I don’t think that gilch answered the question correctly. His two games A and B are both “additive” games (unless I’m misunderstanding him). The wagers are not a percent of bankroll but are instead a constant figure each time. His mention of the Kelly criterion is relevant to questions about the effect of leverage on returns, but is relevant neither to his example games nor to your question of why volatility is used as a “proxy” for risk.
I’d say that to a large extent you are right to be suspicious of this decision to use variance as a proxy for risk. The choice to use volatility as a risk proxy was definitely a mathematical convenience that works almost all of the time, except when it absolutely doesn’t. And when it doesn’t work out, it does so ways that can negate all the time that it does work out for. The most commonly used model of a stock’s movements is Geometric Brownian Motion, which only has two parameters, µ and σ. Since σ is the sole determinant of the standard deviation of the next minute/day/month/year’s move, it is used as the “risk” parameter since it determines the magnitude distribution for how much you can expect to make/lose.
But to get to the heart of the matter (i.e. why people accept and use this model despite it’s failure to take into account “real” risk), I refer you to this stackexchange post.
I don’t think that gilch answered the question correctly. His two games A and B are both “additive” games (unless I’m misunderstanding him). The wagers are not a percent of bankroll but are instead a constant figure each time. His mention of the Kelly criterion is relevant to questions about the effect of leverage on returns, but is relevant neither to his example games nor to your question of why volatility is used as a “proxy” for risk.
I’d say that to a large extent you are right to be suspicious of this decision to use variance as a proxy for risk. The choice to use volatility as a risk proxy was definitely a mathematical convenience that works almost all of the time, except when it absolutely doesn’t. And when it doesn’t work out, it does so ways that can negate all the time that it does work out for. The most commonly used model of a stock’s movements is Geometric Brownian Motion, which only has two parameters, µ and σ. Since σ is the sole determinant of the standard deviation of the next minute/day/month/year’s move, it is used as the “risk” parameter since it determines the magnitude distribution for how much you can expect to make/lose.
But to get to the heart of the matter (i.e. why people accept and use this model despite it’s failure to take into account “real” risk), I refer you to this stackexchange post.