On hedging your combination bets

What is a combination bet?

A combination bet, also known as an accumulator bet, is a bet that is a conjunction of $n$ smaller bets. If there exist single bets with odds $O_i$ for events $E_i$, then a combination bet is a bet with odds $O = \prod O_i = O_1 O_2 \dots O_n$ and event $E = \bigwedge E_i = E_1 \wedge E_2 \wedge \dots \wedge E_n$.

The multiplying of odds only makes sense when the events are uncorrelated $P(E_i \wedge E_j) = P(E_i)P(E_j)$. It then preserves the property that if there are no +EV single bets ($O_i P(E_i) < 1$) then also there is no +EV combination bet ($O P(E) = \prod O_i P(\bigwedge E_i) \stackrel{\perp}{=} \prod O_i \prod E_i = \prod O_i E_i < 1$).

Of course events are never truly uncorrelated in the real world, but if the covariance is small enough then a modest bit of overround will still overwhelm that (exercise for the reader).

Simple static hedging

Assuming we have a bettable partition of all possible outcomes for $i$ given by $P_i = \{ E_i = E_{i. 1} , E_{i, 2}, \dots, E_{i, n} \}$, we can hedge a combination bet by betting on all other possible combinations of outcomes $$\left \{ \bigwedge H_i \mid H_1 \in P_1, H_2 \in P_2, \dots, H_n \in P_n \right \} \setminus \{ E \}$$

However this is impractical due to the combinatorial explosion of the amount of bets ($\prod |P_i| - 1$, at least $2^n - 1$). Calling the overround of each partition $P_i$ as $V_i$, calculating the stakes gives the total overround of this hedging strategy as $V = \prod V_i$.

Legging it

A more simple way to hedge a combination bet is to hedge the events one by one as they happen. E.g. if the outcome of $E_1$ is determined before $E_2$, then only hedge $E_1$ first. After $E_1$ is done, the combination bet reduces either to nothing or a smaller combination bet. Rinse and repeat.

The downside to this strategy is that it’s impossible to do for simultaneous events. Also there is a risk of the odds shifting against you while you wait to place your hedge bets.

The upside is that we use very few bets, and consequently low costs. The expected overround can be calculated to be $$\sum_{j=1}^n V_j \prod_{k=1}^{j-1} P(E_k)$$

Also there is a change that the odds shift in your favor while watiing.

Dynamic hedging

In financial theory, dynamic hedging is a very successful idea for hedging non-linear derivatives. Combination bets are a non-linear derivative. We can see this by taking the second derivates of $O$ with respect to $O_i$ (the Hessian) of a combaination bet with $n = 3$: $$\begin{bmatrix} 0 & O_3 & O_2 \\ O_3 & 0 & O_1 \\ O_2 & O_1 & 0 \end{bmatrix}$$

Interestingly, the diagonal is zero. That means if we try to hedge the combination bet linearly for every $i$, then if the odds of $O_1$ change and then change again we are still okay, but if the odds $O_1$ change and then the odds $O_2$ change our attempted hedge is broken.

So we would have to rebalance our linear hedge every time the odds change. As with options, because of the transaction costs when to do this becomes a stochastic control problem. Given that transaction costs are very high and odds volatility is concentrated around the time of the game that we are betting on, it is likely the optimal strategy will devolve to just legging it.

Optimized static hedging

Taking advantage of the fact that we don’t have to include every $i$ in every hedging bet, we can reduce the amount of bets compared to simple static hedging. The trick is to reduce our hedging problem one index at a time, with a recursive definition of the hedging events $H$ for a set of events $E$: $$H(E) = \{ \{ H_1, E_2, \dots, E_n \} \mid H_1 \in P_1, H_1 \ne E_1 \} \cup H(\{E_2, E_3. \dots, E_n \})$$ This reduces the amount of bets to a more managable $\sum (|P_i| - 1)$.

Experimental results show that this reduces overround paid as expected. Also note that the order of the events is arbitrary, but there is not a clear optimal order. This seems to be the optimal way to hedge statically, but maybe there is an even better way?