# Greeks(theta) of a Down-and-Out barrier option

I am trying to figure out the theta for a down-and-out barrier put option. After some research of my own, I found out that a down-and-out put can be expressed as $$P_V(S_0, S_0)-P_V(S_0, H)-(S_0 - H)P_D(S_0, H) - \frac{H}{S_0}(P_V(\frac{H^2}{S_0}, S_0) - P_V(\frac{H^2}{S_0}, H) + (S_0 - H)P_D(\frac{H^2}{S_0}, H))$$ where $P_V$ is a vanilla put and $P_D$ is a digital put and $H$ is the barrier.

Using this formulation, I used the usual greek formluas for the vanilla and digital options. However, I am not getting the result that I am supposed to.

This made me suspicious about my approach, and I wanted to check whether my approach was correct. If it is incorrect, I would very much appreciate if someone could tell me how to find the greeks (theta) of a barrier options, down-and-out put in particular.

Thank you in advance.

• have your tried testing your formula against bumping? – Mark Joshi Apr 11 '17 at 23:43
• How are the above formula derived? Can you please provide more details? – Gordon Apr 12 '17 at 14:44
• @Gordon The formula above is from page 10 of people.maths.ox.ac.uk/howison/barriers.pdf. The derivation can be found in the previous pages mostly derived for barrier call options. – Byng Apr 12 '17 at 15:33
• I think all the individual premiums of the vanilla and digital puts that add up to the price for the down-and-out put price should be 0 except for $P_V(\frac{H^2}{S_0}, S_0)$ once the barrier is breached – Byng Apr 12 '17 at 15:42