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Inferring CRR Up and Down Factors from Option Prices

Article Quant Q&A · Author: Analysis
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Summary

The document asks how to recover the parameters of a single-period Cox–Ross–Rubinstein binomial model from European call and put prices at two strikes. It first applies put–call parity to the prices to infer the current share value and risk-free rate. The remaining task is to determine the up and down multipliers that define the next-period stock outcomes.

A tentative answer proposes expressing the multipliers relative to the risk-free return, using the risk-neutral probability condition, and imposing a reciprocal relationship between the up and down factors. The response is explicitly uncertain and offers no worked derivation or verification that these equations fit the quoted prices. The example therefore illustrates how parity and no-arbitrage conditions can constrain model parameters, but it does not establish a complete reconstruction. Care is needed to check all option prices against the assumed binomial model and its conventions.

Key ideas

  • Put–call parity across strikes can be used to infer the share value and risk-free rate.
  • The remaining CRR parameters are the up and down multipliers for the share price.
  • A risk-neutral probability condition relates the multipliers to the risk-free return.
  • The proposed reciprocal condition is tentative and is not verified against all quoted option prices.

Tags

OptionsDerivatives pricingArbitrageEquities
Full text
# Reconstructing the CRR model knowing put and call prices


# Reconstructing the CRR model knowing put and call prices












In an arbitrage-free single-period CRR model, the following options on a share are offered:

[They are all European]

(i) Call option at strike price $100$, price: $C_{0,1}=7.44$

(ii) Call option at strike price $110$, price: $C_{0,1}=3.72$

(iii) Put option at strike price $100$, price: $P_{0,1}=23.59$

(iv) Put option at strike price $110$, price: $P_{0,1}=29.49$

Show that given this information the model is fully specified.

Using the Put-Call parity I got $7.44-23.59=S_0-100/(1+r)$ and $3.72-29.49=S_0-110/(1+r)$ which yields $S_0\approx 80$ and $r\approx 0.04$.

How can I get $u,d$?

## Answer by Ledog (score 0)

https://quant.stackexchange.com/a/76475

I suppose by finding martingale measure. If then $u = 1+b, d = 1+a$, the probability of going to the up state is given by $p = \frac{r-a}{b-a}$. $(r \in (a,b))$. The second equation needed to solve for u and d is $ud=1$. Not 100% sure about this approach so please correct me if I'm wrong

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.