# 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.
– user59
Nov 8 '14 at 16:14
• please write your answer in LaTeX (or it will be closed). Nov 8 '14 at 16:35
• so the answer is dC/dk? Nov 8 '14 at 18:35
• Is this the correct answer? dC/dK= e^(-rT)*N(d_2 )= N(d_2 ) as r=0 Nov 8 '14 at 19:32

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? Nov 8 '14 at 23:24
• i can't remember the page but it's chapter 7 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. Nov 9 '14 at 14:59