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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 $K\gg S_t$,...


3

I am assuming you are short EUR and long USD based on your description of your hedges. I am also assuming the size of your hedges and your fx position are the same. In the first example of a hedge of this position, you cannot be delta neutral. A long at-the-money (ATM) EUR call will have a higher delta than the short out-of-the-money (OTM) call--you will ...


3

There is no contradiction at all here. $Ke^{-rT}$ goes to zero for $T$ going to $\infty$ so the relation you mention suggests that $-S\leq p \leq 0$ as $T$ gets bigger. If $r>0$. Put prices can (in theory) not be negative so $p$ goes also to zero according to the relation you mention. Is this consistent with the pricing function of Black-Scholes? Yes. ...


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 ...


1

Please recheck, from what i can understand this question is trying to test pricing for currency options on currency forwards.Spot price is $0.6868$. If underlying is a currency forward, the underlying price $S0$ would be the forward price calculated using the interest rate parity. $$ S0 = 0.6868*\exp((0.028 - 0.0145)*0.5) $$ ATM options means strike price is ...


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