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−δ− σ2 2 ) h+σ √ h , d = e ( r−δ− 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 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 assetpricing model is

Let's address each question step by step.

### Question a: Ratio \( \frac{S_u}{S_d} \)

The ratio \( \frac{S_u}{S_d} \) represents the ratio of the stock price in the "up" state to the stock price in the "down" state in a single period. We can express this as:
\[
\frac{S_u}{S_d} = \frac{S \cdot u}{S \cdot d} = \frac{u}{d}
\]
where \( u \) and \( d \) are the up and down factors, respectively, given by:
\[
u = e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h + \sigma \sqrt{h}}
\]
\[
d = e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h - \sigma \sqrt{h}}
\]
Thus:
\[
\frac{S_u}{S_d} = \frac{e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h + \sigma \sqrt{h}}}{e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h - \sigma \sqrt{h}}}
\]
Using the properties of exponents, we can simplify this to:
\[
\frac{S_u}{S_d} = e^{\left( \left(r - \delta - \frac{\sigma^2}{2}\right) h + \sigma \sqrt{h}\right) - \left(\left(r - \delta - \frac{\sigma^2}{2}\right) h - \sigma \sqrt{h}\right)}
\]
\[
= e^{2 \sigma \sqrt{h}}
\]
Therefore, the ratio \( \frac{S_u}{S_d} = e^{2 \sigma \sqrt{h}} \).

### Question b: Risk-neutral probability of the stock price going up

The risk-neutral probability, \( p \), is determined by ensuring that the expected value of the stock under the risk-neutral measure grows at the risk-free rate \( r \). This gives:
\[
p \cdot S_u + (1 - p) \cdot S_d = S \cdot e^{r h}
\]
Dividing through by \( S \), we have:
\[
p \cdot u + (1 - p) \cdot d = e^{r h}
\]
Solving for \( p \):
\[
p = \frac{e^{r h} - d}{u - d}
\]
Substitute \( u \) and \( d \) from above:
\[
p = \frac{e^{r h} - e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h - \sigma \sqrt{h}}}{e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h + \sigma \sqrt{h} - e^{\left(r - \delta - \frac{\sigma^2}{2}\right) h - \sigma \sqrt{h}}}
\]
This is the expression for the risk-neutral probability \( p \), as simplified as possible given the provided model.

### Question c: No-arbitrage condition for the binomial asset-pricing model

The no-arbitrage condition in the binomial asset-pricing model requires that the risk-neutral probabilities \( p \) and \( 1 - p \) fall within the interval \( [0,1] \). In other words, we require:
\[
0 \leq p \leq 1
\]
This condition

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−δ− σ^2 / 2 ) h+σ √ h , d = e ( r−δ− 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...

Explore the Jarrow-Rudd model in financial mathematics, focusing on calculating the Su/Sd ratio and deriving the risk-neutral probability for a binomial tree model. This guide covers critical financial concepts, including up and down factors, risk-neutral valuation, and the no-arbitrage condition for asset pricing.

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