RDP 8707: Asymmetric Information and Bid-Ask Spreads in the Eurocurrency Markets Appendix 1 Properties of the Function g(h)

Here we derive the properties of g(h) given in section 3. The function is defined by

g(h) – E(t|t > h) where the unconditional distribution of t is a standard normal.

This can be written as

The primitive function of eInline Equation is not known.

Bounds for g(h) are obtained as follows.

It is easily verified that the primitive function of


Since g'(h) is defined for all h > 0, g(h) is a continuous function for h > 0.

Bounds for g'(h)

Differentiating g'(h) we obtain

Suppose for some h > 0 (say h – t), that g'(t) ) 1.

Then from (A3), g”(h) > 0, i.e. the slope is increasing with h. Therefore g•(h) > 1 for all h > t. But this is impossible since the upper bound of g(h) has a maximum slope of one. Therefore g'(h) < 1 for all h > 0.