Hi,I have an equity derivative which I want to value. In this derivative, there are a set of monthly observation dates out till 2 years and if the spot is higher than 160% of the strike on any of these dates, then the option knocks-out and pays a rebate. If the event does not happen, it offers a pay off of call option at maturity.Now, I did not want to use simulation and wanted to see if I can break this down into simpler structures and value it. I calculated the individual probabilities of the spot breaching 160% of strike rate on each of the observation dates (using N(d2)). However, I am not sure, how to calculate the probability of spot being higher than 160% of strike rate on ANY of the observation dates. Its not a simple summation for sure. However, I am not able to figure out the correct way to do this as these look like dependent events to me. Request your views on the same. Thanks.

Since you mentioned N(d2), this implies you want the value under GBM. While there are more elaborate analytical methods, one non-MC approach is to set it up as a simple finite-difference problem on a discrete lattice in [$](x_{min},x_{max})[$],where [$]x = \log S[$], and the endpoints are simply state space truncations. You'll probably want to arrange things sothe barrier is exactly on a node.Then, just do a standard backwards recursion, such as you might do for American-style options, from maturity to time 0,but with appropriate boundary conditions for your problem. It may or may not be useful to also decompose the value function into the sum of a KO with zero rebate plus a one-touch with the given rebate.Then, evaluate both with the difference method and add them and compare with doing the whole problem at once. Just as a check.Also, write the MC anyway as a second check. If you want to see the more elaborate analytical approaches, google something like the keywords:Wiener-Hopf, discretely monitored barrier options. The probability you ask for can likely be expressed exactly interms of the W-H methods, but I don't know it well-enough to just post it without researching it.

The solution can be expressed analytically using higher-order binary options. The usefulness of this is limited though as monthly observations with two years to maturity will yield a solution in terms of a 24-dimensional normal distribution function.See the references that I gave in this post for the background and notation.Let's consider the payoff component only. Let [$]\left( \tau_i \right)_{i = 1}^n[$] be the times-to-maturity at the [$]n[$] monitoring dates with [$]\tau_i < \tau_{i + 1}[$]. For [$]\tau \in \left[ 0, \tau_1 \right)[$], the valuation function is given by[$]\tilde{V}_1(S, \tau) = \mathcal{Q}_K^+(S, \tau) - \mathcal{A}_B^+(S, \tau) + K \mathcal{B}_B^+(S, \tau)[$].For [$]\tau \in \left[ \tau_1, \tau_2 \right)[$] the option value [$]\tilde{V}_2(S, \tau)[$] satisfies the initial value problem[$]\mathcal{L} \left\{ \tilde{V}_2 \right\} (S, \tau) = 0 \qquad \text{for } (S, \tau) \in \mathbb{R}_+ \times \left[ \tau_1, \tau_2 \right)[$],[$]\tilde{V}_2 \left( S, \tau_1 \right) = \tilde{V}_1 \left( S, \tau_1 \right) \mathrm{1} \{ S < B \}[$].The solution is given by[$]\tilde{V}_2(S, \tau) = \mathcal{Q}_{B K}^{- +} (S, \tau) - \mathcal{A}_{B B}^{- +} (S, \tau) + K \mathcal{B}_{B B}^{- +}(S, \tau)[$].Now continue applying this recursively.