Skip to content
All library documents

Interpreting the Complementary Binomial CDF in the CRR Model

Article Quant Q&A · Author: mtris
Read the original

Summary

The note clarifies the meaning of the function Φ in the Cox–Ross–Rubinstein binomial option model. It denotes a complementary binomial distribution function: for a random variable X, this gives the probability that X exceeds a specified value, or one minus its cumulative distribution function. This interpretation helps translate the notation in the original model into an implementation.

A second answer recommends implementing the stated formula and points to a more general result in a continuous-time arbitrage theory text. The excerpt does not reproduce the formula, define its inputs, or provide an implementation example, so it offers a notation clarification rather than a full derivation or practical coding guide.

Key ideas

  • In the CRR model, Φ denotes a complementary binomial distribution function.
  • The complementary probability is one minus the cumulative probability at the threshold.
  • The note refers readers to a broader continuous-time result but does not state it.

Tags

OptionsStatisticsDerivatives pricing
Full text
# What is Phi in Cox-Ross-Rubinstein Binomial Model?


# What is Phi in Cox-Ross-Rubinstein Binomial Model?












I have a question regarding the Cox-Ross-Rubenstein (CRR) model (Cox et al.,1979). While I do understand how the model is constructed, I am now trying to put it into code and am not sure how to translate the phi correctly or what it means in this notation. Could somebody kindly help me on that small matter? Regards,

## Answer by Kevin (score 4)

https://quant.stackexchange.com/a/70264

It is explained just above the box in the original paper on page 239. The function $\Phi$ denotes the complementary binomial distribution function. Complementary means it is $\mathbb{P}(X>x)=1-F_X(x)$.

## Answer by Peter Lind (score 3)

https://quant.stackexchange.com/a/70265

Hi Mtris, I would recommend implementing the formula above. Very easy to implement but I recommend looking up proposition 2.25 in Björk's fourth edition of Arbitrage Theory in Continuous Time. It is a more general proposition.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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