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In a recent post, I asked for help deriving a computable formula for $\text{Var}[\max(V-K,0)]$ based on the approach on p. 262 of ths book. $V$ is a lognormally distributed random variable and $K$ is a constant. And help did I get.

Now I need to extend this to the more general covariance case. I.e. I need a computable expression for

$$\text{Cov}[\max(V_1-K_1,0), \max(V_2-K_2, 0)]$$

where as before $V_1, V_2$ are lognormally distributed while $K_1, K_2$ are constants.

As before, I know the key is to find an expression for

$$E[\max(V_1-K_1,0) \max(V_2-K_2, 0)]$$

But I am stuck.

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This corresponds to the covariance in the payoff between two EU call options. In particular, $C_i$ denote the terminal payment of option $i \in \{1,2\}$. Then we know that $C_{i}= \begin{cases} \begin{array}{c} V_{i}-K_{i}\\ 0 \end{array} & \begin{array}{c} V_{i}>K_{i}\\ else \end{array} \end{cases} $

such that the option pays $C_i = max(V_i-K_i,0)$ with the probability of \begin{equation} P(V_i > K_i) = P( log(\frac{V_i}{K_i}) > 0 ) \end{equation}

Since, $V_i$ follows a log-normal, then $log(V_i)$ follows a normal distribution. In particular, the $log(\frac{V_i}{K_i})$ denotes the change in the underlying asset price with respect to the exercise price, $K_i$. Nevertheless, since the payoff only takes place if the above condition is satisfied, then the payoff, in fact, would follow a truncated normal distribution. Therefore, to find the expectation of the product (hence, the covariance), you wanna look at the conditional expectation Note that since you have two stocks, you wanna consider four dichotomies.

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