2.1 A discrete financial market

We consider a discrete financial market with two nondividend paying stocks over a finite time horizon. Today is time $0$ , the price of stock $i$ $(i = 1 \text{ or } 2)$ , at the future time $t$ ( $t=1,2,\ldots, n$ ), is denoted by $S_i(t)$ .Footnote ³ Given the price of stock $i$ at time $t-1$ , the future stock price at time $t$ can only increase to $e^{u_{i}}S_{i}(t-1)$ or decrease to $e^{d_{i}}S_{i}(t-1)$ . The forward return of stock $i$ at time $t$ is denoted by $R_{i}(t)$ and defined as:

(1) \begin{equation} R_{i}(t)=\log \frac{S_i(t)}{S_i(t-1)},\ i=1\text{ or }2\text{ and }t=1,2,\ldots, n. \end{equation}

The financial market is also home to a bank account, which allows borrowing and lending at a constant, risk-free interest rate $r$ . The time $0$ value of the risk-free asset is $B(0)$ , and its time $t$ value is given by $B(t) = e^{rt}B(0)$ . We assume that $e^{u_{i}}$ and $e^{d_{i}}$ are symmetric with respect to the forward rate $e^r$ :

(2) \begin{equation} e^{u_i} - e^r = -(e^{d_i}-e^r),\text{ for }i=1,2. \end{equation}

Under the real-world probability measure $\mathbb{P}$ , we denote the joint probabilities of the random vector $\left ( R_{1}(t),R_{2}(t)\right )$ as follows:

(3) \begin{align} \mathbb{P}\left [ R_{1}(t)=d_1,R_{2}(t)=d_2 \right ] & =p_{d d}(t),\nonumber\\ \mathbb{P}\left [ R_{1}(t)=u_1,R_{2}(t)=d_2\right ] & =p_{u d}(t),\nonumber \\ \mathbb{P}\left [ R_{1}(t)=d_1,R_{2}(t)=u_2\right ] & =p_{d u}(t),\nonumber \\ \mathbb{P}\left [ R_{1}(t)=u_1,R_{2}(t)=u_2 \right ] & =p_{u u}(t). \end{align}

We assume that the joint probabilities under real-world measure $\mathbb{P}$ are strictly positive. The distribution of the random vector $\left ( R_{1}(t),R_{2}(t)\right )$ is determined by the marginal distributions of $R_{i}(t),$ and the dependence structure connecting $R_{1}(t)$ and $R_{2}(t).$ We assume that for $i=1,2,$ the random variables $R_{i}(1), R_{i}(2), \ldots, R_{i}\left ( n\right )$ are independent of each other. The probability of stock 1 moving up and down from $t-1$ to $t$ is denoted by $p_{u \cdot }(t)$ and $p_{d \cdot }(t)$ , respectively. For stock 2, these probabilities are denoted by $p_{\cdot u}(t)$ and $p_{\cdot d}(t)$ .

Assume the financial market is arbitrage free, hence, there exists at least one probability measure $\mathbb{Q},$ called a risk-neutral probability measure, satisfying the following conditions:

1. 1. $\mathbb{Q}$ and $\mathbb{P}$ are equivalent probability measures;

2. 2. For any traded asset, its future payoff discounted at the risk-free rate $r$ is a martingale with respect to $\mathbb{Q}$ :

   (4) \begin{equation} \text{e}^{-r}\mathbb{E}_{\mathbb{Q}}\left [ S_{i}(t) |\mathcal{F}_{t-1}\right ] =S_{i}\left ( t-1\right ), \text{ for }i=1,2,\text{and }t=1,2,..,n. \end{equation}

If $\mathbb{Q}$ is a probability measure satisfying the above-stated conditions, we say that it is a feasible risk-neutral probability measure. Under $\mathbb{Q}$ , the joint probabilities of the random vector $(R_1(t), R_2(t))$ are denoted by $q_{u u}(t)$ , $q_{u d}(t)$ , $q_{d u}(t)$ , and $q_{d d}(t)$ . Additionally, the risk-neutral marginal probabilities in $\left [t-1,t\right ]$ are denoted by $q_{u \cdot }\left (t \right ),q_{d \cdot }\left (t \right ),q_{\cdot u}\left (t \right )$ , and $q_{\cdot d}\left (t \right ).$ We show in the next subsection that this simple market model is incomplete by characterizing the set of feasible risk-neutral probability measures.

2.2 The set of equivalent martingale measures

Let us now characterize the set of all feasible risk-neutral probability measures. This set is denoted by $\mathcal{M}$ . Each $\mathbb{Q\in }$ $\mathcal{M}$ is characterized by the joint probabilities $ q_{d d}(t)$ , $q_{u d}(t),$ $q_{d u}(t),$ $q_{u u}(t), t=1,2,\ldots, n$ . In Theorem1, we characterize the risk-neutral pricing measure $\mathbb{Q}$ by the correlation coefficients $\rho _{\mathbb{Q}}(t)= \text{Corr}_{\mathbb{Q}}\left [ R_{1}(t), R_{2}(t) \right ]$ , $t=1,2,\dots, n$ . A proof of this Theorem can be found in Appendix A.1.

Theorem 1. Consider the stock price model ( 3 ) satisfying the conditions ( 2 ). The set $\mathcal{M}$ of risk-neutral probability measures can be characterized as follows:

(5) \begin{align} \mathbb{Q\in }\mathcal{M\Leftrightarrow }\text{ } & \text{ }\exists \text{ }\rho _{\mathbb{Q}}(t)\in \left ( -1,1\right ), t=1,2,\ldots, n,\nonumber \\[5pt] & \text{such that }\left \{ \begin{array} [l]{l}q_{u u}(t)=q_{d d}(t) =\frac{1}{4}(1+\rho _{\mathbb{Q}}(t)),\\ q_{u d}(t)=q_{d u}(t) =\frac{1}{4}(1-\rho _{\mathbb{Q}}(t)). \end{array} \right . \end{align}

The risk-neutral measure $\mathbb{Q}$ is not unique and the market is incomplete. It is clear to see from Theorem1 that the risk-neutral marginal probabilities are all equal to $\frac{1}{2}$ . Note, however, that our model can be generalized to situations where the marginal risk neutral probabilities are different from $\frac{1}{2}$ . Specifying a feasible risk-neutral measure $\mathbb{Q}$ under the stock price model (3), requires specifying the correlation efficient $\rho _{\mathbb{Q}}(t),t=1,2,\ldots, n.$ Each risk-neutral probability measure $\mathbb{Q}$ in $\mathcal{M}$ has the same marginal distributions but different dependence structures. For instance, take $\rho _{\mathbb{Q}}(t)\equiv 0$ , then we find the risk-neutral measure $\mathbb{Q}^{\perp }\in \mathcal{M}$ where the marginals are independent.

The situations characterized by the minimal correlation coefficient $\rho _{\mathbb{Q}^{\min }}(t)\equiv -1$ and the maximal correlation coefficient $\rho _{\mathbb{Q}^{\max }}(t)\equiv 1$ correspond with the probability measure $\mathbb{Q}^{\min }$ and the probability measure $\mathbb{Q}^{\max },$ respectively. The random vector $\left (R_1(t), R_2(t)\right )$ under the probability measure $\mathbb{Q}^{\min }$ is counter-monotonic. In this situation, the components of the random vectors are maximum negative dependent. Inversely, the random vector $\left (R_1(t), R_2(t)\right )$ under the probability measure $\mathbb{Q}^{\max }$ is comonotonic and the components of the random vector are maximum positive dependent.

The real-world joint probabilities are assumed to be strictly positive, hence all the risk-neutral probabilities specified by (5) are strictly positive. Using (5), we can directly find that for each $\mathbb{Q} \in \mathcal{M},$ $\rho _{\mathbb{Q}}(t)\in \left ( -1,1\right )$ , which means that the comonotonic and the counter-monotonic cases are not reachable in $\mathcal{M}$ . The larger $\rho _{\mathbb{Q}}(t)$ , the “closer” the risk-neutral probability measure $\mathbb{Q}$ is to the maximum measure $\mathbb{Q}^{\max }$ .

The set $\mathcal{M}$ contains a wide range of dependence structures. Each element $\mathbb{Q}$ in the set $\mathcal{M}$ of risk-neutral probability measures can be expressed as a linear combination of $\mathbb{Q}^{\min }$ and $\mathbb{Q}^{\max }$ :

(6) \begin{equation} \mathbb{Q\in }\mathcal{M\Leftrightarrow }\text{ }\text{ }\exists \text{ }\rho _{\mathbb{Q}}(t)\in \left ( -1,1\right ) \text{ such that }\mathbb{Q=}\text{ }\frac{\left ({ 1\mathbb{-}\rho _{\mathbb{Q}}(t)}\right )}{2} \mathbb{Q}^{\min }+\frac{\left ({ 1\mathbb{+}\rho _{\mathbb{Q}}(t)}\right )}{2}\mathbb{Q}^{\max }. \end{equation}

By increasing the correlation coefficient $\rho _{\mathbb{Q}}(t),$ we can gradually increase the dependence of the components $R_{1}(t)$ and $R_{2}\left (t\right ).$

It directly follows from (5) that the joint risk-neutral cdf $F_t^{\mathbb{Q}}$ of the forward return vector $(R_1(t),R_2(t))$ can be given by:

(7) \begin{equation} F_{t}^{_{\mathbb{Q}}}\left ( x_{1},x_{2}\right ) =\left \{ \begin{array}[c]{cl} 0, & \text{if }x_{1}\lt d_1\text{ or }x_{2}\lt d_2,\\[4pt] \frac{\rho _{\mathbb{Q}}(t)+1}{4}, & \text{if }x_{1}\in \left [ d_1,u_1\right ) \text{ and }x_{2}\in \left [d_2,u_2\right ), \\[4pt] \frac{1}{2}, & \text{if }x_{1}\in \left [ d_1,u_1\right ) \text{ and }x_{2}\geq u_2,\\[4pt] \frac{1}{2}, & \text{if }x_{1}\geq u_1\text{ and }x_{2}\in \left [ d_2,u_2\right ), \\[4pt] 1, & \text{if }x_{1}\geq u_1\text{ and }x_{2}\geq u_2.\end{array} \right . \end{equation}

It follows from (7) that the joint risk-neutral cdf of forward return $F_t^{\mathbb{Q}}(x_1, x_2)$ can be unambiguously determined by the correlation coefficient $\rho _{\mathbb{Q}}(t)$ .

2.3 Comparing different risk-neutral measures

The set $\mathcal{M}$ contains different risk-neutral measures such that the marginals $R_1$ and $R_2$ are always the same. Therefore, the difference between multivariate risk-neutral measures is in dependence structure. To compare different probability measures in $\mathcal{M}$ and identify under which one the dependence is stronger or weaker, we can use multivariate stochastic orders. The notion of multivariate stochastic orders in actuarial science goes back to Yanagimoto & Okamoto (Reference Yanagimoto and Okamoto1969).

We first introduce the Positive Quadrant Dependence and the Negative Quadrant Dependence. The notions of PQD and NQD were introduced in Lehmann (Reference Lehmann1966).

Definition 2 (Quadrant Dependence). The random vector $\left ( R_{1}(t),R_{2}(t)\right )$ is said to be Positive Quadrant Dependent under the probability measure $\mathbb{Q},$ notation $(R_1(t), R_2(t))\sim \mathbb{Q}$ -PQD, in case the vector $\left ( R_{1}(t),R_{2}(t)\right )$ satisfies:

\begin{equation*} \mathbb {Q}\left [ R_{1}(t)\leq x_{1}\right ] \mathbb {Q}\left [ R_{2}(t)\leq x_{2}\right ] \leq F_{t}^{_{\mathbb {Q}}}\left ( x_{1},x_{2}\right ), \,\text{ for all }\left ( x_{1},x_{2}\right ) \in \mathbb {R}^{2}. \end{equation*}

The vector $\left ( R_{1}(t),R_{2}(t)\right )$ is said to be Negative Quadrant Dependent under the probability measure $\mathbb{Q},$ notation $\mathbb{Q}$ -NQD, in case the vector satisfies:

\begin{equation*} \mathbb {Q}\left [ R_{1}(t)\leq x_{1}\right ] \mathbb {Q}\left [ R_{2}(t)\leq x_{2}\right ] \geq F_{t}^{_{\mathbb {Q}}}\left ( x_{1},x_{2}\right ), \,\text{ for all }\left ( x_{1},x_{2}\right ) \in \mathbb {R}^{2}, \end{equation*}

where $F_{t}^{_{\mathbb{Q}}}$ is the joint cdf of $\left ( R_{1}(t),R_{2}(t)\right )$ under the probability measure $\mathbb{Q}$ .

Using Expression (7) for $F_{t}^{_{\mathbb{Q}}}$ results in the following implications:

(8) \begin{align} \rho _{\mathbb{Q}}(t) & \geq 0\Longleftrightarrow \rho _{\mathbb{Q}}(t)\geq \rho _{\mathbb{Q}^{\perp }}(t)\Longleftrightarrow \left ( R_{1}(t),R_{2}(t)\right ) \text{ is }\mathbb{Q}\text{-PQD,} \\ \rho _{\mathbb{Q}}(t) & \leq 0\Longleftrightarrow \rho _{\mathbb{Q}}(t)\leq \rho _{\mathbb{Q}^{\perp }}(t)\Longleftrightarrow \left ( R_{1}(t),R_{2}(t)\right ) \text{ is }\mathbb{Q}\text{-NQD,}\nonumber \end{align}

where $\rho _{\mathbb{Q}^{\perp }}(t)$ is the correlation of the independent copy of the random vector $\left ( R_{1}(t),R_{2}(t) \right )$ , it is straightforward to show that $\rho _{\mathbb{Q}^{\perp }}(t)$ equals $0$ . Using the notion of quadrant dependence, we can measure the joint behavior of two random variables. If the risk-neutral correlation $\rho _{\mathbb{Q}}(t)$ is larger than $0$ , then the random vector $\left (R_{1}(t),R_{2}(t)\right )$ is $\mathbb{Q}\text{-PQD,}$ which means that the two random variables $R_{1}(t)$ and $R_{2}(t)$ are likely to assume small or large values simultaneously. Conversely, if $\rho _{\mathbb{Q}}(t)$ is less than $0$ , the random vector is $\mathbb{Q}\text{-NQD,}$ implying an inverse relationship where $R_{1}(t)$ and $R_{2}(t)$ may move in different directions.

The concept of quadrant dependence measures the association between two random variables. However, under different risk-neutral measures $\mathbb{Q}^{(1)}$ and $\mathbb{Q}^{(2)}$ , the marginals and quadrant dependence of the random vector $(R_1(t), R_2(t))$ may be identical. To distinguish the difference under such two different risk-neutral measures, we can use the correlation order introduced in Dhaene & Goovaerts (Reference Dhaene and Goovaerts1996), since the joint distribution of $(R_1(t), R_2(t))$ is different when using two different risk-neutral measures.

Definition 3 (Correlation order). Consider the risk-neutral probability measures $\mathbb{Q}^{(1) }$ and $\mathbb{Q}^{(2) }$ with correlation parameters $\rho _{\mathbb{Q}^{(1) }}(t)$ and $\rho _{\mathbb{Q}^{(2) }}(t)$ , respectively. We say that the cdf’s $F_{t}^{_{\mathbb{Q}^{(1) }}}$ and $F_{t}^{_{\mathbb{Q}^{(2) }}}$ are ordered in the correlation order, notation $F_{t}^{_{\mathbb{Q}^{(1) }}}\preceq _{\text{Corr}}F_{t}^{_{\mathbb{Q}^{(2) }}}$ if the following holds:

(9) \begin{equation} F_{t}^{_{\mathbb{Q}^{(1) }}}\left ( x_{1},x_{2}\right ) \leq F_{t}^{_{\mathbb{Q}^{(2) }}}\left ( x_{1},x_{2}\right ), \text{ for all }\left ( x_{1},x_{2}\right ) \in \mathbb{R}^{2}. \end{equation}

Intuitively, the inequality $F_{t}^{_{\mathbb{Q}^{(1) }}}\preceq _{\text{Corr}}F_{t}^{_{\mathbb{Q}^{(2) }}}$ implies that the two stock prices in $[t-1,t]$ move stronger together under the probability measure $\mathbb{Q}^{(2) }$ than under the probability measure $\mathbb{Q}^{(1) }$ , i.e., the probability of having simultaneously large/small realizations in $[t-1,t]$ is larger under $\mathbb{Q}^{\left ( 2\right ) }$ , compared to $\mathbb{Q}^{(1) }$ . Moreover, comparing the correlations $\rho _{{\mathbb{Q}^{(1) }}}(t)$ and $\rho _{{\mathbb{Q}^{(2)}}} (t)$ gives information about the correlation order between the probability measures $\mathbb{Q}^{(1) }$ and $\mathbb{Q}^{(2) }.$ Indeed, it follows directly from Expression (7) that the following equivalence relation holds:

(10) \begin{equation} \rho _{_{\mathbb{Q}^{(1) }}}(t)\leq \rho _{_{\mathbb{Q}^{(2) }}}(t)\iff F_{t}^{_{\mathbb{Q}^{(1) }}}\preceq _{\text{Corr}}F_{t}^{_{\mathbb{Q}^{(2) }}}. \end{equation}

There are infinitely many risk-neutral probability measures and each risk-neutral measure models the stock prices using a different dependence structure. If a contingent claim has to be priced, the market will pick a suitable pricing measure.

3.1 A market with a combined asset

From Theorem1, the spot price of stock $i$ and its potential outcomes $d_{i}$ and $u_{i}$ , fully specify the distribution of $R_i$ . However, individual derivatives written on each stock do not give additional information about the joint distributions. To determine the multivariate distribution, a combined asset and derivative prices on this combined asset are necessary.

Consider the market described above, but now also assume that a stock market index is traded. Its price at time $0$ is denoted by $S(0)$ and its price at time $t$ is denoted by $S(t)$ :

(11) \begin{equation} S(t)=S_{1}(t)+S_{2}(t),\text{ for }t=0,1,2,\ldots, n. \end{equation}

We have that $\mathbb{E}_{\mathbb{Q}}\left [ S(t)\right ] =\mathbb{E}_{\mathbb{Q}}\left [ S_{1}(t)+S_{2}(t)\right ] =$ e $^{rt}S\left ( 0\right ).$ The forward return of the stock market index at time $t$ is denoted by $R(t)$ and defined as follows:

(12) \begin{equation} R(t) = \log \frac{S(t)}{S(t-1)},\text{ for }t=1,2,\ldots, n. \end{equation}

Call options on the index with maturity $T=1,2,\ldots, n$ are also traded. The payoff of an index call option is given by $\left ( S\left ( T\right ) -K\right ) _{+}\,$ where $\left (x\right ) _{+}=\max \left \{x,0\right \}$ . The price of an index call with maturity $T$ and strike $K$ is then denoted by $C_{\mathbb{Q}}\left [ K,T\right ]$ and can be expressed as follows:

\begin{equation*} C_{\mathbb {Q}}\left [ K,T\right ] =\text {e}^{-rT}\mathbb {E}_{\mathbb {Q}}\left [ \left ( S(T)-K\right ) _{+}\right ] . \end{equation*}

The cumulative distribution function of the random variable $S\left ( T\right )$ under the risk-neutral measure $\mathbb{Q}$ is denoted by $F_{S\left ( T\right ) }^{\mathbb{Q}}$ . We introduce the stop-loss order between two risk-neutral probability measures in terms of their call option curves.

Definition 4 (Stop-loss order). Consider the stock price model described in ( 3 ) satisfying the conditions ( 2 ) and the stock market index defined in ( 11 ). Consider the risk-neutral probability measures $\mathbb{Q}^{(1) }$ and $\mathbb{Q}^{(2) }$ . We say that $F_{S\left ( T\right ) }^{\mathbb{Q}^{(1) }}$ and $F_{S\left ( T\right ) }^{\mathbb{Q}^{(2) }}$ are ordered in the stop-loss order, notation $F_{S\left ( T\right ) }^{\mathbb{Q}^{\left ( 1\right ) }}\preceq _{\text{sl}}F_{S\left ( T\right ) }^{\mathbb{Q}^{\left ( 2\right ) }}$ if:

(13) \begin{equation} C_{\mathbb{Q}^{(1) }}\left [ K,T\right ] \leq C_{\mathbb{Q}^{(2) }}\left [ K,T\right ] \text{ for all }K\geq 0. \end{equation}

Intuitively, the stop-loss order relation $F_{S\left ( T\right )}^{\mathbb{Q}^{\left ( 1\right ) }}\preceq _{\text{sl}}F_{S\left ( T\right ) }^{\mathbb{Q}^{\left ( 2\right ) }}$ implies that the stock market index $S\left ( T\right )$ is more volatile under the pricing measure $\mathbb{Q}^{(2) }$ than under the pricing measure $\mathbb{Q}^{(1) }.$ Indeed, one can prove the following implication:

\begin{equation*} F_{S\left ( T\right ) }^{\mathbb {Q}^{(1) }}\preceq _{\text {sl}}F_{S\left ( T\right ) }^{\mathbb {Q}^{(2) }}\Longrightarrow \text {Var}_{\mathbb {Q}^{(1) }}\left [ S\left ( T\right ) \right ] \leq \text {Var}_{\mathbb {Q}^{(2) }}\left [ S\left ( T\right ) \right ] . \end{equation*}

Therefore, call options are more expensive under the measure $\mathbb{Q}^{\left ( 2\right ) }$ than under $\mathbb{Q}^{\left ( 1\right ) }.$ The following theorem shows that the correlation $\rho _{\mathbb{Q}}\left ( 1\right )$ determines the variability of the stock market index $S\left ( 1\right ) .$

Theorem 5. Consider the stock price model described in ( 3 ) satisfying the conditions ( 2 ) and the stock market index defined in ( 11 ). Then, the following equivalence relation holds:

\begin{equation*} \rho _{_{\mathbb {Q}^{(1) }}}(1)\leq \rho _{_{\mathbb {Q}^{(2) }}}(1)\iff F_1^{\mathbb {Q}^{(1)}} \preceq _{\text {Corr}} F_1^{\mathbb {Q}^{(2)}}\iff F_{S(1) }^{\mathbb {Q}^{\left ( 1\right ) }}\preceq _{\text {sl}}F_{S(1) }^{\mathbb {Q}^{\left ( 2\right ) }}. \end{equation*}

Proof. From (10), we find that $\rho _{_{\mathbb{Q}^{(1) }}}(1)\leq \rho _{_{\mathbb{Q}^{(2) }}}(1)\iff F_1^{\mathbb{Q}^{(1)}} \preceq _{\text{Corr}} F_1^{\mathbb{Q}^{(2)}}$ . The call option price $C_{\mathbb{Q}}\left [ K,1\right ]$ can be expressed as follows:

(14) \begin{equation} C_{\mathbb{Q}}\left [ K,1\right ] =\text{e}^{-r}\left ( \frac{1}{4}\left ( P_{1}+P_{2}+P_{3}+P_{4}\right ) +\left ( P_{1}-P_{2}-P_{3}+P_{4}\right ) \frac{\rho _{\mathbb{Q}} \left ( 1 \right )}{4}\right ), \end{equation}

where,

\begin{align*} P_{1} & =\left ( S_{1}\left ( 0\right ) e^{u_{1}}+S_{2}\left ( 0\right ) e^{u_{2}}-K\right ) _{+};\, P_{2}=\left ( S_{1}\left ( 0\right ) e^{u_{1}}+S_{2}\left ( 0\right ) e^{d_{2}}-K\right ) _{+};\\ P_{3} & =\left ( S_{1}\left ( 0\right ) e^{d_{1}}+S_{2}\left ( 0\right ) e^{u_{2}}-K\right ) _{+};\, P_{4}=\left ( S_{1}\left ( 0\right ) e^{d_{1}}+S_{2}\left ( 0\right ) e^{d_{2}}-K\right ) _{+}. \end{align*}

It follows from Expression (14) that $C_{\mathbb{Q}}\left [ K,1\right ]$ is an increasing linear function of $\rho _{\mathbb{Q}}(1), $ since $(P_{1}-P_{2}-P_{3}+P_{4})\geq 0.$

The time $0$ market price of a call option with strike $K$ and maturity $1$ year, denoted by $C_{\mathbb{Q}}\left [ K,1\right ]$ , can then be used to extract the correlation $\rho _{\mathbb{Q}}\left ( 1\right )$ associated with the risk-neutral measure $\mathbb{Q}$ , as demonstrated in (14). Indeed, we have that:

(15) \begin{equation} \rho _{\mathbb{Q}} \left ( 1\right )=\frac{4 \text{e}^{r}C_{\mathbb{Q}}\left [ K,1\right ] - \left ( P_{1}+P_{2}+P_{3}+P_{4}\right ) }{\left ( P_{1}-P_{2}-P_{3}+P_{4}\right ) }. \end{equation}

From Theorem1, the joint probabilities $q_{d d}(1), q_{d u}(1), q_{u d}(1), q_{u u}(1)$ in this case are fully specified. For the single period case, Theorem 5 also indicates the equivalence relation between the correlation of two stocks and the volatility of the stock index. Moreover, we provide the following Theorem6 to show that having available the option prices $C_{\mathbb{Q}}\left [ K,T\right ], $ for $T=1,2,\ldots, n,$ one can back out the pricing measure $\mathbb{Q}$ .

Proof. First, we already determined $\rho _{\mathbb{Q}}(1)$ from expression (15). Next, knowledge about the price $C_{\mathbb{Q}}[K,2]$ enables us to back out the correlation $\rho _{\mathbb{Q}}(2)$ . Indeed, we can write:

(16) \begin{align} C_{\mathbb{Q}}\left [ K,2\right ] &=\text{ } \text{e}^{-2r}\mathbb{E}_{\mathbb{Q}}\left [ \mathbb{E}_{\mathbb{Q}}\left [ \left ( S(2) -K\right ) _{+}\mid \mathcal{F}_{1}\right ] \right ] \nonumber \\ &=\text{ } \text{e}^{-2r}\left [ q_{u u}(1) \mathbb{E}_{\mathbb{Q}}\left [ \left ( S(2) -K\right ) _{+}\mid S_{1}(1) =S_{1}\left ( 0\right ) e^{u_{1}},S_{2}(1) =S_{2}\left ( 0\right ) e^{u_{2}}\right ] \right . \nonumber \\ \text{ } &\quad +q_{u d}(1) \mathbb{E}_{\mathbb{Q}}\left [ \left ( S(2) -K\right ) _{+}\mid S_{1}(1) =S_{1}\left ( 0\right ) e^{u_{1}},S_{2}(1) =S_{2}\left ( 0\right ) e^{d_{2}}\right ] \nonumber \\ &\quad \text{ }+q_{d u}(1) \mathbb{E}_{\mathbb{Q}}\left [ \left ( S(2) -K\right ) _{+}\mid S_{1}(1) =S_{1}\left ( 0\right ) e^{d_{1}},S_{2}(1) =S_{2}\left ( 0\right ) e^{u_{2}}\right ] \nonumber \\ &\quad \text{ }\left . +\text{ }q_{d d}(1) \mathbb{E}_{\mathbb{Q}}\left [ \left ( S(2) -K\right ) _{+}\mid S_{1}(1) =S_{1}\left ( 0\right ) e^{d_{1}},S_{2}(1) =S_{2}\left ( 0\right ) e^{d_{2}}\right ] \right ] . \end{align}

From Expression (14), we find that the conditional expectations are given by:

\begin{equation*} \mathbb {E}_{\mathbb {Q}}\left [ \left ( S(2) -K\right ) _{+}\mid S_{1}(1) =s_{1},S_{2}(1) =s_{2}\right ]=\frac { (P_{1}+P_{2}+P_{3}+P_{4}) + \left ( P_{1}-P_{2}-P_{3}+P_{4}\right )\rho _{\mathbb {Q}\left ( 2\right )}}{4}, \end{equation*}

where the constants $P_{j}$ , $j=1,2,3,4,$ are now defined as follows:

\begin{align*} P_{1} & =\left ( s_{1}e^{u_{1}}+s_{2}e^{u_{2}}-K\right ) _{+};\,P_{2}=\left ( s_{1}e^{u_{1}}+s_{2}e^{d_{2}}-K\right ) _{+};\\ P_{3} & =\left ( s_{1}e^{d_{1}}+s_{2}e^{u_{2}}-K\right ) _{+};\,P_{4}=\left ( s_{1}e^{d_{1}}+s_{2}e^{d_{2}}-K\right ) _{+}. \end{align*}

The probabilities $q_{u u}(1), q_{u d}(1), q_{d u}(1), q_{d d}(1)$ are already determined from the call option price $C_{\mathbb{Q}}[K,1]$ . As a result, the only unknown parameter in the Expression (16) is $\rho _{\mathbb{Q}}(2)$ and observing the option price $C_{\mathbb{Q}}\left [ K,2\right ]$ allows to solve for $\rho _{\mathbb{Q}}(2).$ Suppose we have derived the correlations $\rho _{\mathbb{Q}}(t), t=1,2,\ldots, i$ , where $i\lt n$ , we can then apply the same approach to acquire the correlation $\rho _{\mathbb{Q}}\left ( i+1\right )$ from the correlations $\rho _{\mathbb{Q}}(1)$ , $\rho _{\mathbb{Q}}(2)$ , $\dots, \rho _{\mathbb{Q}}\left ( i\right )$ and the index option price $C_{\mathbb{Q}}\left [ K,i+1\right ]$ . The probability distributions of $S_1(i)$ and $S_2(i)$ can be specified using $\rho _{\mathbb{Q}}\left ( 1\right )$ , $\rho _{\mathbb{Q}}\left ( 2\right ), \dots, \rho _{\mathbb{Q}}\left ( i\right )$ . Hence, by using the option price $C_{\mathbb{Q}}\left [ K,i+1\right ]$ , we can solve for $\rho _{\mathbb{Q}}\left ( i+1\right )$ . Therefore, we conclude that having available the option prices $C_{\mathbb{Q}}\left [ K,T\right ], $ for $T=1,2,\ldots, n$ , one can back out the pricing measure $\mathbb{Q}$ .

Theorem6 showed that adding index call options with maturities $t=1,2,\ldots, T,$ completes the market described in (3). Indeed, one can back out the risk-neutral correlation using index call options, and therefore the risk-neutral probability measure is unique in this market setting. The idea of extracting the implied correlation from available multivariate option prices is widely discussed in the literature. For example, Linders & Schoutens (Reference Linders and Schoutens2014) used basket option prices, Ballotta et al. (Reference Ballotta, Deelstra and Rayée2017) used quanto options, and Garcia et al. (Reference Garcia, Goossens, Masol and Schoutens2009) derived implied correlations using CDO spreads.

Prices of traded stocks and options can be used to back out the corresponding risk-neutral pricing measure, the correlation of the stock returns and the volatility of the stock market index. We refer to these quantities as implied measures, implied correlation, implied volatility. Note that market prices are expectations under the pricing measure $\mathbb{Q}$ and therefore the implied correlation and the implied volatility have to be understood as correlation and volatility levels with respect to the risk-neutral probability measure $\mathbb{Q}$ . Correlation and volatility under the probability $\mathbb{P}$ are referred to as real-world correlation and volatility, respectively.

3.2 Example: A single period model

We consider a one-period financial market as described in Section 2. The risk-free rate $r$ is assumed to be $0$ and consider $e^{u_1}=1.4$ , $e^{u_2}=1.7$ . The time $0$ spot prices of the traded assets are given by $S_{1}\left ( 0\right ) =100$ and $S_{2}\left ( 0\right ) =200$ . The dynamics of the financial market under the real-world measure $\mathbb{P}$ are described by the following equations:

(17) \begin{align} p_{d d}(1) & =0.3, p_{u d}(1)=0.2,\\ p_{d u}(1) & =0.2, p_{u u}(1) =0.3.\nonumber \end{align}

Notice that we choose the marginals such that the $\mathbb{P}$ -marginals are the same as the $\mathbb{Q}$ -marginals, this is to make the example simpler and is not required in general case.

Consider the stock market index $S(t)=S_{1}(t) +S_{2}(t), $ $t=0,1.$ We find that the real-world correlation $\rho _{\mathbb{P}}(1) = \text{Corr}[R_1(1), R_2(1)] = 0.2,$ and the real-world volatility $\sigma _{\mathbb{P}}(1) =\sqrt{\text{Var}_{\mathbb{P}}\left [ R(1) \right ]} = 0.585.$ Since $\rho _{\mathbb{P}}(1) =0.2\gt 0$ , the return vector $\left ( R_{1}(1), R_{2}(1) \right )$ is Positive Quadrant Dependent under the real-world probability measure $\mathbb{P}$ :

(18) \begin{equation} \left ( R_{1}(1), R_{2}(1) \right ) \text{ is }\mathbb{P}-\text{PQD.} \end{equation}

We conclude from (18) that stocks are positively dependent under the real-world measure $\mathbb{P}$ specified by (17). However, the dependence structure under the risk-neutral measure $\mathbb{Q}$ can be different, even opposite from the dependence under $\mathbb{P}$ . The following Proposition1 is presented to show that the stock prices can be negatively dependent under the risk-neutral measure.

Proposition 1. A call option written on the stock market index $S(1)$ , with strike $K = 300$ , is traded and its time $0$ price $\hat{C}_{\mathbb{Q}}$ can be observed in the market. Then we have the following equivalence relations:

(19) \begin{align} \hat{C}_{\mathbb{Q}} \geq 70 & \Longleftrightarrow \mathbb{Q}-\text{PQD},\\ \hat{C}_{\mathbb{Q}} \leq 70 & \Longleftrightarrow \mathbb{Q}-\text{NQD} \nonumber \end{align}

Proposition1 shows that the market decides if the stocks are positive or negative dependent under the risk-neutral measure. Indeed, different market situations result in different risk-neutral measures, which can be different with the real-world measure $\mathbb{P}$ . Here we provide three different market situations in Table 1.

From Table 1, we can also conclude that the risk-neutral pricing measures $\mathbb{Q}^{(1) }$ , $\mathbb{Q}^{(2) }$ , and $\mathbb{Q}^{\left ( 3\right ) }$ are ordered in the correlation order:

\begin{equation*} F_{t}^{\mathbb {Q}^{(1)}}{\preceq }_{\text {Corr}}F_{t}^{\mathbb {Q}^{(2)}}{\preceq }_{\text {Corr}}F_{t}^{\mathbb {Q}^{\left ( 3\right )}}. \end{equation*}

This example shows that deriving the implied correlation and volatility to learn about the future dynamics of the stock prices is only part of the story. Implied measures are giving information about the risk-neutral dynamics and these statements cannot be directly translated to statements under the real-world probability measure. If the risk-neutral measure $\mathbb{Q}^{(1)}$ is chosen by the market, then implied volatility and correlation are substantially different with the real-world volatility and correlation. It might feel counter-intuitive to use a negative dependence structure to price the index option while under the real-world probability measure, the stocks are positive dependent. Notice that, however, the corresponding index option price does not lead to arbitrage and is consistent with other derivative prices.

Table 1. Three different market situations given that $\rho _{\mathbb{P}} = 0.2$ and $\sigma _{\mathbb{P}} = 0.585$

3.3 Example: A multiperiod model

Consider the financial market described above, but now we consider the future times $t=1,2,\ldots, 10.$ Assume that the real-world correlation $\rho _{\mathbb{P}}(t)$ is given by $\rho _{\mathbb{P}}(t) = 0.92 - 0.08t, \text{ for } t=1,2,\ldots 10.$ Similar to the one-period case, we still assume the real-world marginals are the same as the risk-neutral marginals, then the real-world dynamics can be expressed as:

(20) \begin{align} p_{d d}(t) =\frac{1}{4}(1+\rho _{\mathbb{P}}(t))&, p_{u d}(t) =\frac{1}{4}(1-\rho _{\mathbb{P}}(t)), \\ p_{d u}(t) =\frac{1}{4}(1-\rho _{\mathbb{P}}(t))&, \nonumber p_{u u}(t) =\frac{1}{4}(1+\rho _{\mathbb{P}}(t)).\nonumber \end{align}

We show real-world correlations in the second column of Table 2.

Table 2. Real-world and risk-neutral correlations

Assume at time 0, at-the-money call options with maturities $T=1,2,\ldots, 10$ are traded. The market prices are denoted by $\widehat{C}\left [ 300,T\right ], \text{ where }T=1,2,\ldots, 10,$ and are given in the third column of Table 2. The market prices can be used to determine the unique risk-neutral measure $\widehat{\mathbb{Q}}$ . Indeed, Theorem6 shows that implied correlations $\rho _{\widehat{\mathbb{Q}}}(t)$ can be uniquely determined by index call option prices $\widehat{C}[300,t]$ for $t=1,2,\dots, n$ . We present implied correlations $\rho _{\widehat{\mathbb{Q}}}(t)$ in the fourth column of Table 2. We can find from Table 2 that there always exists a gap between real-world and implied correlations. We call this gap the correlation gap. The following Fig. 1 shows the plot of the correlation gap, $\rho _{\mathbb{P}}(t) - \rho _{\widehat{\mathbb{Q}}}(t)$ for time $t = 1, 2, \dots, 10.$

Figure 1 The correlation gap $\rho _{\mathbb{P}}(t) - \rho _{\widehat{\mathbb{Q}}}(t)$ with respect to time $t$ .

Implied correlations capture the aggregate view of the market about future risk-neutral correlations. Table 2 and Fig. 1 clearly show that these views cannot always be directly translated to real-world statements. Indeed, in our example, the risk-neutral correlation is increasing: market participants anticipate a increase in correlation. However, the real-world correlation is moving in the opposite direction and decreases over time. Moreover, we can see from Fig. 1 that the correlation gap $\rho _{\mathbb{P}}(t) - \rho _{\widehat{\mathbb{Q}}}(t)$ is decreasing over time. The correlation gap is highly positive at time $1$ and becoming strongly negative at time $10$ . Notably, a nonzero correlation gap may persist in the financial market over time. In the next Section 4, we demonstrate that this nonzero correlation gap not only manifests in the financial market but also carries implications for the purchase of unit-linked insurance products.

4.1 Illustration: A single period model

Example 3.2 illustrates that there exists a correlation gap in the financial market. Consider the same market setting as in Example 3.2, we show how the correlation gap determines the expected excess return for buying the unit-linked insurance product.

In the single period market with $r=0$ , as in Example 3.2, let us consider a one-period unit-linked insurance product with $T = 1$ . The payout per policy for this unit-linked insurance contract, given by (23), is denoted as $h(1)$ . Assume the threshold $K = 300,$ from (24), a market-consistent valuation of this one-period unit-linked insurance product is denoted as $\rho [h(1)]$ and given by:

\begin{equation*} \rho [h(1)] = \mathbb {E}^{\mathbb {Q}}[\text {max}(S_1(1)+S_2(1), 300)]p_x = (\hat {C}_{\mathbb {Q}} + 300)p_x, \end{equation*}

where $\hat{C}_{\mathbb{Q}}$ represents the time- $0$ index call option price with strike $K=300$ and maturity $T=1$ , and $p_x$ stands for the probability that a policyholder aged exactly $x$ survives to time $1$ . Subsequently, the expected excess return for each policyholder at time- $1$ can be written as:

(29) \begin{equation} \text{Expected ER} = p_x \left (\mathbb{E}^{P}[(S(1) -300)_+] - \hat{C}_{\mathbb{Q}}\right ), \end{equation}

where $S(1)$ is the market index defined in (11).

Using the real-world dynamics in (17), the expectation $\mathbb{E}^{P}[(S(1) -300)_+]$ can be expressed as follows:

\begin{equation*}\mathbb {E}^{P}[(S(1) -300)_+] = 180 \times \frac {1}{4}(1+\rho _{\mathbb {P}}(1)) + 100 \times \frac {1}{4}(1-\rho _{\mathbb {P}}(1)) = 70 + 20\rho _{\mathbb {P}}(1). \end{equation*}

Additionally, Proposition1 implies that $\hat{C}_{\mathbb{Q}} = 70+20\rho _{\mathbb{Q}}(1).$ Hence in this market scenario, we can find that:

(30) \begin{equation} \text{Expected ER} = 20 \cdot p_x(\rho _{\mathbb{P}}(1) - \rho _{\mathbb{Q}}(1)). \end{equation}

From (30), we can clearly see that different correlation gaps give rise to different expected excess returns. Indeed, a positive (negative) correlation gap implies a positive (negative) expected excess return for buying the unit-linked insurance. Table 3 compares the expected excess return of the unit-linked insurance policyholder across the three different market scenarios presented in Example 3.2.

Table 3 clearly reveals that the presence of a nonzero correlation gap results in a nonzero expected excess return for the purchase of the unit-linked insurance product. Therefore, a policyholder of the unit-linked insurance contract is also exposed to a correlation risk in the financial market. In the following Section 5, we show how one can trade and hedge the correlation risk in the financial market.

Table 3. Expected excess return for each policyholder in three different market situations

5.1 Variance swaps

We first introduce a new derivative in our financial market: A variance swap on the stock market index. At maturity $T,$ the holder of the floating leg of a variance swap receives the difference $ RV[T] -SR[T], $ where $RV[T]$ is defined as:

(31) \begin{equation} RV[T] =\frac{1}{T}\sum _{t=1}^{T} R(t)^2. \end{equation}

The quantity $RV[T]$ is called the annualized realized variance and represents the floating leg of the contract. The fixed leg of the contract is $SR[T]$ , also called the swap rate. The swap rate of the contract is a constant which is determined at inception such that the contract is fair and no money has to be exchanged at inception. Denote by $\mathbb{Q}$ the pricing measure used by the market to price derivatives. The swap rate is then given by:

(32) \begin{equation} SR[T] =\mathbb{E}_{\mathbb{Q}}\left [ RV[T] \right ]. \end{equation}

The holder of a variance swap will pay the swap rate $SR[T]$ at maturity and will receive the realized variance $RV[T]$ . At time $0$ , the swap rate is known but the realized variance is random. Meanwhile, investors are willing to pay a premium to hedge against the variance risk in the financial market, called the variance risk premium (VRP); see, e.g., Carr & Wu (Reference Carr and Wu2009), Bollerslev & Zhou (Reference Bollerslev and Zhou2007), and Alibeiki & Lotfaliei (Reference Alibeiki and Lotfaliei2021). Indeed, the VRP is the expected profit for the holder of the variance swap, which is given by:

\begin{equation*} VRP= \text {Expected Profit}=\mathbb {E}_{\mathbb {P}}\left [ RV[T] \right ] -SR[T] . \end{equation*}

The VRP can be expressed as follows:

(33) \begin{equation} VRP=\frac{1}{T}\sum _{t=1}^{T}\left ( \sigma _{\mathbb{P}}^{2}(t) -\sigma _{\mathbb{Q}}^{2}(t) \right ) +\frac{1}{T}\sum _{t=1}^{T}\left ( \left ( \mathbb{E}_{\mathbb{P}}\left [ R(t) \right ] \right ) ^{2} - \left ( \mathbb{E}_{\mathbb{Q}}\left [ R(t) \right ] \right ) ^{2} \right ), \end{equation}

where $\sigma _{\mathbb{Q}}(t)$ is the $t-$ year forward volatility. Expression (33) clearly shows that the holder of the variance swap longs the real-world volatility and shorts the risk-neutral volatility. The summation in the expression of the VRP arises because a variance swap is actually trading the second moment, rather than the variance. Investing in a variance swap becomes attractive if the gap between real and risk-neutral volatility is substantially large. If this gap is positive, it may lead to a positive VRP, while in the other case, it may yield a negative risk premium. Investors who are exposed to variance risk may be willing to buy the variance swap, even if the risk premium is negative, since it can provide insurance against adverse variance risk scenarios. Indeed, investors who buy the variance swap receive a hedge for their exposure to variance risk and are willing to accept a negative expected profit in return.

5.2 Dispersion swaps

In Example 3.2, it is clear that there exists a correlation gap. However, variance swaps do not allow us to directly trade the difference between the real-world correlation and the implied correlation. Indeed, two prevalent approaches have been proposed for exploiting or hedging the correlation risk within the financial market. First, investors can implement a dispersion trading strategy, wherein they construct a portfolio by taking a position in an index option and the opposite position in the options written on the constituents of the index. Such dispersion strategies have been established using calls, puts, straddles, and variance swaps, see, e.g., Jacquier & Slaoui (Reference Jacquier and Slaoui2007), Bossu (Reference Bossu2014), and Meissner (Reference Meissner2015). In the second approach, one directly trades in correlation derivatives such as correlation swaps or covariance swaps. Correlation swaps were first introduced in the early 2000s as a derivative to hedge against the correlation risk exposure inherent in exotic derivatives trading. However, unlike implied volatility, implied correlations are not directly observable from option markets, making the pricing and hedging of correlation swaps a challenging task. For literature on the pricing of the correlation swap, we refer to Bossu (Reference Bossu2005, Reference Bossu2007). In addition to correlation swaps, covariance swaps are a generalization of the variance swap, providing an alternative method to hedge correlation risk within a two-asset framework. The floating leg of the covariance swap is the realized covariance between the log returns of two underlying assets. Discussions on the pricing and hedging strategies of covariance swaps can be explored in Carr & Corso (Reference Carr and Corso2001) and Salvi & Swishchuk (Reference Salvi and Swishchuk2014).

Instead of employing existing approaches to capitalize on the correlation risk between several assets, we propose a new derivative called the dispersion swap. The floating leg of the contract relies solely on the product of forward returns of the two underlying stocks, establishing an intrinsic link to the correlation between the assets, and simplifying the determination of the fixed leg. Theoretical analysis and numerical results will be presented to demonstrate that the dispersion swap provides an opportunity to trade the correlation gap. Moreover, we also show that the existence of a CRP in the financial market is justified.

The holder of the floating leg of a dispersion swap receives at maturity the following payoff:

\begin{equation*} RD[T] -P[T], \end{equation*}

where

(34) \begin{equation} RD[T] =\frac{1}{T}\sum _{t=1}^{T}R_{1}(t) R_{2}(t) . \end{equation}

We call $RD[T]$ the realized dispersion. $P[T]$ is the fixed leg of the contract, determined at inception such that the contract is fair and with zero initial payment. The following proposition is presented to show the realized dispersion can be approximated by the realized variance of stock index and the realized variances of individual stocks. A proof of this proposition can be found in Appendix A.2.

Proposition 2. The realized dispersion can be approximated by the difference between the realized variance of the stock market index and the sum of realized stock variances:

(35) \begin{equation} RD[T] \approx \frac{1}{2}\left (RV[T] - RV_1[T] -RV_2[T]\right ), \end{equation}

where $RV_i[T]$ is the realized variance for stock $i$ and is given by:

\begin{equation*}RV_i[T] = \frac {1}{T}\sum _{t=1}^{T}R_i(t)^2, i = 1,2.\end{equation*}

From Proposition2, the realized dispersion can be approximated by subtracting marginal variances from the basket variance. It implies that the realized dispersion is indeed linked to the dependence. Additionally, there is a more refined approximation approach based on Taylor series expansion, which can be found in Appendix A.5. This approximation allows to determine the approximation error.

At time $0,$ the realized dispersion is unknown and random, while the strike is fixed and known. The fixed leg $P[T]$ is determined as follows:

\begin{equation*} P[T] =\mathbb {E}_{\mathbb {Q}}\left [ RD[T] \right ] . \end{equation*}

In our market setting, the fixed leg $P[T]$ can be expressed as follows:

(36) \begin{equation} P[T] =\frac{1}{4T}\sum _{t=1}^{T}\left [\rho _{\mathbb{Q}}(t)(u_1-d_1)(u_2-d_2)+(u_1+d_1)(u_2+d_2)\right ]. \end{equation}

A proof of Expression (36) is given in Appendix A.3.

The expected profit for the holder of the floating leg of a dispersion swap is given by:

\begin{equation*} \text {Expected Profit}=\mathbb {E}_{\mathbb {P}}\left [ RD[T] \right ] -P[T] . \end{equation*}

Hence it is straightforward to show that the expected profit of the dispersion swap can be expressed as:

(37) \begin{align} \text{Expected Profit} = & \frac{1}{T}\sum _{t=1}^{T}\left (\sigma _{\mathbb{P}}[R_1(t)]\sigma _{\mathbb{P}}[R_2(t)]\rho _{\mathbb{P}}(t) - \sigma _{\mathbb{Q}}[R_1(t)]\sigma _{\mathbb{Q}}[R_2(t)]\rho _{\mathbb{Q}}(t)\right )\nonumber \\ &+ \frac{1}{T}\sum _{t=1}^{T} \left (\mathbb{E}_{\mathbb{P}}\left [R_1(t)\right ]\mathbb{E}_{\mathbb{P}}\left [R_2(t)\right ]-\mathbb{E}_{\mathbb{Q}}\left [R_1(t)\right ]\mathbb{E}_{\mathbb{Q}}\left [R_2(t)\right ]\right ). \end{align}

Expression (37) clearly indicates that the buyer of the dispersion swap longs the real-world correlation and shorts the risk-neutral correlation. To be more specific, in Appendix A.4, it is proven that the expected profit of the dispersion swap can also be written as follows:

(38) \begin{align} \text{Expected Profit} = &\frac{(u_1-d_1)(u_2-d_2)}{T}\sum _{t=1}^{T}\left (\sqrt{p_{u \cdot }(t)p_{d \cdot }(t)p_{\cdot u}(t)p_{\cdot d}(t)} \rho _{\mathbb{P}}(t)-\nonumber \frac{1}{4}\rho _{\mathbb{Q}}(t)\right )\nonumber \\ & + \frac{1}{T} \sum _{t=1}^{T} \mathbb{E}_{\mathbb{P}}\left [R_1(t)\right ]\mathbb{E}_{\mathbb{P}}\left [R_2(t)\right ] -\frac{1}{4}\left (u_1+d_1\right )\left (u_2+d_2\right ). \end{align}

From Expressions (37) and (38), we can find that the expected profit of the dispersion swap is closely associated with the correlation gap. The CRP is also defined using the correlation gap. In our market model, the CRP is defined as follows:

(39) \begin{equation} CRP = \frac{1}{T}\sum _{t=1}^{T} (\rho _{\mathbb{P}}(t) - \rho _{\mathbb{\mathbb{Q}}}(t) ). \end{equation}

By assuming the real-world marginals are same as the risk-neutral marginals in this market setting, i.e., $p_{u \cdot }\left ( t \right )= p_{\cdot u}\left ( t \right ) =\frac{1}{2},$ the expected profit of the dispersion swap given by (38) can be written as:

(40) \begin{align} \text{Expected Profit Dispersion Swap} &=\frac{(u_1-d_1)(u_2-d_2)}{4T}\sum _{t=1}^{T}\left (\rho _{\mathbb{P}}(t)- \rho _{\mathbb{Q}}(t)\right )\nonumber \\ &=\frac{(u_1-d_1)(u_2-d_2)}{4} CRP. \end{align}

When the marginal distributions are the same under $\mathbb{P}$ and $\mathbb{Q}$ , the expected profit of buying the dispersion swap is solely determined by the correlation gap. It is proportional to the CRP, which allows us to directly trade the weighted difference of the correlation gap $\rho _{\mathbb{P}}(t)- \rho _{\mathbb{Q}}(t)$ . A larger positive correlation gap results in a higher expected profit, increasing the contract’s appeal to potential investors.

5.3 Illustration: One-period example

In this subsection, we still consider the same market setting as in Example 3.2, and will demonstrate the potential use of dispersion swaps in exploiting the correlation gap. In this one-period market with $r=0$ , it follows directly from the Expression (33) that the expected profit of the variance swap (EPVS) is given by:

(41) \begin{equation} \text{EPVS} = VRP = \sigma _{\mathbb{P}}^{2}(1)-\sigma _{\mathbb{Q}}^{2}(1) + (\mathbb{E}_{\mathbb{P}}[R(1)])^2 - (\mathbb{E}_{\mathbb{Q}}[R(1)])^2. \end{equation}

Additionally, the expected profit of the dispersion swap (EPDS) given by Expression (40) can be expressed as follows:

(42) \begin{equation} \text{EPDS} = \frac{(u_1-d_1)(u_2-d_2)}{4} CRP = \frac{(u_1-d_1)(u_2-d_2)(\rho _{\mathbb{P}}(1)- \rho _{\mathbb{Q}}(1))}{4}. \end{equation}

We use Table 4 to compare the expected profits of the variance swap and the dispersion swap, the VRPs and the CRPs, concerning three different market situations in Example 3.2.

In Example 3.2, the first market situation featured a significant gap between the real-world and risk-neutral scenarios. Variance and dispersion swaps serve as tools to exploit this gap. As shown in Table 4, both the VRP and the CRP are positive, suggesting potential benefits from taking long positions in the variance swap and the dispersion swap. However, the gap between the real-world and the risk-neutral volatility and correlation is much smaller in the second market situation, which is reflected in the expected profit of the variance and the dispersion swap, as shown in Table 4. Expected profits turn out to be limited in case. Since these are only expectations, fluctuations can result in losses when stepping in a variance or dispersion trade, making these products less investment-worthy in this market situation than the first market situation. In the third market situation, both the volatility and correlation gaps are strongly negative, resulting in significantly negative variance and CRPs. Rational investors are willing to short the variance swap or the dispersion swap to gain a profit. But who is willing to long the dispersion swap or the variance swap? Market participants who are afraid that the correlation and volatility risk will have significant negative impact on their portfolios are willing to accept a negative expected profit to long the dispersion swap or the variance swap. Indeed, they are more afraid of the correlation and volatility risk than they are of the possible losses from the dispersion and volatility swap.

Table 4. Variance risk premium (VRP) and correlation risk premium (CRP) in three different market situations