Risk neutral pricing and arbitrage

We start by defining what we call a risk neutral probability measure. Suppose that a probability measure $Q$ which is equivalent to $P$ [that is to say they agree to the sets of zero probability] exists, such that the discounted price process

$$\left\{ \frac{S_{k}}{R_{k}}\right\} _{k=0}^{n}$$

is a $Q$-martingale. Then we call $Q$ a risk neutral [probability] measure.

It follows directly that under a risk neutral measure the discounted wealth process is always a martingale too. The fact that it does not depend on the portfolio process might seem to be counterintuitive, therefore we give its trivial proof: We have

$$\mathbf{E}_{Q}\left[ \frac{X_{k}}{R_{k}}\vert\mathscr{F}_{k-1}\ri... ...k-1}}{R_{k-1}}\right) +%% \frac{X_{k-1}}{R_{k-1}}\vert\mathscr{F}_{k-1}\right]$$

now all elements with subscripts $\left( k-1\right)$ are $\mathscr{F}_{k-1}$-measurable and therefore nonrandom conditional on $\mathscr{F}_{k-1}$, hence

$$\mathbf{E}_{Q}\left[ \frac{X_{k}}{R_{k}}\vert\mathscr{F}_{k-1}\ri... ...cr{F}_{k-1}%% \right] -\frac{S_{k-1}}{R_{k-1}}\right) +\frac{X_{k-1}}{R_{k-1}}$$

but the discounted price process is a martingale which implies

$$\mathbf{E}_{Q}\left[ \frac{S_{k}}{R_{k}}\vert\mathscr{F}_{k-1}\right] =\frac{%% S_{k-1}}{R_{k-1}}\text{,}$$

therefore the expected discounted wealth under $Q$ is

$$\mathbf{E}_{Q}\left[ \frac{X_{k}}{R_{k}}\vert\mathscr{F}_{k-1}\right] =\frac{%% X_{k-1}}{R_{k-1}}\text{,}$$

and the proof is concluded.

Now suppose that one wants to price a contract which promises a payoff $%% W_{n}=W_{n}\left( \omega _{1},\ldots ,\omega _{n}\right)$, which is $%% \mathscr{F}_{n}$-measurable. A hedging strategy would be

1. Sell the contract at time 0 and receive $X_{0}$,

2. Construct a portfolio $\mathbf{H}$ which starts with $X_{0}$ and ends with $X_{n}=W_{n}$,

3. If a risk neutral measure exists, then $X_{0}$ [the price of the contract] should satisfy
   $$X_{0}=\mathbf{E}_{Q}\left[ \frac{X_{n}}{R_{n}}\vert\mathscr{F}_{0}\right] =%% \mathbf{E}_{Q}\left[ \frac{W_{n}}{R_{n}}\vert\mathscr{F}_{0}\right]$$   .

The above analysis implies that if all steps were feasible, then the price of the contract should be equal to the discounted cashflows under the risk neutral measure. The problem is that in discrete time, such perfect hedges are generally not feasible. This is just due to the fact that there are infinite possible future prices but only a finite time of trading dates. We have therefore forget market completeness [and unique risk neutral measures] and look for a second best solution: Prices that do not allow arbitrage.

We call arbitrage a portfolio $\mathbf{H}$ which starts with $X_{0}=0$ and ends with $X_{n}$, where

$$P\left[ X_{n}\geq 0\right] =1$$ and $$P\left[ X_{n}>0\right] >0$$.

Now the fundamental theorem of asset pricing [one directional] states that: If there is a risk neutral probability measure, then there exists no arbitrage. This proof is also trivial. Suppose that a risk neutral measure $%% Q$ exists. If for all portfolios $\mathbf{H}$ which start with $X_{0}=0$ we have that $P\left[ X_{n}\geq 0\right] <1$, then there is no arbitrage. Now suppose that there exists a portfolio $\mathbf{H}$ which starts with $%% X_{0}=0$ and $P\left[ X_{n}\geq 0\right] =1$. We will use that fact that $P$ and $Q$ are equivalent in order to write

$$P\left[ X_{n}\geq 0\right] =1 \Rightarrow P\left[ X_{n}<0\right] =0 \Rightarrow Q\left[ X_{n}<0\right] =0 \Rightarrow Q\left[ X_{n}\geq 0\right] =1$$.

On the other hand we know that the discounted wealth process is a martingale, implying

$$\mathbf{E}_{Q}\left[ \frac{X_{n}}{R_{n}}\vert\mathscr{F}_{0}\right] =X_{0}=0 \Rightarrow \mathbf{E}_{Q}\left[ X_{n}\vert\mathscr{F}_{0}\right] =0$$.

The above two relationships will give that $Q\left[ X_{n}=0\right] =1$, or

$$Q\left[ X_{n}>0\right] =0 \Rightarrow P\left[ X_{n}>0\right] =0$$

and there is no arbitrage. This concludes the proof.