Equity Basket
Valuation (Cont)  It is well known that the sum of a series of lognormal random variables is not a lognormal random variable. The weighted summation R is approximated by a shifted lognormal random variable (SLN). đ?‘?−đ?‘Ž đ?‘…~đ?‘†đ??żđ?‘ = đ?‘? + đ?‘‘ ∙ đ?‘’đ?‘Ľđ?‘? đ?‘? where đ?‘?~đ?‘ (0,1) follows a standard normal distribution.
 We solve for a, b, c, d by matching central moments between.  The central moments of SLN are 1 2đ?‘?2 1 2đ?‘Ž − đ?‘?2 đ?‘?
đ?‘€1 = đ?‘? + đ?‘‘ ∙ đ?‘’đ?‘Ľđ?‘? đ?‘€2 = đ?‘‘2 đ?‘’đ?‘Ľđ?‘? đ?‘€3 =
đ?‘? 3 đ?‘’đ?‘Ľđ?‘?
3 2đ?‘?2
−
3đ?‘Ž đ?‘?
−
đ?‘Ž đ?‘?
đ?‘’đ?‘Ľđ?‘? đ?‘’đ?‘Ľđ?‘?
1 đ?‘?2 1 đ?‘?2
−1 −1
2
đ?‘’đ?‘Ľđ?‘?
1 đ?‘?2
+2