# Derivative: Delta of a Down and Out Call Option with Barrier=Debt(K)

I am trying to compute the derivative of this function with respect to V0:

This is the price of a down and out call option, assuming the barrier equal to the level of debt K. In other terms, I need to compute the Delta of this DOC Option, in the case of Barrier=K (neither Barrier higher than K nor Barrier lower than K) and I cannot find this case anywhere in the literature. Furthermore, the derivative of the first two terms of the equation equals N(d1), the delta of a plain vanilla call option. Therefore, I just need the derivative of what is in the parenthesis [...] with respect to V_0.

Can someone of you help me?

Anything will be really appreciated!

• Your question is not readable. Can you please use Latex? – Gordon Apr 21 '16 at 14:05
• Can you read it now? I apologize, but I am not able to write it in Latex, can you still help me? – Livia G Apr 21 '16 at 16:06
• See Page 8 in people.maths.ox.ac.uk/howison/barriers.pdf. Your formula does not look correct to me. – Gordon Apr 21 '16 at 18:21
• what I did was taking the formula of a down and out call option in the case of the barrier below the strike K and, as I am going for the case where barrier=K, I substituted the barrier with K. papers.ssrn.com/sol3/… page 15 – Livia G Apr 22 '16 at 8:37
• Your formula is not the same as Formula (3) on Page 15 of the paper you referred to. Please double check and revise. Is the + sign before the last term a typo? – Gordon Apr 22 '16 at 12:54

Let \begin{align*} C(S, K, t) = SN(d_1) - e^{-rt}KN(d_2) \end{align*} denote the Black-Scholes call option price with initial asset value $S$, strike $K$, and maturity $t$. Note that \begin{align*} \frac{\partial C}{\partial S} = N(d_1). \end{align*}