The Jarrow-Rudd model (aka, the lognormal binomial tree) is a binomial tree in which the up and down factors are defned as follows u = e ! r− σ2 2 " h+σ √h , d = e ! r− σ2 2 " h−σ √h , where • r stands for the continuously-compounded, risk-free interest rate, • δ is the stock’s dividend yield, • σ denotes the volatility parameter, and • h stands for the length of a single period in a tree. Answer the following questions: a. (2 points) What is the ratio Su/Sd? b. (2 points) What is the (as simplifed as possible) expression for the risk-neutral probability of the stock price going up in a single step? c. (5 points) As was the case with the forward tree, the no-arbitrage condition for the binomial asset pricing model is satisfed for the Jarrow-Rudd tree regardless of the specifc values of σ, δ, r and h. True or false?

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,
\[
\frac{S_u}{S_d} = e^{2 \sigma \sqrt{h}}
\]

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### 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 \( h \).

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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.

The Jarrow-Rudd model (aka, the lognormal binomial tree) is a binomial tree in which the up and down factors are defined as follows u = e^( (r − σ^2/2) h + σ√h ), d = e^( (r − σ^2/2) h − σ√h ), where • r stands for the continuously-compounded, risk-free interest rate, • δ is the stock’s dividend yield, • σ denotes the volatility parameter, and • h stands for the length of a single period in a tree. Answer the following questions: a. (2 points) What is the ratio Su/Sd? b. (2 points) What is the (as simplified as possible) expression for the risk-neutral probability of the stock price going up in a single step? c. (5 points) 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 σ, δ, r and h. True or false?

Explore the Jarrow-Rudd model for binomial tree analysis, including calculations for the ratio Su/Sd and the risk-neutral probability of price movement. This resource covers up and down factors, risk-neutral valuation, and the no-arbitrage principle, ideal for undergraduate students in finance or mathematics.

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