# Price a contingent claim with payoff $(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$

I'd like to price the following contingent claim using a copula model. $$V_T = (S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}$$

where $$S$$ and $$X$$ are two stock price processes which follow a non-flat vol model. In particular, $$S$$ and $$X$$ are not standard GBM models with flat vols. Suppose you are given the distributions of $$S_T$$ and $$X_T$$ as a black box. Moreover, suppose you are given a copula density function, $$f_{(S,X)}$$, which is very good at approximating the joint distribution. I would like to derive a semi-analytic formula for the price, $$V_t$$. Below is my attempt. Does it look correct? What should I do next? Can I simplify further? \begin{align*} V_t & = e^{-r(T-t)} E[(S_T-K)1_{\{S_T>K\}}1_{\{L\leq X_T\leq U\}}] \\ &= e^{-r(T-t)}\int_{L}^{U}\int_{K}^{\infty}(a-K)f_{S,X}(a,b)dadb \\ &= e^{-r(T-t)}\int_{K}^{\infty}(a-K)\int_{L}^{U}f_{S,X}(a,b)dbda \\ &= e^{-r(T-t)}\int_{K}^{\infty}(a-K)\left(F_{S,X}(a,U) - F_{S,X}(a,L)\right)da \\ &= e^{-r(T-t)}\left( (a-K)\int_{K}^{\infty}F_{S,X}(a,U) - F_{S,X}(a, L) da - \int_{K}^{\infty} \int_{K}^{\infty}F_{S,X}(a,U) - F_{S,X}(a, L)dada \right) \end{align*}

• Oh nevermind, X =/= S! Dec 15 '20 at 13:18

I don't know whether you are studying in the same class with the author of this question Copula analytic formula for $$max(S_T^1−K,0)1_{L.
The idea is to decompose the payoff into 2 parts. The first part is $$1_{S_T>K}1_{L and the second one is $$S_T 1_{S_T>K}1_{L .
The value of the first part is equal to $$P(\{(W_T^1-W_t^1) > d_1 \}\cap \{ d_2<(W_T^2-W_t^2)) and you can use the density function $$f_{S,X}$$ to have its closed form solution.
• If the volatility is a deterministic but time-dependent functionk, for example $\sigma_S = \sigma_S(t)$, you can use the same argument. If the volatility is not a deterministic function, for example, the volatility is a function of $t$ and $S_t$ as following $\sigma_S = \sigma_S(S_t,t)$, I don't think you can have closed form solution, even if the payoff were just $(S_T-K)^+$.