4 deleted 2 characters in body
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In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$$$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)^2 p(S_\tau) dS_\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS_\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)^2 p(S_\tau) dS_\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS_\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)^2 p(S_\tau) dS_\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS_\tau. $$

3 spelling correction, TeX formatting
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In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic funcitonfunction of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$$$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)^2 p(S_\tau) dS\tau $$$$ \int_0^\infty (BSM( S_\tau ) - \mu_O)^2 p(S_\tau) dS_\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$$$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS_\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic funciton of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)^2 p(S_\tau) dS\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)^2 p(S_\tau) dS_\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS_\tau. $$

2 Fix BS prob density and a couple typos
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In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic funciton of stock price. For example, the mean of the option value for time $\tau$ is $$ \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$$$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \mu_O = \frac{e^{-rT} n( d_2 )} {K \sigma \sqrt{T} }. $$$$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)^2 p(S_\tau) dS\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$$$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic funciton of stock price. For example, the mean of the option value for time $\tau$ is $$ \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \mu_O = \frac{e^{-rT} n( d_2 )} {K \sigma \sqrt{T} }. $$

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)^2 p(S_\tau) dS\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic funciton of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The function $p(S_\tau)$ would typically be the Black-Scholes probability density $$ \frac{n( d_2(S_0,S_\tau) )} {S_\tau \sigma \sqrt{T-\tau} }. $$ with $S_\tau$ taking the place of strike $K$ in the formula for $d_2()$.

Similarly, the variance of the option value for time $\tau$ is $$ \int_0^\infty (BSM( S_\tau ) - \mu_o)^2 p(S_\tau) dS\tau $$ and covariance of option value with stock price would be $$ \int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS\tau. $$

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