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

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

  • $\begingroup$ Mark Thanks for your input. Are you referring to The concepts and practice of Mathematical Finance. Which page? $\endgroup$
    – Riser
    Nov 8 '14 at 23:24
  • $\begingroup$ i can't remember the page but it's chapter 7 $\endgroup$
    – Mark Joshi
    Nov 9 '14 at 6:19
  • $\begingroup$ 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. $\endgroup$
    – Riser
    Nov 9 '14 at 14:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.