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First, my notation. $K$ is the strike price, $S$ is the stock price, $r$ is the continuously compounded risk-free rate, $T$ is time at expiration, $t$ is time at issue, $\sigma$ is volatility, $\delta$ is continuously compounded dividend rate. The Black-Scholes formula for a European call is $C = Se^{-\delta (T-t)} N(d_1) - Ke^{-r(T-t)} N(d_2)$ $d_1 = ...


You're buying a call and selling a put, both are directional up bets and your delta will be positive regardless. Therefore you're net long


A variance swap can be replicated with vanilla European options. If you take derivative with respect to variance, you need to do the same thing on both sides. That is, you need also take derivative with respect to variance on those vanilla options. However, the resulting derivative is not the vega in the usual sense, which is the derivative with respect to ...


Long options have a positive gamma because as price increases, call Delta approaches 1 from 0 put Delta approaches 0 from -1 (think of $S=0\to+\infty$). Based on below numerical example, theta and gamma can have equal signs for both put and call. If we set $r=0$, they have different signs.

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