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: where:
- is the volatility parameter.
- is the length of a single period in the binomial tree.
Part (a): Ratio
The ratio of the stock price after an up move to that after a down move can be calculated as follows:
So, the answer is:
Part (b): Risk-Neutral Probability of an Up Move
The risk-neutral probability 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 . The probability is: Substituting the values of and :
This expression is the simplified form for the risk-neutral probability of an up move.
Part (c): Express in Terms of , , and
In a binomial tree, represents the stock price after one up move followed by one down move. The value can be calculated as: Substituting and :
Thus, .
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 , , and .
Answer: True.
The CRR model ensures that the up and down factors and are set in a way that the model is arbitrage-free under any values of , , and . 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
- How does the CRR model differ from other binomial models in option pricing?
- Can we generalize the CRR model to other time increments beyond one period?
- What are the implications of changing volatility in the CRR model?
- How is the risk-neutral probability interpreted in the context of pricing options?
- What role does the risk-free rate 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 (, , and ) 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
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