Prove that, under the equivalent martingale measure \(\mathbb{Q}_{n}\) of Section 2.3.4, the ratio \(f_{t} B_{t}\) is a martingale.

Data From Section 2.3.4,

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The mathematics of arbitrage relies on stochastic models for asset prices, inter- est rates, and other risk factors. In Section 2.3.4, we have seen how, in a simple binomial setting, where we assume market completeness, we may replicate any payoff by a portfolio consisting of two spanning assets. Market completeness in the binomial case means that the payoffs of the two spanning assets are lin- early independent and are a basis for the two-dimensional space R. Clearly, two states do not make an excellent model of uncertainty. If we increase the number of scenarios to m, we may represent uncertainty more accurately, but if we pursue the same approach as before, we would need a set of m spanning assets forming a basis for R. This does not seem practically sensible, if we are considering an option written on a single underlying asset. As we shall see in Chapter 13, the trick is to introduce dynamic replicating portfolios, based on cash and the underlying asset. If we allow for trading in continuous time, we may even be able to cope with a continuous random variable modeling un- certainty. We will deal with such models in Chapter 11, but developing the mathematics of arbitrage in that context requires tools from stochastic calcu- lus and functional analysis, beyond the scope of this book. Here, we restrict the analysis to static trading strategies and finite-dimensional models, which al- low us to use simpler tools from linear algebra and linear programming. This is sufficient to get the fundamental insights, without bothering too much with advanced mathematical machinery. Nevertheless, the uninterested reader may safely skip this section. We consider a single-period market, where trading occurs at dates t = 0 and t = T. The sample space ? consists of m possible states of the world (scenarios) = {wi, w2,..., wm), with probability measure p(w) > 0, Vw E 2. Hence, we deal with quite simple stochastic processes with sample paths Y(t,w), t = {0,T}.35 Since the initial state is known, when we refer to time t = 0 we may suppress dependence of the process on w to improve readability. Thus, we will write Y (0) and Y (T, w) when referring to the initial and terminal states, respectively. On this market, n + 1 securities are traded: