(a) Describe the (Cox-Ross-Rubinstein) binomial model. What are the necessary and sufficient conditions on the model parameters for it to be arbitrage-free? How is the equivalent martingale measure $\mathbb{Q}$ characterised in this case?

(b) Consider a discounted claim $H$ of the form $H=h\left(S_{0}^{1}, S_{1}^{1}, \ldots, S_{T}^{1}\right)$ for some function $h$. Show that the value process of $H$ is of the form

$$V_{t}(\omega)=v_{t}\left(S_{0}^{1}, S_{1}^{1}(\omega), \ldots, S_{t}^{1}(\omega)\right)$$

for $t \in\{0, \ldots, T\}$, where the function $v_{t}$ is given by

$$v_{t}\left(x_{0}, \ldots, x_{t}\right)=\mathbb{E}_{\mathbb{Q}}\left[h\left(x_{0}, \ldots, x_{t}, x_{t} \cdot \frac{S_{1}^{1}}{S_{0}^{1}}, \ldots, x_{t} \cdot \frac{S_{T-t}^{1}}{S_{0}^{1}}\right)\right]$$

You may use any property of conditional expectations without proof.

(c) Suppose that $H=h\left(S_{T}^{1}\right)$ only depends on the terminal value $S_{T}^{1}$ of the stock price. Derive an explicit formula for the value of $H$ at time $t \in\{0, \ldots, T\}$.

(d) Suppose that $H$ is of the form $H=h\left(S_{T}^{1}, M_{T}\right)$, where $M_{t}:=\max _{s \in\{0, \ldots, t\}} S_{s}^{1}$. Show that the value process of $H$ is of the form

$$V_{t}(\omega)=v_{t}\left(S_{t}^{1}(\omega), M_{t}(\omega)\right)$$

for $t \in\{0, \ldots, T\}$, where the function $v_{t}$ is given by

$$v_{t}(x, m)=\mathbb{E}_{\mathbb{Q}}\left[g\left(x, m, S_{0}^{1}, S_{T-t}^{1}, M_{T-t}\right)\right]$$

for a function $g$ to be determined.