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I am trying to understand a paper on options titled "OPTION PRICING: A SIMPLIFIED APPROACH" [1]. In it option price is calculated as the expected payoff from the possible states of stock prices by binomial distribution approach.
I am stuck at one step.
What does the following sentences exactly mean?

Now, the latter bracketed expression is the complementary binomial distribution function $\Phi[a; n,p]$. The first bracketed expression can also be interpreted as a complementary binomial distribution function $\Phi[a; n, p’]$.

Thank you in advance.

Bibliography

[1] John C. COX and Stephen A. ROSS and Mark RUBINSTEIN "Option Pricing: A Simplified Approach", Journal of Financial Economics 7 (1979) 229-263. DOI: 10.1016/0304-405X(79)90015-1

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    $\begingroup$ Please take the effort to copy the "bracketed expressions" in your question. $\endgroup$
    – Federico Poloni
    Commented Dec 14, 2019 at 10:12
  • $\begingroup$ Hi, the format of expression is not getting pasted properly. I tried attaching image of it also, but it says I don't have sufficient reputation points. Can you please suggest an alternative. $\endgroup$
    – Chetan
    Commented Dec 14, 2019 at 12:31
  • $\begingroup$ Thanks Daniele for the edits. $\endgroup$
    – Chetan
    Commented Dec 14, 2019 at 19:18

1 Answer 1

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The complementary binomial distribution is defined by $$\Phi(a;n,p)=\sum _{j=a}^n \binom{n}{j} p^j(1-p)^{n-j},\;\;0<p<1,\;\;0\leq a\leq n.$$ see The derivation of diffusion-jump modes for power plant projects under risk. A more correct terminology would be to call $\Phi$ the complement of the binomial cumulative distribution function, which is how it is called in MatLab.

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  • $\begingroup$ Hi, thanks for the answer. I will checked it, but it contains same step like the paper I mentioned and I am trying to understand that only. What exactly is complementary binomial distribution function and how this is derived: $\endgroup$
    – Chetan
    Commented Dec 14, 2019 at 12:32
  • $\begingroup$ p′ = (u/r)p and 1 – p′ = (d/r)(1 – p) $\endgroup$
    – Chetan
    Commented Dec 14, 2019 at 12:38
  • $\begingroup$ 1) $\Phi(a;n,p)$ is the probability that the binomial random variable takes a value in the range $a,a+1,\ldots n$. $\endgroup$
    – Carlo Beenakker
    Commented Dec 14, 2019 at 13:15
  • $\begingroup$ 2) the cited paper contains two instances of $\Phi$, one instance with probability $p$ (which is the formula I wrote down), and one instance with a different probability $p'=(u/r)p$; then $1-p'=(d/r)(1-p)$ for $d=(r-pu)/(1-p)$. $\endgroup$
    – Carlo Beenakker
    Commented Dec 14, 2019 at 13:17
  • $\begingroup$ Hi, "Φ(a;n,p) is the probability that the binomial random variable takes a value in the range a,a+1,…n", it means it's complementary probability distribution function as per the MatLab site you posted right? $\endgroup$
    – Chetan
    Commented Dec 14, 2019 at 13:24

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