This article shows how to implement synthetic Asian handicap bets, and how to construct a Dutch book so that any arbitrage opportunities arising between these synthetic bets and actual bets can be locked in to generate profits, irrespective of the outcome of the match.

Expected return is of the order of 0.2% per match, so some degree of automation is vitally important to cover large numbers of matches, as is availability of sufficient liquidity to match bets.

Asian handicap bets: the basics

An Asian handicap bet is one in which a handicap is assigned to one of the competing teams. The handicap is always given when referring to the bet; for instance, Liverpool -1.5&-2.0 means that Liverpool have a handicap of -1.75 goals (and their opponents a handicap, or an advantage, of +1.75 goals).

This handicap can be a multiple of a half in order to ensure that the result of a bet is always a win or a loss. For instance, a handicap of -1.5 goals for the home team means that the home team has to win by at least two goals in order to win an Asian handicap bet. If it wins by one goal, draws, or loses then it loses the Asian handicap bet.

Handicaps can also be integer valued, in which case a draw result is possible. In this case a draw results in the bettor’s stake being returned, so the net profit on the match is always zero.

Lastly, handicaps can be given in terms of quarter goals. A handicap of 0.25 goals is regarded as two separate bets, with equal stakes assigned to bets of h=0.0 and 0.5.

Match handicaps are usually chosen so as to ensure that the odds of the Asian handicap bet are as close to 50/50 as possible. This ensures a more exciting betting experience for punters who may support a weaker team.

Replicating Asian handicap bets

One of the simplest Asian handicap bets is to give the home team a handicap of -0.5 goals. With this handicap, the home team wins the Asian handicap bet if it wins, and loses the Asian handicap bet if it draws or loses. The odds of a home win in the market odds market are therefore exactly the same as the odds of a home win in the Asian handicap market.

Since the payoff for these two strategies is the same, we can say that the market odds home win bet replicates an Asian handicap home win bet at h=-0.5. It is therefore possible, in principle, to look for cases where the market odds of backing a home win are greater than the Asian handicap odds (at h=-0.5) of laying a home win. If such a situation occurs, then both sides of the trade may be taken to lock in a profit. The same situation occurs if the market odds of a home win are less that the Asian handicap odds, but in this case the trades should be reversed.

This arbitrage strategy appears to be well known, and as a result very few cases are seen in which market odds and Asian handicap odds diverge as described. However, we show below that it is possible to replicate exactly any Asian handicap bet with a handicap of between -0.5 and +0.5 using at most two market odds bets. This range of handicaps covers more than 60% of matches in the UK Premier League.

Arbitrage strategies

Denote

• the back and lay odds of a home win as;  H_B, H_L

• the back and lay odds of an away win as;  A_B, A_L

• the back and lay odds of a draw as;  D_B, D_L

• the back and lay odds of an Asian handicap home win for the current handicap as;  H*_B, H*_L

• the back and lay odds of an Asian handicap away win for the current handicap as;  A*_B, A*_L

Home team handicap H=-¼

There are four cases to consider, each of which must be examined to search for arbitrage opportunities. The payoff matrices are as shown:

Match result	Back home win	Lay away win	Lay Asian handicap home win
Home win	HB – 1	1	1 – H*L
Draw	– 1	1	0.5
Away win	– 1	1 – AL	1

 

Match result	Lay home win	Back away win	Back Asian handicap home win
Home win	1 – HL	– 1	H*B – 1
Draw	1	– 1	0.5
Away win	1	AB – 1	1

 

Match result	Lay home win	Back away win	Lay Asian handicap away win
Home win	1 – HL	-1	1
Draw	1	-1	( 1 – A*L ) / 2
Away win	1	AB – 1	1 – A*L

 

Match result	Back home win	Lay away win	Back Asian handicap away win
Home win	HB – 1	1	1
Draw	– 1	1	( A*B -1 ) / 2
Away win	– 1	1 – AL	A*B -1

Home team handicap H=0

Here we use the fact that a combination of backing a home win and laying an away win results in a zero return for a draw, which is the same behaviour as an Asian handicap bet.

Again there are four cases to consider, each of which must be examined to search for arbitrage opportunities. The payoff matrices are as shown (note that they are identical to the case where H=-¼, except for the return from the Asian handicap bet when the match results in a draw):

Match result	Back home win	Lay away win	Lay Asian handicap home win
Home win	HB – 1	1	1 – H*L
Draw	– 1	1	0
Away win	– 1	1 – AL	1

 

Match result	Lay home win	Back away win	Back Asian handicap home win
Home win	1 – HL	– 1	H*B – 1
Draw	1	– 1	0
Away win	1	AB – 1	– 1

 

Match result	Back away win	Lay home win	Lay Asian handicap away win
Home win	– 1	1 – HL	1
Draw	– 1	1	0
Away win	AB – 1	1	1 – A*L

 

Match result	Lay away win	Back home win	Back Asian handicap away win
Home win	1	HB – 1	– 1
Draw	1	– 1	0
Away win	1 – A*L	– 1	A*B – 1

 

Home team handicap H=+¼

This case is essentially symmetrical with the H=-¼ handicap.

Home team handicap H=+½

Match result	Back home win	Lay away win	Lay Asian handicap home win
Home win	HB – 1	– 1	1
Draw	– 1	DB – 1	1
Away win	– 1	– 1	1 – H*L

 

Match result	Lay home win	Back away win	Back Asian handicap home win
Home win	1 – HL	1	H*B – 1
Draw	1	1 – DL	H*B – 1
Away win	1	1	1 – H*L

Home team handicap H=-½

This case is essentially symmetrical with the H=½ handicap.

Dutching an Asian handicap book

By considering the payoffs of all outcomes, we determine if a combination of bets exists that always generates a positive return, irrespective of the result of the match. The aim of this section is to determine how much should be put on each bet to ensure that the return of the strategy is the same. One way to address this problem is to regard it as a matrix equation in three unknowns. If the payoff matrix is denoted by M and the amounts of capital w1, w2, w3 to be placed on each bet are denoted by w, then the ‘constant return’ condition is given by

M_w = 1

where is the (constant) profit from the strategy and 1 is the unit vector.

The solution is

(1.2)

Consider the following payoff matrix:

	Back H	Lay A	Lay AH home-%
HW	1.02	1	-0.73
DR	-1	1	0.5
AW	-1	-3.4	1

In this case, the matrix equation to solve is

1.02	1	– 0.73	W1		1
– 1	1	– 0.50	W2	=	1
– 1	-3.40	1	W3		1

This forms a set of three equations in four unknowns; the fourth constraint is given by

W₁ + W₂ + W₃ = 1

where each

Solving and normalising the

W₁

to one gives

W = (0.3529, 0.0674, 0.5796)

which are the appropriate relative amounts to bet on each market in order to lock in a profit of 0.42% per dollar staked.

Discussion

Although fleeting, several opportunities per week of this type arise in UK Premier League football, and others probably occur in the European markets. The opportunities appear to last for periods between minutes and hours.

To take advantage of these requires appropriate technology to perform the above arbitrage calculations for each market, and availability of liquidity, particularly in Asian handicap markets.

The profits seen from this strategy are small in relation to those generated by others, but they are almost completely risk-free by design.