Risk-Neutral Valuation and Replicating Portfolios in Option Pricing

# What is the purpose of risk neutral valuation vs replicating portfolio approach


# What is the purpose of risk neutral valuation vs replicating portfolio approach












I'm currently writing a bachelor's thesis on GPU accelerated option pricing algorithms. As a CS major I'm not knowledgeable on the higher level math, but I have tried learning the basics of option pricing, mainly the replicating portfolio argument under no-arbitrage assumptions, and how that can be reformulated into the risk-neutral probability expectation form.

What I don't really understand is the purpose of this, or why this is preferred in e.g. binomial lattice models? Earlier, my understanding was that conceptually, one could imagine a world with only risk-neutral investors, where all assets grow at and are discounted by the risk-free rate. But after more digging, I suppose this is perhaps not really accurate, as a risk-neutral investor would still use real-world probabilities, as opposed to these fixed risk-neutral probabilities. So I have now given up on this conceptual understanding idea.

I then considered the possibility of this simply being more computationally efficient, but can't seem to convince myself. Doesn't the replicating portfolio approach have a closed form solution for ∆ and B over each time step, that would simply need recalculation using a few updated parameter values, not unlike the risk neutral approach?

Any light on the matter is appreciated!

## Answer by Achrbot (score 1)

https://quant.stackexchange.com/a/83555

Risk-neutral probabilities answer the question of "which probabilities would a risk-neutral investor be facing, to arrive at the current prices". Their usefulness lies in providing the link between a derivates payoff-function, and its price.

The replicating portfolio on the other hand, provides an explicit way to hedge derivatives and so are useful to market makers. Assuming no arbitrage, it also proves the existence of risk-neutral probabilities (through the Feynman Kac theorem). However, computing the replicating portfolio for complicated payoffs can be difficult, and depending on the model, it may not even exist.

Thus, most option-pricing models explicitly assume no arbitrage, and specify their parameters directly under the $\mathbb{Q}$ measure. These can be calibrated to vanilla derivatives, and can be linked to the physical probabilities, by estimating $\frac{d\mathbb{P}}{d\mathbb{Q}}$.