Quantitative Analysis, Risk Management, Modelling, Algo Trading, and Big Data Analysis

## Dutch Book: Making a Riskless Profit

If you think there is no way to make a riskless profit, think again. This concept is known as Dutch Book. First, let me remind the definition of the probability stated as odds. Given a probability $P(E)$ of an event $E$, odds for $E$ are equal
$$E=P(E)/[1-P(E)] .$$ Given odds for $E$ of a:b, the implied probability of $E$ is $a/(a+b)$. Reversely speaking, odds against $E$ are $$E=[1-P(E)]/P(E).$$ Thus, given odds against $E$ of a:b, the implied probability of $E$ is $b/(a+b)$.

Now, suppose John places $\$100$bet on$X$at odds of 10:1 against$X$, and later he is able to place a$\$600$ bet against $X$ at odds of 1:1 against $X$. Whatever the outcome of $X$, that person makes a riskless profit equal to $\$400$if$X$occurs or$\$500$ if $X$ does not occur because the implied probabilities are inconsistent.

John is said to have made a Dutch Book in $X$.