An easy guide to closing out trades

Andrew Colin breaks down the mathematics of hedging your pre-game position, in-play.

Categories: All Sports, Prices, Statistical models, The basics

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Closing out a trade to lock in your profit is one of those basic techniques that everyone should understand, but is surprisingly hard to find explained. So here is a simple account.

Closing out a back trade

Suppose that you’ve backed a soccer team to win at odds of 10. They do well during the match and the odds drop to 5. The value of your bet has increased in value, but you’re worried that the match could go against you. How do you close, or hedge, the bet before the game ends to lock in your profit?

You can do this by putting on a lay bet against the same team that gives equal but opposite exposure to your first bet. How much should you lay to lock in the profit?

The trick here is to look at how much each of the two bets will make at the end of the match. Suppose that

–          the odds at which you made your initial back bet were ODDS1;

–          the odds at which you  made your subsequent bet were ODDS2, where ODDS2 < ODDS1;

–          you bet $1 on the initial back bet;

–          you bet $S on the subsequent lay bet, where we don’t know S yet.

If the team wins, profit from the back bet will be -1 + ODDS1, and the loss from the lay bet will be (1-ODDS2) * S. The total amount won will be (-1+ODDS) + (1-ODDS2) * S.

If the team draws or loses, profit from the back bet will be -1, and the profit from the lay bet will be 1 * S. Therefore the total amount won will be -1 + S.

To lock in a guaranteed profit, irrespective of the outcome of the match, set the two expressions for winning amounts whether you win or lose to be the same: in other words

(-1 + ODDS1) + (1-ODDS2) * S = -1 + S

The only unknown in this expression is S. A little algebra shows that

S = ODDS1 / ODDS2

and the profit is S-1 for each dollar staked.

Example: In the above case, the value of S is 10/5  = 2.  To lock in a profit of S-1 – 2-1 = $1, you should hedge by placing a $2 lay bet at odds of 5.

Closing out a lay trade

Consider the opposite case, where we have placed a lay bet at ODDS1, seen the odds decrease, and now want to place a back bet at ODDS2 to take profits. As before, set the amount you back to be S.

If the team loses, profit from the lay bet will be 1, and the loss from the back bet will be -1 * S. The total amount won will be 1-S.

If the team wins, loss from the lay bet will be 1-ODDS1, and the profit from the back bet will be (-1 + ODDS2) * S. The total amount won will be (1-ODDS1) + (-1 + ODDS2)*S.

Setting the two expressions to give the same outcome implies

(1-ODDS1) + (-1 + ODDS2) * S = 1 – S

which gives

S = ODSS1/ODDS2

and a profit of 1-S.

For example, suppose you initially placed a lay bet on the team at odds of 5, and the odds move to 10. In this case S is 5/10 = 0.5. To lock in a profit of 1 – 0.5 = $0.5, hedge by placing a back bet of $0.5 at odds of 10.

Conclusion

The one result to take away from this is that the amount to hedge is always ‘first odds divided by second odds’, whatever your initial bet.

Happy hedging!

About Andrew Colin

Andrew holds a PhD in Mathematics from the University of St Andrews, is a Fellow of the Institute of Mathematics and its Applications, and holds Chartered Mathematician accreditation. Andrew is also Adjunct Professor at the Faculty of Business at the University of Tasmania.
Andrew’s previous experience includes design and implementation of forecasting and arbitrage algorithms for soccer, horse racing and greyhound racing. He has also working in defence, academia, as a proprietary trader and a bond fund manager.
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