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 4 deleted 2 characters in body edited Mar 5 '12 at 14:57 Brian B 12.3k2020 silver badges4848 bronze badges 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 edited Mar 4 '12 at 17:05 Brian B 12.3k2020 silver badges4848 bronze badges 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 edited Mar 1 '12 at 13:33 Brian B 12.3k2020 silver badges4848 bronze badges 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 $$\int_0^\infty (BSM( S_\tau ) - \mu_O)(S_\tau-\mu_S) p(S_\tau) dS\tau.$$ 1 answered Feb 29 '12 at 23:05 Brian B 12.3k2020 silver badges4848 bronze badges