Solving a Mean-Reverting Log-Price Process with Ito's Lemma

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


# 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?

## Answer by LocalVolatility (score 3, accepted)

https://quant.stackexchange.com/a/34548

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:

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

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

- 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