Rational for Continuous Risk-Neutral Pricing

[quotes]

Hi everyone,

I have a serious doubt:

What is the rational for risk-neutral pricing in continuous Geometric Brownian Motion?

I know the techniques -Girsanov transformation and discounting expectation

But I don't know why

Why this can price an option?

Why this can price an European, American, Barrier etc?

Any walkthrough? Thank you guys.

 

[DStahl]

I could be wrong, but I think the risk neutral measure is the only measure that makes both the discounted asset and the discounted replicating portfolio agree on expectation. Therefore the expectation of the discounted asset is the correct price since it agrees with the replicating portfolio at every point in time. That is, ( \mathbb{E}[D_t S_t |\mathcal{F}_s]=\mathbb{E} [D_t X_t |\mathcal{F}_s] ) only if the expectation is given under the risk neutral measure, where D is the discounting factor, S is the asset, and X is the replicating portfolio.

 

[Siddharth Singh]

I think under the risk neutral measure, the rate of return of replicated portfolio becomes interest rate which is deterministic and the system is not stochastic anymore.

 

[DStahl]

The risk neutral measure removes "drift" when discounted, it does NOT remove the underlying stochastic motion (it alters it, but under the risk neutral measure the "stochastic" part is still Brownian motion). If it was not stochastic the expectation would be meaningless.

 

[quotes]

Thank you DStahl. I checked some textbooks - Feyman Kac Theorem guarantees martingale pricing would work for any European Options whose boundary condtion is at maturity. However, the applicability of the martingale approach to path-dependent options remain plausible.