Implication of unique risk neutral measure

Question

I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing),

This is the part that confuses me :

suppose there is only one risk-neutral measure. This means first of all that the filtration for the model is generated by the $d$-dimensional Brownian motion driving the assets. If that were not the case (i.e., if there were other sources of uncertainty in the model besides the driv­ing Brownian motions), then we could assign arbitrary probabilities to those sources of uncertainty without changing the distributions of the driving Brow­nian motions and hence without changing the distributions of the assets. This would permit us to create multiple risk-neutral measures

This seems to be a proof by contradiction of the statement: only one risk-neutral mesure $\implies$ the driv­ing Brownian motions are the only source of uncertainty.

I think the example of market model is the GBM: $$dS_i(t) = \alpha_i(t)S_i(t)dt+\sigma_i(t)S_i(t)dW_i(t) \quad i=1,...,d$$ Here the driving Brownian motion seems to refer to $W_i(t), i=1,...,d$, so I wonder what could be "other sources of uncertainty in the model besides the driv­ing Brownian motions", could they be $\alpha_i(t) $ or $\sigma_i(t) $ or some $dB_i(t) $ for some random variable B added to the model?

Then the writer says we could "assign arbitrary probabilities to those sources of uncertainty", does it refer to the risk neutral probability measure we construct? But that measure only acts on the (preimage of ) random stock price, we cannot assign probabilities to any uncertainty...

I think "to create multiple risk-neutral measures", is related to the definition of risk neutral probability measure namely (i) It must be equivalent to the original probability measure and (ii) it must make the discounted stock price process a martingale

Anyone understand what the writer is saying?

Answer 1

I think you can think of that part of Shreve's by noting that you're using $d$ stocks (and a bank account, with the instantaneous rate being an adapted process, i.e. you know at time $t$ what the rate covering the interval $[t, t+dt]$ would be) to hedge away the risks in the portfolio. In other words, you can design a perfect hedging strategy by using what you have at hand (the $d$ stocks and the bank account).

Now imagine that you have another source of uncertainty (or risk) in your model. As Shreve's notes a bit further in that text, it would imply that the market price risk equations would have a different solution and that this has modified your hedging strategy. In fact, you would have a different solution per possible change you can make in the probability measure (i.e., a different solution depending on how you assign probabilities in your new uncertainty source). This would also imply that assets have different prices depending on which RN-measure you choose, and this is incompatible with having a set of well defined price equations, as an initial (before you added the new source) RN-measure must have.

Therefore the new source of uncertainty is either redundant to the filtration and does not provide new information to the hedging process (and therefore all the RN-measure you can define from it are all equivalent), or your RN-measure was not well defined initially, as you were missing information.

A good way of thinking about a new redundant source, as in the first case of the previous paragraph could be having $d$ stocks (and the associated $d$-dimensional BM) and throwing a dice. The outcome of the dice tells you nothing about what your hedging strategy should be for your portfolio, and therefore it does not matter how you assign probabilities to the outcome of that dice, all the probability measures you define are equivalent, as the market price of risk equations remain unchanged.