The problem in your provided image involves computing sigma-fields ($\mathcal{F}_{S_1}$ and $\mathcal{F}_{S_2}$) associated with the random variables $S_1$ and $S_2$ in a 2-step CRR model. Let me analyze it and solve step by step.

Sigma-field $\mathcal{F}_{S_1}$

$\mathcal{F}_{S_1}$ is generated by the random variable $S_1$, which takes values $s_0 \cdot u$ and $s_0 \cdot d$ based on the outcomes of the first coin flip ($e_1$). Hence, the partition is:

• $\{HH, HT\} \rightarrow s_0 \cdot u$
• $\{TH, TT\} \rightarrow s_0 \cdot d$

$\mathcal{F}_{S_1}$ is thus the sigma-field generated by these sets: $\mathcal{F}_{S_1} = \{\emptyset, \Omega, \{HH, HT\}, \{TH, TT\}\}.$

Sigma-field $\mathcal{F}_{S_2}$

$\mathcal{F}_{S_2}$ is generated by $S_2$, which depends on the outcomes of both coin flips ($e_1$ and $e_2$) and takes four possible values: $s_0 \cdot u^2$, $s_0 \cdot ud$, $s_0 \cdot d^2$, and $s_0$. The partitions are:

• $\{HH\} \rightarrow s_0 \cdot u^2$
• $\{HT\} \rightarrow s_0 \cdot ud$
• $\{TH\} \rightarrow s_0 \cdot ud$
• $\{TT\} \rightarrow s_0 \cdot d^2$

$\mathcal{F}_{S_2}$ is the sigma-field generated by the individual sets: $\mathcal{F}_{S_2} = \{\emptyset, \Omega, \{HH\}, \{HT\}, \{TH\}, \{TT\}, \{HH, HT\}, \{TH, TT\}, \{HH, TH\}, \{HT, TT\}, \dots\}.$

Would you like a detailed explanation of the derivations?
Here are some further questions to delve deeper:

1. How does the filtration $\mathcal{F}_t$ evolve over time in this CRR model?
2. How does the value of $u$ and $d$ affect the structure of the sigma-field?
3. Can we explicitly compute probabilities for each leaf in this tree model?
4. What happens if the coin is biased instead of fair?
5. How do these sigma-fields relate to martingales in this model?

Tip: Visualizing partitions of outcomes often helps in better understanding sigma-fields!