Risk-Neutral Probabilities, Variance, and Completeness in a Three-State Model

# Understanding completeness in this simple one-period exercise


# Understanding completeness in this simple one-period exercise












Let's consider a one period model (t=0, 1) with one risk-free asset that yields r, and one risky asset. $S_t^j$ will be the value of the asset j=0,1 at time t=0,1, where j=0 is the risk-free asset and j=1 the risky asset. I will use a probability space with three states at t=1, being $\Omega = \{w^{-}, w, w^{+}\}$, all of them with strictly positive probability.

Now, let's consider exercise 1.6 of Nicolas Privault's Notes on Stochastic finance, chapter 1:

In this exercise, $w^{-}$ would correspond to a return of -b, $w$ a return of 0, and $w^{+}$ a return of +b. Solving a), we get that all possible risk-neutral probability measures are defined by $p^* = q^* = \frac{1-\theta^*}{2}$. Specifically, this means that there is at least one risk-neutral measure, so by The First Fundamental Theorem of Asset Pricing this market is without arbitrage. Also, by the second fundamental theorem, it is not a complete market since there are multiple risk-neutral measures.

Also, in part b), it puts a condition on the variance such that it can be shown that there is a unique risk-neutral measure. By the second fundamental theorem, this implies that the market is complete.

However, let's study this fact by elementary calculations:

Let's consider a contingent claim C, and let $C(w^{-})$, $C(w)$ and $C(w^{+})$ the payoff of the contingent claim in those states. Let's call $a^{-}:=S_1^1(w^{-})$, $a:=S_1^1(w)$ , $a^{+}=S_1^1(w^{+})$ the values of the risky asset at each final state in t=1. To build a portfolio ($\xi_0$, $\xi_1$) that replicates this contingent claim, we have to solve the system of equations:

$$ (1+r)\xi_0 + \xi_1a^{-} = C(w^{-}) $$ $$ (1+r)\xi_0 + \xi_1a = C(w) $$ $$ (1+r)\xi_0 + \xi_1a^{+} = C(w^{+}) $$

For the market to be complete, this system has to have solutions for all contingent claims C. Now, this system of equations does not depend on any probability of the states, nor on the variance of the risky asset's returns. How is it possible that, if we impose the variance of the risky asset's returns, now the system has solutions for all contingent claims? What am I doing wrong?

## Answer by MrLCh (score 0, accepted)

https://quant.stackexchange.com/a/77679

I don't think you need the contingent claim calculation but the problem can be solved a little bit easier. (For the future, don't use b twice when meaning different things :)).

I think you can solve it like this:

So as you know by a) $\mathbb{E}^*(R_1)=0$. This means that $Var^*(R_1)=\mathbb{E}^*(R_1^2)$ (Assuming the star means with regards to the risk-neutral $P^*$ similar to the notation of expected value). So:

$$Var^*(R_1)=(-b)^2q^* + 0^2 \theta^* + b^2p^* = 2b^2 {p^*}^2 (= 2b^2 {q^*}^2)$$.

For a given $\sigma$ and $b$ you can solve $p^*$ (or $q^*$)(which given a) yields all other probabilities).

$$p^* = \frac{\sigma^2}{2b^2}$$

Because $\sigma^2 > 0, b>0$ this has a unique solution and defines one probability measure.