Minimal Market Spread Under Correlation Uncertainty
Two events $A$ and $B$ have known marginal probabilities $P(A) = p$ and $P(B) = q$. Define the indicator $Y = \mathbf{1}\{A \oplus B\}$, i.e., $Y = 1$ if exactly one of $A$ or $B$ occurs (exclusive or). The catch: you do not know the joint distribution. You only know that the correlation $\rho = \operatorname{Corr}(\mathbf{1}_A, \mathbf{1}_B)$ lies in a known interval $[\rho_L, \rho_U]$.
- Derive tight (attainable) bounds on $E[Y]$ over all joint laws consistent with the given marginals and the constraint $\rho \in [\rho_L, \rho_U]$.
- You must post a single symmetric market $(b, a)$ with mid $m$ and half-spread $s/2$, so $b = m - s/2$ and $a = m + s/2$. An informed trader knows the true $\rho \in [\rho_L, \rho_U]$ and will trade against you whenever your quote is off. Find the minimal spread $s$ (and the corresponding mid $m$) that guarantees nonnegative expected P&L against this worst-case informed trader.
Hints
- Write $Y = \mathbf{1}_A + \mathbf{1}_B - 2\,\mathbf{1}_A\mathbf{1}_B$ and express $E[Y]$ in terms of $P(A \cap B)$. How does $P(A \cap B)$ relate to the correlation $\rho$?
- $E[Y]$ turns out to be a linear function of $\rho$. So optimizing over $\rho \in [\rho_L, \rho_U]$ is trivial -- the extremes are at the endpoints.
- For part (ii), the market maker avoids losses exactly when the bid-ask interval covers the entire range of possible fair values. Set $s = E[Y]_{\max} - E[Y]_{\min}$ and center the market at their midpoint.
Worked Solution
How to Think About It: This is a classic robust market-making problem. You are pricing a digital on an XOR event, but you do not know how the two underlying events are correlated -- you just have bounds on the correlation. Part (i) is really about Frechet bounds restricted by a correlation constraint, and part (ii) asks: how wide does your spread need to be to avoid getting picked off by someone who knows more than you? The answer to part (ii) falls out immediately once you have part (i) -- the spread has to cover the full range of fair values.
Quick Estimate: Take $p = q = 0.5$ and $\rho \in [-0.5, 0.5]$. Then $\sigma_A = \sigma_B = 0.5$, so $c = 0.25$. The XOR base rate is $p + q - 2pq = 0.5$. The range of $E[Y]$ is $0.5 - 2(-0.5)(0.25) = 0.75$ down to $0.5 - 2(0.5)(0.25) = 0.25$. So the fair value swings between 0.25 and 0.75 depending on correlation, and the spread must be at least $0.75 - 0.25 = 0.50$. That is a very wide spread -- correlation uncertainty is expensive.
Approach: Express $E[Y]$ as a linear function of $\rho$, then optimize over the constraint interval.
Formal Solution:
*Part (i): Bounds on $E[Y]$*
Write $Y = \mathbf{1}_A + \mathbf{1}_B - 2\,\mathbf{1}_A \mathbf{1}_B$, so:
$$E[Y] = p + q - 2\,P(A \cap B)$$
The correlation between the two indicators is:
$$\rho = \frac{P(A \cap B) - pq}{\sqrt{p(1-p)\,q(1-q)}}$$
Let $c = \sqrt{p(1-p)\,q(1-q)}$ denote the product of marginal standard deviations. Then $P(A \cap B) = pq + \rho\, c$, and:
$$E[Y] = (p + q - 2pq) - 2\rho\, c$$
This is linear and decreasing in $\rho$. Therefore over $\rho \in [\rho_L, \rho_U]$:
$$E[Y]_{\max} = (p + q - 2pq) - 2\rho_L\, c \quad (\text{attained at } \rho = \rho_L)$$
$$E[Y]_{\min} = (p + q - 2pq) - 2\rho_U\, c \quad (\text{attained at } \rho = \rho_U)$$
These bounds are tight because for any $\rho$ in the Frechet-feasible range, there exists a $2 \times 2$ joint law with the given marginals achieving that correlation.
*Part (ii): Minimal spread*
The market maker posts bid $b = m - s/2$ and ask $a = m + s/2$. The informed trader knows the true fair value $V(\rho) = (p+q-2pq) - 2\rho\,c$ and acts as follows:
- If $V(\rho) > a$: trader buys at $a$, MM loses $V(\rho) - a$ per contract.
- If $V(\rho) < b$: trader sells at $b$, MM loses $b - V(\rho)$ per contract.
- If $b \leq V(\rho) \leq a$: no trade (or zero expected loss).
For the MM to have nonnegative expected P&L against all $\rho \in [\rho_L, \rho_U]$, we need:
$$b \leq V(\rho) \leq a \quad \forall\, \rho \in [\rho_L, \rho_U]$$
That is, the bid-ask interval must contain the entire range $[E[Y]_{\min},\, E[Y]_{\max}]$. The minimal spread that achieves this is:
$$s^{*} = E[Y]_{\max} - E[Y]_{\min} = 2(\rho_U - \rho_L)\,c = 2(\rho_U - \rho_L)\sqrt{p(1-p)\,q(1-q)}$$
with mid:
$$m^{*} = \frac{E[Y]_{\max} + E[Y]_{\min}}{2} = (p + q - 2pq) - (\rho_L + \rho_U)\,c$$
Answer:
The tight bounds on $E[Y]$ are:
$$E[Y] \in \Big[(p+q-2pq) - 2\rho_U\,c,\;\; (p+q-2pq) - 2\rho_L\,c\Big]$$
where $c = \sqrt{p(1-p)\,q(1-q)}$. The minimal spread is:
$$s^{*} = 2(\rho_U - \rho_L)\sqrt{p(1-p)\,q(1-q)}$$
with mid $m^{*} = (p+q-2pq) - (\rho_L + \rho_U)\,c$.
Intuition
The key insight is that XOR is just a linear function of the joint probability $P(A \cap B)$, which in turn is linear in the correlation $\rho$. So the entire problem reduces to: $E[Y]$ is an affine function of $\rho$, and you are optimizing an affine function over an interval. That is about as simple as robust optimization gets -- the extrema are always at the endpoints.
The market-making part illustrates a general principle: when you face model uncertainty (here, unknown correlation), the width of your spread is determined by the range of fair values under that uncertainty. The more parameter uncertainty you have, the wider you must quote. In practice, this is exactly why correlation products (basket options, dispersion trades) tend to have wide bid-ask spreads -- the correlation parameter is hard to pin down, and the pricing is sensitive to it. The factor $\sqrt{p(1-p)\,q(1-q)}$ tells you that the spread is largest when both events have probability near 0.5 (maximum Bernoulli variance) and shrinks when either event is near-certain or near-impossible.