In reference to this thread a couple of people have asked how to properly calculate the variance on ternary outcomes events, specifically those that may end only in a win, lose, or push (so this wouldn't apply to Asian handicap).
Without further ado:
Without further ado:
Let pw = win probability
Let pl = loss probability
Let pd = 1 - pw - pl = draw probability
Let f = fractional odds
Let x = bet size
Let E = % edge
E = pw * f - pl
σ2/x2 = (1 - pd + E) * (f - E) - pd * E
You'll note that for a push probability of zero (i.e., for pd = 0) this reduces to the familiar formulation of variance:
σ2/x2 {no push} = (1 + E) * (f - E)
Let pl = loss probability
Let pd = 1 - pw - pl = draw probability
Let f = fractional odds
Let x = bet size
Let E = % edge
E = pw * f - pl
σ2/x2 = (1 - pd + E) * (f - E) - pd * E
You'll note that for a push probability of zero (i.e., for pd = 0) this reduces to the familiar formulation of variance:
σ2/x2 {no push} = (1 + E) * (f - E)