How to check if arbitrage is possibile in a recombining Binomial tree?

Question

Consider the recombining Binomial tree below; knowing that:

• $S_0 = 100$ is the cost of an asset at $t=0$ (now),
• $∆t$ is the distance between two time points, e.g. $∆t = 0.5 =$ six months,
• $u = e^{σ\sqrt{∆t}},\ d = 1/u,\ σ = 0.25,$
• In the market there is a risk-free asset that grows as $B_{t_{i+1}} = B_{t_i}e^{r∆t}$ with $r = 0.01,\ B_0 = 1,$

perform the following:

Check if it is possible to create arbitrage opportunities by trading on the underlying asset and the bank account. If the market is arbitrage free, deduce the probability measure $Q$.

Looking at the image below, it seems like $\Delta t=1$, since looking at the indeces of $S_0,\ S_1,\ S_2$ I guess $\Delta t=2-1=1-0=1$, is this correct?

Does "...by trading on the underlying asset and the bank account" means that I have to create a portfolio made of risky assets and risk-free assets and check if arbitrage is possible?

Anyway, from the book "Arbitrage theory in continuous time" by Bjork, I found the following ways to check if arbitrage is possibile in the binomial model:

1. the binomial model is free of arbitrage iff $d < 1 + r < u;$
2. the binomial model is free of arbitrage iff $\Pi(T;X)=X$ ($T$ time of expiry), where $\Pi(t;X)$ is the price of a claim $X$ at time $t;$
3. the market model is arbitrage free iff there exists a martingale measure $Q.$

I don't know how to use the second approach, moreover the third one needs the computation of the probability measure $Q$ (i.e. to compute the probabilities $p_u$ of an "up" and $p_d$ of a "down") which has to be done after checking if the market is arbitrage free.

So I'm trying the first approach, i.e. to check if $$e^{-\sigma\sqrt{∆t}} < 1+r < e^{\sigma\sqrt{∆t}}.$$ My first doubt here is that the middle term is a constant, while the first and third terms depend on $\Delta t$. So, should I check the inequality only one time using $\Delta t=1$, or since the model has two periods, have I also to check the inequality with $\Delta t=2$?

Using $\Delta t=1$ we get $$e^{-0.25} < 1.01 < e^{0.25} \quad\implies\quad 0.78<1.01<1.28.$$ Then the market is arbitrage free. Is this procedure correct?

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Assuming the previous approach is correct, notice that from the previous inequality it follows that $1+r$ can be written as convex combination of $u$ and $d$: $$1+r=p\cdot u+(1-p)\cdot d,\quad p\in[0,1]$$ which leds to the formula for $p$: $$p=\frac{(1+r)-d}{u-d}, \quad 1-p=\frac{u-(1+r)}{u-d}.$$ Using the exercise's input (and $\Delta t=1$) we find $p=p_u=0.46$ (prob. of an "up") and so $1-p=p_d=0.54$ (prob. of a "down"). Is this the right answer to the question "deduce the probability measure $Q$" ?

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[fig.: the recombining binomial tree]

Answer 1

1. You can't tell from the image what $\Delta t$ is. It is also not really important because we can just work with arbitrary $r$, $d$ and $u$ (not necessarily given by an equation like $u = e^{\sigma{\sqrt \Delta t}}$).
2. Yes, you have to make a portfolio containing a number of shares of the risky and risk-free assets. For example, you can look at a portfolio of $1$ risky asset and $-100$ risk-less assets and see what happens if we have $1 + r \leq d$. Using this technique, you can arrive at the conclusion that there is no arbitrage iff $d < 1+r < u$. Then you can go back to your specific example (where you have specific $d$, $r$ and $u$) and check if this is true.
3. You already seem to have a formula for the probabilities of an up- and down-movement under the risk-free measure $Q$, so yes, you just need to plug in your specific $r$, $d$ and $u$.