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As already answered, your first question is call-put parity and this is an arbitrage relation independent of model assumption. Your second question (under zero rates and dividends, in the Black-Scholes model) relates to call-put symmetry : $$Call(spot=S_0,strike=K)=Put(spot=S_0,strike= \frac{S_0^2}{K})\times \frac{K}{S_0}$$ It can be easily derived from the ...

A logical way of answering this question is proof by contradiction. First note that if it is not optimal to exercise an american option prior to expiry, then the option should have the same value as the european option. So assume that it is not optimal to exercise the option prior to expiry, i.e. assume that the american option has the same value as the ...

@RemusStanescu Question 2) was answered quite intuitively but incorrectly by Freddy (he'd be right if he focused on conditional expectations rather than probabilities: indeed, P(s>110) < P(s<90) assuming lognormal dynamics for the underlying stock.) This follows from its negative skewness, which is key to your question. First note that call and put ...

A simple intuitive answer why the OTM Call is more expensive than the OTM Put is because of the skewness of the log-normal distribution. Think about it, what is the probability that the stock price is above 110 at expiration and what is the probability it is below 90? This should answer your question. Written in probability terms: The median of the ...

As SRKX mentioned the price relation between the Put and Call (with equal strikes) follows from the Put-Call parity. The whole point behind Put-Call parity is that it does not depend on the underlying distribution which describes your stock price. Put-Call parity is consequence of no arbitrage. It makes no reference to whatever model you are using to ...

The put call parity is given as follows: $$c_t-p_t = S_t - \frac{X}{e^{r(T-t)}}$$ If you assume $r=0$, you get $$c_t-p_t = S_t - X$$ So, $c_t \neq p_t$. The rationale behind it is much more financial than mathematical. You have to look at the payoff on both side of the equation, and you see that both portfolio will give the same payoff at time $T$ (the ...