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?

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.

• Hi. Thank you for the answer. But this really does not answer my question, as I am not asking about how to solve the problem. I am asking about how the way of computing it with contingent claims gives a result that does not depend on the variance of the assets returns Dec 11, 2023 at 13:23
• @ConfusedQuant Okay, sorry for that misunderstanding. I had another look at your proposed system of equations. Aren't you missing another equation that relates the final payoff of $C$ to the current price of the contingent claim or am I now the confused quant? Dec 11, 2023 at 16:19
• I think I am not missing it, because I only want to find the portfolio that hedges the contingent claim. The current price of the contingent claim has to already match the current price of the hedging portfolio, because there is no arbitrage (a risk-neutral measure exists in this context). Dec 11, 2023 at 16:24
• I thought about the question again. I don't think you are able to replicate every (non-linear) payoff, in a three-state model with only two assets right? Dec 11, 2023 at 18:08
• Yes. That is correct. It is exemplifyied by the 3 equations that I included: 3 equations with two variables does not generally have a solution. However, if the variance is fixed (as in part b of the exercise), it should have always a replicating portfolio, by the uniqueness of the risk-neutral measure. Which is very strange, since the equations do not depend on the variance of the risky asset. Dec 11, 2023 at 18:17