# Binary option expression

Given r=0, σ(K)=const Binary=lim┬(ε→0)⁡〖((C(K,σ(K))-C(K+ε,σ(K+ε))))/ε〗 What is the analytical expression for the binary option value?

σ(K)=const Therefore, Binary=lim┬(ε→0)⁡〖((C(K)-C(K+ε)))/ε〗

What is the next step? Thank you

• This isn't how I would compute the value of a binary option, but your next step would appear to be taking the limit itself as epsilon goes to 0, no? You are effectively looking at the derivative of C(K) by the definition of derivative. – barrycarter Nov 8 '14 at 16:14
• please write your answer in LaTeX (or it will be closed). – emcor Nov 8 '14 at 16:35
• so the answer is dC/dk? – Quin Lai Nov 8 '14 at 18:35
• Is this the correct answer? dC/dK= e^(-rT)*N(d_2 )= N(d_2 ) as r=0 – Quin Lai Nov 8 '14 at 19:32

## 1 Answer

if you let the implied vol depend on K you get two terms the first is

$N(d_2)$

but you get a correction term which is the slope times the vega

$$\frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$

(see eg my book)

• Mark Thanks for your input. Are you referring to The concepts and practice of Mathematical Finance. Which page? – Quin Lai Nov 8 '14 at 23:24
• i can't remember the page but it's chapter 7 – Mark Joshi Nov 9 '14 at 6:19
• Mark, Thank you. I had one more, how can you show that th e price of a binary is not monotonous with volatility for OTM call. – Quin Lai Nov 9 '14 at 14:59