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Instructions:
Data analytics is the process of examining large and complex data sets to extract meaningful insights that can inform business decisions. Data analytics involves using various techniques such as statistical analysis, machine learning, and data visualization to analyze data and identify patterns, correlations, and trends. The insights obtained from data analytics can help businesses make more informed decisions, identify opportunities for growth, and optimize operations.
Here are some ways that data analytics can be used for business decision making:
- Market research: Data analytics can be used to gather and analyze data on consumer behavior, preferences, and trends. This information can be used to develop marketing strategies and product offerings that better align with customer needs and desires.
- Performance tracking: By analyzing key performance indicators (KPIs), such as sales revenue, customer satisfaction, and employee productivity, businesses can identify areas of strength and weakness in their operations. This information can be used to make data-driven decisions that optimize performance and increase profitability.
- Risk management: Data analytics can be used to identify potential risks and vulnerabilities in a business’s operations, such as fraud, security breaches, or supply chain disruptions. By monitoring and analyzing data in real-time, businesses can detect potential issues early on and take proactive steps to mitigate risk.
- Financial analysis: Data analytics can be used to analyze financial data, such as cash flow, revenue, and expenses, to identify trends and patterns that can inform financial decision making. This information can be used to optimize cash flow, reduce expenses, and improve profitability.
- Resource allocation: By analyzing data on resource usage and performance, businesses can make informed decisions about how to allocate resources such as time, money, and personnel. This can help optimize operations, reduce waste, and increase efficiency.
Overall, data analytics is a powerful tool that can help businesses make better decisions and stay ahead of the competition. By leveraging data analytics, businesses can gain valuable insights that inform strategy, optimize operations, and drive growth.
Price a lookback option with payoff at $t=3$ equal to $\left(\max _{0 \leq t \leq 3} S_t\right)-S_3$ using risk-neutral probability.
The following binomial tree describes the evolution of stock price and the
bold-face numbers next to the final stock price are the payoffs from the lookback option:
Recalling from part (a) that the risk-neutral probabilities of an up-movement and a down-movement are both $\frac{1}{2}$, all terminal nodes have the same Q-probability of $\left(\frac{1}{2}\right)^3=$ $\frac{1}{8}$. Therefore the price of the lookback option is given by
$$
\begin{aligned}
C & =\left(\frac{1}{1+r}\right)^3 E^Q\left[D_3\right] \
& =\left(\frac{4}{5}\right)^3 \cdot \frac{1}{8}\left(8+6+2+2+\frac{7}{2}\right) \
& =\frac{64}{125} \cdot \frac{1}{8} \cdot \frac{43}{2} \
& =\frac{172}{125}
\end{aligned}
$$
Show that, under the risk-neutral measure, the discounted gain process
$$
\hat{G}t=\frac{P_t}{B_t}+\sum{s=1}^t \frac{D_s}{B_s}
$$
is a martingale (i.e. $E_t^Q\left[\hat{G}{t+1}\right]=\hat{G}_t$ ) from the definition of risk-neutral measure in lecture notes $$ P_t=E_t^Q\left[\sum{u=t+1}^T \frac{B_t}{B_u} D_u\right]
$$
That is the reason why the risk-neutral measure is also called the “equivalent martingale measure” (EMM).
We want to show $E_t^Q\left[\hat{G}_{t+1}\right]=\hat{G}_t$.
$$
E_t^Q\left[\hat{G}{t+1}\right]=E_t^Q\left[\frac{P{t+1}}{B_{t+1}}+\sum_{s=1}^{t+1} \frac{D_s}{B_s}\right]
$$
Now recall that
$$
P_{t+1}=E_{t+1}^Q\left[\sum_{u=t+2}^T \frac{B_{t+1}}{B_u} D_u\right]
$$
Substituting this into the first expression, we have
$$
\begin{aligned}
E_t^Q\left[\hat{G}{t+1}\right] & =E_t^Q\left[\frac{P{t+1}}{B_{t+1}}+\sum_{s=1}^{t+1} \frac{D_s}{B_s}\right] \
& =E_t^Q\left[\frac{1}{B_{t+1}} E_{t+1}^Q\left[\sum_{u=t+2}^T \frac{B_{t+1}}{B_u} D_u\right]+\sum_{s=1}^{t+1} \frac{D_s}{B_s}\right] \
& =E_t^Q\left[E_{t+1}^Q\left[\sum_{u=t+2}^T \frac{D_u}{B_u}\right]+\sum_{s=1}^{t+1} \frac{D_s}{B_s}\right] \
& =E_t^Q\left[\sum_{s=1}^T \frac{D_s}{B_s}\right] \
& =E_t^Q\left[\sum_{u=t+1}^T \frac{D_u}{B_u}\right]+\sum_{s=1}^t \frac{D_s}{B_s} \
& =\frac{1}{B_t} E_t^Q\left[\sum_{u=t+1}^T \frac{B_t}{B_u} D_u\right]+\sum_{s=1}^t \frac{D_s}{B_s} \
& =\frac{P_t}{B_t}+\sum_{s=1}^t \frac{D_s}{B_s} \
& =\hat{G}_t
\end{aligned}
$$
This shows that $\hat{G}_t$ is a martingale under the risk-neutral measure.
Suppose that uncertainty in the model is described by two independent Brownian motions, $Z_{1, t}$ and $Z_{2, t}$. Assume that there exists one risky asset, paying no dividends, following the process
$$
\frac{d S_t}{S_t}=\mu\left(X_t\right) d t+\sigma d Z_{1, t}
$$
where
$$
d X_t=-\theta X_t d t+d Z_{2, t}
$$
The risk-free interest rate is constant at $r$.
What is the price of risk of the Brownian motion $Z_{1, t}$ ?
Let the price of risk for $Z_{1, t}$ and $Z_{2, t}$ be $\eta_t=\left[\begin{array}{cc}\eta_{1, t} & \eta_{2, t}\end{array}\right]^T$. Then we must have
$$
\mu\left(X_t\right) S_t-\left[\begin{array}{ll}
\sigma S_t & 0
\end{array}\right] \eta=r S_t
$$
This gives
$$
\mu\left(X_t\right) S_t-\sigma \eta_{1, t} S_t=r S_t
$$
whereas there is no constraint for $\eta_{2, t}$. Hence, the price of risk of $Z_{1, t}$ is
$$
\eta_{1, t}=\frac{\mu\left(X_t\right) S_t-r S_t}{\sigma S_t}=\frac{\mu\left(X_t\right)-r}{\sigma}
$$
