15. (General risk-neutral pricing) We can transform the log-optimal pricing formula into a risk-neutral pricing equation From the log-optimal pricing equation we have P = E () where R is the return on the log-optimal portfolio. We can then define a new expectation operation by (x) = E Rr R+ This can be regarded as the expectation of an artificial probability Note tha: the usual rules of expectation hold. Namely:

(a) If x is certain, then (x)=x. This is because E(1/R') = 1/R.

(b) For any random variables x and y, there holds (ax + by) = a(x) + b(y)

(c) For any nonnegative random variable x, there holds E(x) 0 Using this new expectation operation, with the implied artificial probabilities, show that the price of any security dis This is risk neutral pricing

(d) R