Cox-Ross-Rubinstein (CRR)
The Cox-Ross-Rubinstein model is a binomial tree in which the up and down factors are given as
u = eσ
√h, d = e−σ
√h,
where σ denotes the volatility parameter and h stands for the length of a single period in a tree.
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. (2 points) Express Sud in terms of S(0), σ and h in a CRR tree.
d. (5 points) As was the case with the forward tree, the no-arbitrage condition for the binomial assetpricing model is satisfed for the CRR tree regardless of the specifc values of σ, r and h. True or
false?

Let’s address each part based on the Cox-Ross-Rubinstein (CRR) model:

### Given Information
The up and down factors in the CRR model are:
\[
u = e^{\sigma \sqrt{h}}, \quad d = e^{-\sigma \sqrt{h}}
\]
where:
- \(\sigma\) is the volatility parameter.
- \(h\) is the length of a single period in the binomial tree.

---

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

The ratio \( \frac{S_u}{S_d} \) of the stock price after an up move to that after a down move can be calculated as follows:
\[
\frac{S_u}{S_d} = \frac{S_0 \cdot u}{S_0 \cdot d} = \frac{e^{\sigma \sqrt{h}}}{e^{-\sigma \sqrt{h}}} = e^{2\sigma \sqrt{h}}
\]

So, the answer is:
\[
\frac{S_u}{S_d} = e^{2\sigma \sqrt{h}}
\]

---

### Part (b): Risk-Neutral Probability of an Up Move

The risk-neutral probability \(p\) of an up move in the CRR model is derived by ensuring that the expected price under the risk-neutral measure is discounted at the risk-free rate \(r\). The probability \(p\) is:
\[
p = \frac{e^{r h} - d}{u - d}
\]
Substituting the values of \(u\) and \(d\):
\[
p = \frac{e^{r h} - e^{-\sigma \sqrt{h}}}{e^{\sigma \sqrt{h}} - e^{-\sigma \sqrt{h}}}
\]

This expression is the simplified form for the risk-neutral probability of an up move.

---

### Part (c): Express \( S_{ud} \) in Terms of \( S(0) \), \( \sigma \), and \( h \)

In a binomial tree, \( S_{ud} \) represents the stock price after one up move followed by one down move. The value can be calculated as:
\[
S_{ud} = S(0) \cdot u \cdot d
\]
Substituting \( u = e^{\sigma \sqrt{h}} \) and \( d = e^{-\sigma \sqrt{h}} \):
\[
S_{ud} = S(0) \cdot e^{\sigma \sqrt{h}} \cdot e^{-\sigma \sqrt{h}} = S(0) \cdot e^{\sigma \sqrt{h} - \sigma \sqrt{h}} = S(0)
\]

Thus, \( S_{ud} = S(0) \).

---

### Part (d): No-Arbitrage Condition for the CRR Tree

**True or False?** The no-arbitrage condition for the binomial asset-pricing model is satisfied for the CRR tree regardless of the specific values of \( \sigma \), \( r \), and \( h \).

**Answer:** True.

The CRR model ensures that the up and down factors \(u\) and \(d\) are set in a way that the model is arbitrage-free under any values of \( \sigma \), \( r \), and \( h \). This construction guarantees the existence of a risk-neutral probability and thus, the no-arbitrage condition is satisfied without needing additional constraints on these parameters.

---

Would you like further clarification on any part?

---

### Additional Questions
1. How does the CRR model differ from other binomial models in option pricing?
2. Can we generalize the CRR model to other time increments beyond one period?
3. What are the implications of changing volatility \( \sigma \) in the CRR model?
4. How is the risk-neutral probability interpreted in the context of pricing options?
5. What role does the risk-free rate \( r \) play in determining the no-arbitrage condition in the CRR model?

#### Tip
When working with binomial models like CRR, understanding the impact of each parameter (\( \sigma \), \( r \), and \( h \)) helps in assessing how option prices might respond to market changes.

Cox-Ross-Rubinstein (CRR) model with given up and down factors u = e^σ√h, d = e^−σ√h. Find: (a) the ratio Su/Sd, (b) the simplified expression for risk-neutral probability of an up move, (c) express Sud in terms of S(0), σ, and h, and (d) determine if the no-arbitrage condition is satisfied regardless of σ, r, and h.

Explore the Cox-Ross-Rubinstein (CRR) model with a focus on calculating up/down price ratios, simplified risk-neutral probability, and the no-arbitrage condition. Detailed solutions provided for understanding volatility, period length, and risk-free rates in financial mathematics.

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