# How to use reflection principle to solve the analytic solution of double barrier-out-call

We consider up/down-out-call whose payment $$V(T,S_T) = \Psi(S_T)\mathbb{II}(S_T),\ V(t,B) = 0.$$ Here the range constraint function is indictor function such that $\mathbb{II}(S_T)$ = \begin{cases} \mathbb{II}_{\{S_T < B\}} & \textrm{up-out-call}\\ \mathbb{II}_{\{S_T > B\}} & \textrm{down-out-call}\\ \end{cases} We see that the only difference between the barrier option and vanilla option is the barrier condition: $V(t,B) = 0,$ so we can suppose $$V(t,S) = \widehat V(t,S) - \widetilde V(t,S)$$ s.t $$\widehat V(T,S) = \Psi(S_T),\quad \widetilde V(T,S) = 0$$ $$\widehat V(t,B) = \widetilde V(t,B)$$ And use the reflection principle $$\widetilde V(t,S) = \left(\dfrac{S}{B}\right)^{2\alpha}\widehat V(t,\dfrac{B^2}{S})$$ Since if $\mathbb{II}(S)$ is not zero, $\mathbb{II}(\dfrac{B^2}{S})$ must be zero, then $\widetilde V(T,S) = 0.$

But how to use this method to deal with the double barrier-out-call i.e $$V(t,B_1) = V(t,B_2) = 0,\quad B_1 < B_2$$ or is it possible to use down-out-call with barrier $B_1$ and up-out-call with barrier $B_2$ to construct double barrier?

• No, the pricing of a double barrier knock-out option cannot be decomposed into single barrier options. See Buchen and Konstandatos (2009) "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries". A very clear exposition can also be found in Chapter 3.5 the Ph.D. thesis by Konstandatos (2003), University of Sydney. If you don't have access to that, then I believe it is also reproduced in his book Konstandatos (2008) "Pricing Path Dependent Exotic Options". – LocalVolatility Jun 4 '17 at 13:04
• @LocalVolatility this is an answer! Could you please post it as such? – SRKX Jun 4 '17 at 16:17

No, the pricing of a double barrier knock-out option cannot be decomposed into single barrier options.

Here are a few references that apply the method of images to the valuation of double barrier options:

1. A very clear and easy to follow exposition can be found in Chapter 3.5 of the Ph.D. thesis by Konstandatos (2003).

2. If you don't have access to that, then I believe it is also reproduced in the author's book Konstandatos (2008).

3. Another reference is the paper Buchen and Konstandatos (2008). Here, they consider exponential barriers. You seem to be interested in the special case when the exponential "bending" is zero.

References

Buchen, Peter W. and Otto Konstandatos (2009) "A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries," Applied Mathematical Finance, Vol. 16, No. 6, pp. 245-259

Konstandatos, Otto (2003) "A New Framework for Pricing Barrier and Lookback Options," Ph.D. Thesis, University of Sydney

Konstandatos, Otto (2008) Pricing Path Dependent Exotic Options: VDM Verlag Dr. Müller

• Could you offer me a link of the book Pricing Path Dependent Exotic Options – A.Oreo Jun 6 '17 at 1:35
• Not sure if links to commercial book sellers are appreciated here. But searching for the title on the usual online stores immediately returns the desired result for me. You could also try to contact Otto Konstandatos at UTS and ask for a copy of his Ph.D. thesis..? I have one but I am not in a position to distribute it. – LocalVolatility Jun 6 '17 at 1:42