Explain various explanations regarding risk-neutral pricing

Question

Explain various explanations regarding risk-neutral pricing.

Answer 1

Here are several further explanations of risk-neutral pricing.

Explanation 1: When you hedge correctly within a Black–Scholes world then all risk is removed. If there is no risk after that we must not expect any compensation for risk. We can therefore work in a measure wherein everything grows at the risk-free interest rate.

Explanation 2: When the model for the asset is dS = µSdt + σSdX then the µs cancel within the derivation of such Black–Scholes equation.

Explanation 3: Two measures are equal if they have similar sets of zero probability. Since zero probability sets don’t change, a portfolio is an arbitrage in one measure if and only if this is one under all equivalent measures. Thus a price is non-arbitrageable within the real world if and only if this is non- arbitrageable in the risk-neutral world. There in risk-neutral price is always non-arbitrageable. When everything has a discounted asset price process that is a martingale then there can be no arbitrage. Therefore if we change to a measure in that all the fundamental assets, for illustration the stock and bond, are martingales after discounting, and then explain the option price to be the discounted expectation making this in a martingale also, we have that everything is a martingale within the risk-neutral world. Hence there is no arbitrage in the real world.

Explanation 4: We can synthesize arbitrarily suitably any payoff with similar expiration, if we have calls with a continuous distribution of strikes from zero to infinity. But these calls explain the risk-neutral probability density function for such expiration, and therefore we can interpret the synthesized option in terms of risk-neutral random walks. If such a static replication is possible then this is model independent, we can price complex derivatives in terms of vanillas. (Obviously, the continuous distribution requirement does spoil it argument to some extent.) This must be noted that risk-neutral pricing only works under assumptions of zero transaction costs, continuous hedging and continuous asset paths. Once we move away from such simplifying world we may get that this doesn’t work.