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4

If you hold an option, you're always vega long, i.e. if volatility increases, your position increases as well - regardless of moneyness and the option type (put or call). 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). Let now $K\gg S_t$,...


4

I think it is far easier to understand by just drawing the payoffs. You have two put options: A European put option on a non-dividend paying stock with strike price 80 is priced at 8 dollars, and a put option on the same stock with strike price 90 dollars is priced at 9 dollar The difference between the two payoffs is equal to 10 dollars (90 strike puts ...


3

You need an implied volatility assumption in addition to the price drop assumption to compute that. With a higher implied volatility increase the "profitability peak" you have will gravitate towards lower strikes. Vega is a more important pnl factor in that situation than pure delta, it's not surprising that an option which is still OOM would be more ...


3

Maybe it will help your intuition if you think in terms of log-moneyness $\ln S/K$ instead of $S/K$. Let's look at a `deep' in the money put $K=100, S=10$. That sounds really deep in the money, but the value of log-moneyness for this situation is only $-2.3$, which is not that much if you consider the possible range of $\ln S/K$ is $(-\infty,\infty)$. So ...


2

I firstly fill in the gaps from Slade whose comments outline the answer and then I provide an alternative approach. Let $K_1\leq K_2$. You want to prove $P(S_t,K_1,T)\leq P(S_t,K_2,T)$. Recall firstly that $P(S_t,K,T)=e^{-r(T-t)} \mathbb{E}^\mathbb{Q}[\max\{K-S_T,0\}\mid\mathcal{F}_t] $ which is the result from risk-neutral pricing. Probably you know that ...


1

You can find the answers to most of your questions in the Taylor's series and/or approximation theory articles, but I will add a bit more detail below (in order): A simplistic example would be $y=a+bx$ vs $z=bx$, so greeks being equal does not necessarily mean that the prices will be equal. But you can use hedging/replicating argument, though it needs more ...


1

Answer was provided by Chris Taylor: the formula for the risk-neutral probability was off by a minus sign, it should be $$ p = \frac{e^{r \Delta t} S - S_m}{S_p - S_m} $$


1

I just want to add a simple piece to this reasoning, that is very intuitive and not excessively mathematic, since the mathematic explanation has already been given in the other answers (I like to base my mathematical understanding on logic intuitive reasoning). Just consider what a put option is: it is a contract to sell at the strike K and buy at the ...


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