Math Problem Statement

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?

Solution

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=eσh,d=eσhu = e^{\sigma \sqrt{h}}, \quad d = e^{-\sigma \sqrt{h}} where:

  • σ\sigma is the volatility parameter.
  • hh is the length of a single period in the binomial tree.

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

The ratio SuSd\frac{S_u}{S_d} of the stock price after an up move to that after a down move can be calculated as follows: SuSd=S0uS0d=eσheσh=e2σh\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: SuSd=e2σh\frac{S_u}{S_d} = e^{2\sigma \sqrt{h}}


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

The risk-neutral probability pp 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 rr. The probability pp is: p=erhdudp = \frac{e^{r h} - d}{u - d} Substituting the values of uu and dd: p=erheσheσheσhp = \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 SudS_{ud} in Terms of S(0)S(0), σ\sigma, and hh

In a binomial tree, SudS_{ud} represents the stock price after one up move followed by one down move. The value can be calculated as: Sud=S(0)udS_{ud} = S(0) \cdot u \cdot d Substituting u=eσhu = e^{\sigma \sqrt{h}} and d=eσhd = e^{-\sigma \sqrt{h}}: Sud=S(0)eσheσh=S(0)eσhσh=S(0)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, Sud=S(0)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 σ\sigma, rr, and hh.

Answer: True.

The CRR model ensures that the up and down factors uu and dd are set in a way that the model is arbitrage-free under any values of σ\sigma, rr, and hh. 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 σ\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 rr 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 (σ\sigma, rr, and hh) helps in assessing how option prices might respond to market changes.

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Math Problem Analysis

Mathematical Concepts

Financial Mathematics
Binomial Option Pricing Model
Risk-Neutral Valuation

Formulas

u = e^(σ√h), d = e^(-σ√h)
Su/Sd = e^(2σ√h)
Risk-neutral probability p = (e^(r h) - d) / (u - d)
Sud = S(0)

Theorems

No-Arbitrage Condition

Suitable Grade Level

Undergraduate-Level Finance