  We develop two alternate approaches to arbitrage-free, market-complete,
option pricing. The first approach requires no riskless asset. We develop the
general framework for this approach and illustrate it with two specific
examples. The second approach does use a riskless asset. However, by ensuring
equality between real-world and risk-neutral price-change probabilities, the
second approach enables the computation of risk-neutral option prices utilizing
expectations under the natural world probability P. This produces the same
option prices as the classical approach in which prices are computed under the
risk neutral measure Q. The second approach and the two specific examples of
the first approach require the introduction of new, marketable asset types,
specifically perpetual derivatives of a stock, and a stock whose cumulative
return (rather than price) is deflated.
