The Binomial No-Arbitrage Pricing Model
Exercise 1.1 Assume in the one-period binomial market of Section 1.1 (The stock price $S$ has two possible outcomes in the next period: it can either go up to $uS_0$ or down to $dS_0$.) that both H and T have positive probability of occurring. Show that condition $0<d<1+r<u$ precludes arbitrage. In other words, show that if $X_0 = 0$ and $$X_1=\Delta_0 S_1 + (1+r)(X_0 - \Delta_0 S_0),$$ then we cannot have $X_1$ strictly positive with positive probability unless $X_1$ is strictly negative with positive probability as well, and this is the case regardless of the choice of the number $\Delta_0$....