To answer these questions on the Jarrow-Rudd model, let's go through each question systematically.

Question (a)

To find the ratio $S_u / S_d$, where:

• $S_u = S_0 \cdot u$ and $S_d = S_0 \cdot d$
• $u = e^{(r - \frac{\sigma^2}{2}) h + \sigma \sqrt{h}}$
• $d = e^{(r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}}$

The ratio $S_u / S_d$ is: $\frac{S_u}{S_d} = \frac{S_0 \cdot u}{S_0 \cdot d} = \frac{u}{d}$

Now substitute the expressions for $u$ and $d$: $\frac{u}{d} = \frac{e^{(r - \frac{\sigma^2}{2}) h + \sigma \sqrt{h}}}{e^{(r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}}}$

Since both terms share the base $e$, we can combine them as: $\frac{u}{d} = e^{\left((r - \frac{\sigma^2}{2}) h + \sigma \sqrt{h}\right) - \left((r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}\right)}$

Simplifying, we get: $\frac{u}{d} = e^{2 \sigma \sqrt{h}}$

Thus, $\frac{S_u}{S_d} = e^{2 \sigma \sqrt{h}}$

---

Question (b)

The risk-neutral probability $p$ of the stock price going up in a single step in the binomial model is given by the formula: $p = \frac{e^{(r - \delta) h} - d}{u - d}$

Substituting $u = e^{(r - \frac{\sigma^2}{2}) h + \sigma \sqrt{h}}$ and $d = e^{(r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}}$, we proceed to simplify the expression.

Since $u - d = e^{(r - \frac{\sigma^2}{2}) h + \sigma \sqrt{h}} - e^{(r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}}$, it can be simplified as: $u - d = e^{(r - \frac{\sigma^2}{2}) h} \left( e^{\sigma \sqrt{h}} - e^{-\sigma \sqrt{h}} \right)$

Using the identity $e^{\sigma \sqrt{h}} - e^{-\sigma \sqrt{h}} = 2 \sinh(\sigma \sqrt{h})$, we get: $u - d = e^{(r - \frac{\sigma^2}{2}) h} \cdot 2 \sinh(\sigma \sqrt{h})$

Then, $p = \frac{e^{(r - \delta) h} - e^{(r - \frac{\sigma^2}{2}) h - \sigma \sqrt{h}}}{e^{(r - \frac{\sigma^2}{2}) h} \cdot 2 \sinh(\sigma \sqrt{h})}$

After simplifying, we find that $p$ depends on $r, \delta, h, \sigma$, and remains in terms of $e$-based expressions.

---

Question (c)

True or False: "As was the case with the forward tree, the no-arbitrage condition for the binomial asset pricing model is satisfied for the Jarrow-Rudd tree regardless of the specific values of $\sigma$, $\delta$, $r$, and $h$."

Answer: True.

In the Jarrow-Rudd model, the no-arbitrage condition is inherently satisfied by the construction of $u$ and $d$, ensuring that the stock price will align with the risk-neutral valuation framework, regardless of the values of $\sigma$, $\delta$, $r$, and $h$.

---

Would you like more details on any part of these explanations or need further clarification? Here are some related questions that might also interest you:

1. How does the Jarrow-Rudd model differ from the Cox-Ross-Rubinstein (CRR) model in terms of up and down factors?
2. Why is it essential for the no-arbitrage condition to hold in binomial tree models?
3. What role does the dividend yield $\delta$ play in adjusting the risk-neutral probability?
4. Can we apply the Jarrow-Rudd model to options with dividends, and if so, how?
5. How does the period length $h$ influence the accuracy of the model in approximating continuous stock price movements?

Tip: When working with binomial trees, always check if the model requires adjustments for dividends to maintain accurate risk-neutral probabilities.