Assumptions Behind Discounted Risk-Neutral Pricing

# Assumptions in using risk-neutral pricing formula


# Assumptions in using risk-neutral pricing formula












The well-known risk-neutral pricing formula goes as follows (extracted from Shreve's Volume 2, section $5.2.4$ (Pricing Under the Risk-Neutral Measure)):

Given any $T>0$ and any $t\in[0,T],$ if $V(T)$ denotes the payoff of a derivative security at time $T$ which is $\mathscr{F}(T)$-measurable. Let $R(t)$ be the interest rate process. Then Steven Shreve indicates that $$V(t)=\tilde{\mathbb{E}}[e^{-\int_t^TR(s)ds}V(T)|\mathscr{F}(t)], \quad 0\leq t\leq T.$$

In the proof, it seems that he does not use any assumption at all. But from this post, it seems that we need to remove arbitrage opportunity for the formula above to hold.

> Question: What assumptions do we need to fulfill when applying the risk-neutral pricing formula above?

## Answer by Kevin (score 4)

https://quant.stackexchange.com/a/50094

Shreve is a bit naughty here but, of course, he is right. When you have the risk-neutral measure $\mathbb{Q}$ or $\tilde{\mathbb{P}}$, you can price derivatives as discounted expectation by the very definition of the risk-neutral measure (better called: equivalent martingale measure). So indeed, once you have $\tilde{\mathbb{P}}$, you can price derivatives without any further assumptions.

But Shreve has not (yet) told you how do you find a risk-neutral measure? When does such a measure exist? The first Fundamental Theoream of Asset Pricing states that the absence of arbitrage possibilities is equivalent to the existence of (at least) one risk-neutral measure. The second FTAP states that a complete market can have at most one risk-neutral measure.

Thus, the main assumption is merely the absence of arbitrage (strategies with zero initial cost, positive probability of a positive payoff and zero probability of loosing money).

Note that you do not need further assumptions such as continuous trading (it applies equally to discrete models (Cox-Ross-Rubinstein, see Shreve's first book) and to time continuous models. The risk-neutral pricing formula also applies to jumps models which have discontinuous sample paths (see jump-diffusions like Merton and Kou or more advanced models like Variance Gamma, Normal Inverse Gaussian, CGMY etc.). In general, you don't need any assumptions on the dynamics of the stock price or the interest rate.

## Answer by siou0107 (score 3)

https://quant.stackexchange.com/a/50089

For Risk-neutral Pricing to “work”, you need assumptions where risk elimination by trading financial instruments is possible : no counterparty risk, no transaction costs, continuous trading, continuous asset paths.

If such assumptions are not fulfilled (which is the case in real markets ; however, for large banks they are sufficiently near from reality), risk-neutral pricing cannot work.

A great reference on this topic is Joshi's The Concepts and Practice of Mathematical Finance. And, of course, the books of Paul Wilmott, who is always very keen to explain where finance theory breaks down ;)