The Black-Scholes formula depends of many parameters, is easy to note that it is increasing respect to the parameter $S_t$, $\sigma$ and $r$, it means to fix a all parameters and vary only one. Is possible to say the same for the strike price $K$ or the time $t$, I think BS-formula is decreasing respect to these parameters, it is easy to see for $K$ at time T because the formula gives $(S_T-K)^+$ but not sure. If anyone has an explicit proof of this I really appreciate it. Thank you!.
1 Answer
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In both cases, for European options, it's a matter of determining whether the first derivative of the call price w.r.t. the parameter is always positive or negative. That said, intuition and good financial arguments are always helpful and can often be more enlightening than using the math.
In the case of the strike price, I like your argument -- to make it very slightly more formal, you can say that if Call A had a higher strike than Call B, but also cost more, there would be an arbitrage opportunity. (Again, you can inspect the derivative to see also.) Of course, using the same argument, we can see that the opposite holds for puts.
For theta, I'll link this answer as well as this link giving the formulas for theta. Particularly, we can see by inspection that for non-negative interest rates (and for 0 dividend yield), theta for a call is always negative (in this case, in the formula we are adding two terms that are only ever negative), this is not the case for puts. Put theta can be positive or negative even with positive interest rates, but in practical terms, theta will be negative for most traded options.
