages are present!

> **Comments:**
> **username_0**: re the last paragraph, what exactly do you mean? the $\psi_i$'s are not the probabilities. the $q_i$'s are. They are not nonnegative but they sum to 1  
> **username_2**: I think I misunderstood some of your notation. Let me think on it a bit and rewrite it.  
> **username_0**: Thanks BKay. Wasn't notified though I think. Will read later  
> **username_0**: how to run it w/o downloading python? ['ImportError: No module named numpy  '](http://www.tutorialspoint.com/execute_python_online.php)  
> **username_2**: Try pasting the code in here: https://www.pythonanywhere.com/try-ipython/  
> **username_0**: OKay BKay, this is what I got: [0.37499999999999994](https://www.pythonanywhere.com/try-ipython/). Is that what it's supposed to be? What is that exactly? The least upper bound? What algorithm is this exactly? Not a computational finance person. Also, I don't suppose you know how to find positive lower bounds do you? :P  
> **username_2**: Please see my further update/  
> **username_0**: Post answer in [QF](http://quant.stackexchange.com/questions/22228) too? :P I can give you the bounty there  

---

