Get expected joint-payoff price of digital options from individual payoffs
I am trying to model a joint distribution $ f(X_1,X_2) $
(where $ X_1 $ and $ X_2 $ are market prices of the options) and then find from it the value of joint payoff price:
$ F(X_1, X_2; B_1, B_2) = E[ max(X_1-B_1,0) * max(B_2 -X_2,0)] $
where $ B_1 $ and $ B_2 $ are corresponding strike prices.
I have limited samples of individual payoffs $ Q_1 $ and $ Q_2 $ for different values of $ B_1 $ and $ B_2 $
$ Q_1(X_1,X_2;B2) = E[X_1 * max(B_2-X_2,0)] $
$ Q_2(X_1,X_2;B1) = E[X_2 * max(X_1-B_1,0)] $
$ C(X;k) = E[max(0,X_T - k)] $
Is there a way to solve this with or without assuming that $ X_1 $ and $ X_2 $ are independent.
I am new to modelling option-prices.
Suppose you would like to compute
$$ \begin{align} Q_1(x_1,x_2;B) &= \Bbb{E}[X_1\max(B-X_2,0)]\ Q_2(x_1,x_2;B) &= \Bbb{E}[X_2\max(X_1-B,0)] \end{align} $$ where you know the marginal probability density functions $ p_{X_1}(u) $ and $ p_{X_2}(v) $ .
Let’s start by focusing on $ Q_1 $ . By definition, the expectation equivalently writes:
$$ Q_1(x_1,x_2;B) = \int_0^\infty \int_0^\infty u \max(B-v,0), p_{X_1X_2}(u,v) du dv $$ where $ p_{X_1X_2} $ figures the joint probability density function of random variables $ X_1 $ and $ X_2 $ : $$ p_{X_1X_2}(u,v) = p_{X_1 \vert X_2 = v}(u) p_{X_2}(v) $$ $$ Case 1: $X_1$ and $X_2$ are independent $$
Then by definition
$$ p_{X_1 \vert X_2 = v}(u) = p_{X_1}(u) $$ and we have (Fubini) $$ \begin{align} Q_1(x_1,x_2;B) &= \int_0^\infty \int_0^\infty u \max(B-v,0), p_{X_1}(u) p_{X_2}(v) du dv \ &= \int_0^\infty u p_{X_1}(u) du \int_0^\infty \max(B-v,0) p_{X_2}(v) dv \ &= \Bbb{E}[X_1] \Bbb{E}[\max(B-X_2,0)] \end{align} $$ which you know how to solve since you know the marginals.
$$ Case 2: $X_1$ and $X_2$ are not independent $$
The marginals do not suffice: you need an assumption concerning the dependence structure of $ X_1 $ and $ X_2 $ . Your question amounts then to saying, “can I infer a unique dependence structure from the knowledge of $ Q_1 $ and $ Q_2 $ “, the answer is no. The intuition behind that is given in this SE question.