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For American options there is no parity rule, as I stated in the comments. However, there is the following disequality: $$S_0 - D - K \leq C - P \leq S_0 - K e^{-rT}$$ where $C$ and $P$ are prices of American call and put respectively, $S_0$ is the spot price today, $K$ is the strike price, $D$ is present value of the cash dividend (not as percentage), $r$ ...

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First of all, if you are new in quantitative finance, I suggest to read the Hull'book, that's the basic for who wants to get topic fundamentals. Your evaluation is correct if you assume that linear relationship, but on real prices anything is linear; so, it depends on whath you're looking for: if you have to conclude a project work at your university, it is ...

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These equations gives the probability of a successful trade for a European put finishing in the money (that is, the probability that the strike price is above the market price at maturity). European call finishing in the money (that is, the probability that the strike price is below the market price at maturity) Probability of a Successful Option Trade ...

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Forward implied volatility smile is implied from forward start options. For example call options have payoff $$g_{T+\theta} = \left( \frac{S_{T+\theta}}{S_T} -K\right)_+$$ If you are in a stochastic volatility model this can be rewritten  g_{T+\theta} = \left( e^{ \int_T^{T+\theta} r - \frac{1}{2}\sigma_t^2 dt + \int_T^{T+\theta}\sigma_tdW^S_t } ...

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I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations ...

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the length of rate which closest corresponds to the maturity of the option. This will be true opportunity cost of having capital tied up in option positions with regard to the risk free rate.

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Actually BS model is still applicable in the market where the upwards/downwards move is much more probable than move in the opposite direction. The Black-Scholes price process model has the form: $\frac{dS}{S} = \mu dt + \sigma dW$ And with significantly non-zero $\mu$ (called drift) it will capture just what you are talking about. Quite surprisingly, the ...

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Well if you think that this model represents reality more accurately than the Black-Scholes assumptions. A lot of people do indeed think so. But I wouldn't say you're "tweaking" Black-Scholes... you're just assuming another model altogether and you will use risk-neutral pricing to compute the fair value of the option at time $t$, just like BS. Frankly, I'm ...

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There is something called a 'capped option' which does have a restricted domain. There are a couple versions of this, some of which use the method of images combined with automatic exercise at the barrier. It doesn't tell you how existing options behave under the introduction of a barrier in the context of put call parity

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Negative probabilities under Heston have been discussed elsewhere in the Stack. Please check this post as it could be of interest.

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I'm not an equities guy so I don't know anything about volume-weighting, but if I was set this problem the approach that I would take would be to work out the implied volatility of each strike for that day, so that I have a graph of implied volatilty against strike, then interpolate on that graph to get the implied volatility for the ATM strike. Do that for ...

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