The Probabilistic Sharpe Ratio

x xxxxxx xxxxx xxxxxxxx xxxx x xxxxxx xxxxxx xx xx xxxxxxxx xxxx xxxxx xxxx, xxx xxx xxxxx xxxx xxxxx xxxx xxxxxxx xxx xxxxxx xxx xxx-xxxxxx. xxx xxxxxxxxxxxxx xxxxxx xxxxx xxxxx xxxx xxxxxxxx xxxx x xxxxxx: xxx xxxxxxxxxxx xxxx xxx xxxx xxxxxx xxxxxxx x xxxxxxxxx, xxxxx xxx xxxxx xxxxxx'x xxxxxx xxx xxxxx. xxxxxxxxxx xxxxx xxxxxx xxxxxxxxxxx xxxxxxxxx xx xxxxxxx xxxx xxxxxxxxx ("xxx xxxxx xxxx xxxxxx xx xxxxx?"), xxx xxxx xx xxx xxxxxxxxxxx xxxxxx xxx xxxxxx xxxxxx.

xxxxxxxxx xxxx_xxxxxx(xx_xxx, xx_xxxx, x, xxxx, xxxx).

• xx_xxx: xxx xxxxxxxx xxx-xxxxxx xxxxxx xxxxx.
• xx_xxxx: xxx xxxxxxxxx xx xxxx.
• x: xxx xxxxxx xx xxxxxx xxxxxxxxxxxx (( xxx x )).
• xxxx: xxx xxxxxxxx xx xxx xxxxxxx.
• xxxx: xxx xxxxxxxx xx xxx xxxxxxx (xxxxxx xxxxxxx xxxx xxxx = x).

xxxxxxx xxx xxxxxxxx xxxxx xx xxx xxxxxx xxxxxxxxx (xxxxxxx):

[ xxxxx_{xxxxxxx{xx}} = xxxx{xxxx{x - xxxxx_x,xxxxxxx{xx}

• xxxx{xxxxx_x - x}{x},xxxxxxx{xx}^x}{x - x}} ]

xxx xxxxxx

[ xxx = xxx!xxxx(xxxx{xxxxxxx{xx} - xx^*}{xxxxx_{xxxxxxx{xx}}}xxxxx) ]

xxxxxxx xx x xxxxxxxx. xx xxxxx: xxxxx xxx xxxxxxxx xxxxxx xxx xxxx xxxx.xxx, xxxxx ( xxx(x) = xxxxx{x}{x}xxxx(x + xxxxxxxxxxxx{xxx}(x/xxxx{x})xxxxx) ).

xxxxxxxx

xxxx_xxxxxx(x.x, x.x, xxx, x.x, x.x)
# x.xxx

xxxx_xxxxxx(x.xx, x.xx, xxx, -x.x, x.x)
# x.x

x xxxx xx xxxxx xxxx xx x xxxxx xxxxxx xx x.x xx xx% xxxxxx xx xxxx xxxx xxxxx xxxxxx xxxxxxx. x xxxxxxxxx xxxxx xx xxx xxxxxxxx xxxxx xx x xxxx xxxx xx xxxxxxxxxxxx, xxxxxxxx xxx xxxxx xxxxxxxxxx xx.