# What mathematical characteristics are required from the asset price process in order to stay within the RNP framework?

I'm currently doing a course in derivatives pricing and I'm having some trouble wrapping my head around the sweet spot where theory meets reality in terms of Risk Neutral Pricing.

I know that the first and second fundamental theorems of asset pricing lays the foundation for the normal risk neutral pricing argument, and that part is fine. However my lecturer said that for example if you have certain types of stochastic volatility, or the asset price follows a jump process you can no longer hedge all the risks from holding the option. But the way I understand it you need to be able to hedge all the risks from the asset price process in order to use the RNP framework.

So my question is this; when building a model for an asset price process (that you ideally want to be as realistic as possible), what specific characteristics (assumptions about sources of risk etc.) need to be in place in order to keep you within the RNP framework. Conversely, at what point does this machinery break down?

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I assume that by "this machinery breaks down" you mean when it breaks down as theory, but not as a practical tool.

I would say that the exact point where risk neutral pricing approach fails is when the payoff is no more attainable. There exist a precise mathematical characterization for attainable payoffs (see the book of Hans Föllmer, Alexander Schied, "Stochastic Finance: An Introduction in Discrete Time"). And as far as I know, there is no standard approach to valuation and hedging non-attainable payoffs (for example). Note also that there are many reasons for payoff to be non-attainable and it's hard to define them all precisely.

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This depends. I am not aware of a general risk neutral pricing framework applying to all asset classes and/or stochastic processes. In order to reach more general statements about risk neutral pricing you need to consider jumps and autocorrelation depending on the asset regarded, maybe stochastic interest rates and/or volatility. If I remember correctly, in case of (autocorrelated) fractional Brownian Motion arbitrage opportunities occur. Thus standard assumptions do not hold anymore. So you need to compare with the assumptions of "your" risk neutral pricing framework. In case of the Black-Scholes-Model simply deviation from its assumptions need to be accounted for: normally distributed returns do not exhibit jumps nor autocorrelation, constant volatility does not account for shifts or clusters of volatility. Your asset might not be liquid (think of options on small single stocks) or interest rate might not be constant - gets more important with interest rate derivatives, but applies to longer dated euqity (index) options as well.

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Thanks for this. I should have specified that what I meant with RNP-framework is the fact that you can discount whatever payoff you are able to conjure up, at the risk free rate instead of some rate that takes into account the uncertainty of this payoff. So my question really is, what must be in place in order for it to be ok to do this. – L1meta Apr 22 '12 at 12:10
If you can find a risk-free portfolio that replicates the conjured payoff, return must equal the risk-free interest rate. Else, an arbitrage opportunity exists. – Konsta Apr 22 '12 at 15:28
Aha, so does this imply that all models need to evaluated using PDE methods before one finds more practical pricing methods? – L1meta Apr 23 '12 at 15:30
Practicality is a difficult quality. I think you do not need to over-engineer a model. It should fit your underlying, payoff and purpose enough. Maybe it must be calculated in 1 ms instead of being accurate to the 7th decimal point. And no, PDE methods do not always help you. Their approximations might suffer from the "curse of dimensionality". So they might be incomputable for complicated underlying and payoff. – Konsta Apr 23 '12 at 17:53