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It is fairly standard to hedge a sold option as follows: at any time $t$ buy $\alpha(t)=\frac{\partial}{\partial S}c(t,S(t))$ amounts of stock $S(t)\,,$ and invest $\beta(t)=\frac{c(t,S(t))-\alpha(t)S(t)}{B(t)}$ into the money market account $B(t)=e^{rt}$ By definition, the hedge portfolio $X(t)=\alpha(t)S(t)+\beta(t)B(t)$ exactly matches the option value ...


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First, let's go back to basics to answer why theta can be both positive and negative, and why it's referred to as time decay? At it's core, an option's value is composed of two components: intrinsic value, and time value. As time passes, the proportion of the 'time value' gradually decreases until the option is worth exactly its intrinsic value at its ...


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It’s just the effect of interest. If you are long a deep ITM European put, it is worth the PV of K minus the stock price. But one day later the PV of K has grown a bit. That’s it. It’s the opposite for calls because you have to pay the K, so bringing the date closer costs you money. This is all assuming interest rates are positive.


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