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 $X$.

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