106 Frequently Asked Questions In Quantitative Finance
are martingales after discounting, and then define the
option price to be the discounted expectation making
it into a martingale too, we have that everything is a
martingale in the risk-neutral world. Therefore there is
no arbitrage in the real world.
Explanation 4: If we have calls with a continuous dis-
tribution of strikes from zero to infinity then we can
synthesize arbitrarily well any payoff with the same
expiration. But these calls define the risk-neutral proba-
bility density function for that expiration, and so we can
interpret the synthesized option in terms of risk-neutral
random walks. When such a static replication is possi-
ble then it is model independent, we can price complex
derivatives in terms of vanillas. (Of course, the contin-
uous distribution requirement does spoil this argument
to some extent.)
It should be noted that risk-neutral pricing only works
under assumptions of continuous hedging, zero transac-
tion costs, continuous asset paths, etc. Once we move
away from this simplifying world we may find that it
doesn’t work.
References and Further Reading
Joshi, M 2003The Concepts and Practice of Mathematical
Finance.CUP
Neftci, S 1996An Introduction to the Mathematics of Financial
Derivatives. Academic Press
