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A hedged stock position means that you own the stock and an option to reduce the risk. For owning the stock, you pay $S_0$ but receive $S_T$ in 3 months. Similarly, your option costs $1.50$ but will g …

\end{align*} As an alternative to this answer, you may want to consider a classical no arbitrage argument and look at a portfolio which owns one put option with strike price $K_1$ and is short one put … It always boils down to the intuitive idea that a put option with larger strike price has a higher payoff and thus needs to cost more than a put option with a lower strike price. …

Note firstly that by the model-free put-call parity, put and call options have the same vega (i.e. changes in volatility affect put and call prices in an identical way). … should increase the call price'' but that would then also imply increasing put prices. …

I don't think one can answer your question. Suppose $X=e^{\mu+\sigma Z}$ is log-normal, i.e. positive. Thus, $\mathbb{E}[\max\{0,X\}]=\mathbb{E}[X] $ and $\mathbb{E}[\min\{0,X\}]=0$. From just knowing …