A certain underlying state graph is a tree where each node has three successor nodes, indexed $a$, $b$, $c$. There are two assets defined on this tree which pay no dividends except at the terminal time $T$. At a certain period it is known that the prices of the two accets are multiplied by factors, depending on the successor node. These factors are shown in the table below:
 | | a | b | c
security | 1 | 1.2 | 1.0 | 0.8
 | 2 | 1.2 | 1.3 | 1.4

 Is there a short-tem riskless asset for this period? Answer True or False.
user
Sure, based on the given information, there doesn't seem to be a short-term riskless asset in this scenario.

In financial mathematics, a short-term riskless asset is typically a cash instrument or asset that has zero market-risk in the short-term. In this case, the security prices given are the only assets available, and they are not riskless, as their prices depend on the underlying states of the graph, i.e., the node they reach.

Furthermore, cash instruments such as deposits or government bills are not mentioned in the given scenario, so they are not a part of your options. It's only the security returns available.

Another criterion for measuring short-term riskless asset may be on the presence of a replicating portfolio in a European option context. In this case, there is no option to create a replicating portfolio or a hedging strategy, as it requires an underlying security -one of three- to utilize.

So, examining on the present case, I'm afraid you can't identify a Short-Term-riskless asset instrument from this arrangement.
assistant
A certain underlying state graph is a tree where each node has three successor nodes, indexed $a$, $b$, $c$. There are two assets defined on this tree which pay no dividends except at the terminal time $T$. At a certain period it is known that the prices of the two accets are multiplied by factors, depending on the successor node. These factors are shown in the table below:
 | | a | b | c
security | 1 | 1.2 | 1.0 | 0.8
 | 2 | 1.2 | 1.3 | 1.4

 Is there a short-tem riskless asset for this period? Answer True or False.
user
True.

A riskless asset is one whose return is known and constant. In this case, if you have a portfolio consisting of both security 1 and 2, you can create a riskless asset for the next period.

Let's denote the amount of security 1 as x and the amount of security 2 as y. To create a riskless asset, we want the portfolio to have the same return regardless of whether we move to node a, b, or c.

So, we want to solve the following equations:

1.2x + 1.2y = r (for node a)
1.0x + 1.3y = r (for node b)
0.8x + 1.4y = r (for node c)

where r is the return of the riskless asset.

We can solve these equations to find the values of x and y that create a riskless asset. The fact that we can find a solution indicates that it is possible to create a riskless asset for this period.
assistant