5
There are several derivations and interpretations for $N(d_1)$ and $N(d_2)$. As you know,
\begin{align*}
C(t,S_t)=S_te^{-q(T-t)}N(d_1) -Ke^{-r(T-t)}N(d_2).
\end{align*}
We can also show that
\begin{align*}
\mathbb{Q}_S[\{S_T\geq K\}]&=e^{-q(T-t)}N(d_1), \\
\mathbb{Q}[\{S_T\geq K\}] &=e^{-r(T-t)}N(d_2).
\end{align*}
Thus, $N(d_i)$ may be seen as ...
4
It depends very much on the individual option you are pricing.
Sometimes you can get a Black-Scholes PDE with some extended state and boundary conditions.
up-and-out barrier option will have virtually the same pricing PDE, and zero boundary condition at $S=0$ and $S=B$ (barrier level). The up-and-in barrier option can be priced by $$C_\text{up-and-in} + C_\...
3
The resolution is that the GBM that is assumed in Black-Scholes is continuous, and the hedge is riskless only if it is rebalanced continuously. Now it is true that a GBM with any vol could produce any price history, but if you hedge at discrete intervals, the sampled path history and its observed point-to-point volatility becomes very important for the ...
2
You got to be careful with $\mathbb{P}$ and $\mathbb{Q}$. Indeed, $N(d_2)$ is the probability of the event $\{S_T\geq K\}$ in the risk-neutral world. Note that $r$ (or $r-q$) is the drift in the risk-neutral world and hence this variable occurs in $d_2$. Since time to maturity and volatility are typically small numbers, i.e. $d_1=d_2+\sigma\sqrt{T-t}\approx ...
2
I will try to be as concise as possible. The sum of log returns over long time horizons for many assets tends to be less divergent from a normal than the distribution of log return at daily level, and the proximity gets better as your horizon increases (i.e the frequency of returns decreases). Indeed you can view low-frequency log returns as the sum of ...
1
If I understand the question correctly, you have the implied vol by delta, and you would like to calculate the price using the Black Scholes formula. And I assume you know the other inputs-e.g., underlying price, interest rate and maturity. Very typical problem in the FX world, so what you can do is first convert delta (using the other inputs and vol) to ...
1
In the Black-Scholes' setting, as we discussed in this question, the portfolio $\Pi = \Delta S -V$, where $V= \frac{\partial V}{\partial S}= N(d_1)$, is not self-financing. Moreover,
\begin{align*}
\Pi = \Delta S -V=Ke^{-r(T-t)}N(d_2)
\end{align*}
does not satisfy the equation
\begin{align*}
d\Pi = r\Pi dt.
\end{align*}
In fact, let
\begin{align*}
\...
1
Here's a simple way to get the price of the call on the forwards price using risk neutral pricing.
Suppose we have a European call that pays at $t = T$, $(For(T,T^*) - K)^+$, where $T^* \geq T$. Further assume interest rates are constant and are represented by "$r$". Let $c^{For}(t, s)$ be the price of the call where $S(t) = s$.
Then if the stock pays no ...
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