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Results tagged with geometric-brownian
Search options answers only
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user 58786
3
votes
Accepted
GBM - How to make make annualized dividend reflected in one quarter
What follows is a discrete dividend version of my answer in
this post:
When the stock pays dividends it is not true that the deflated stock price
$$
e^{-\int_0^tr(s)\,ds}S_t
$$
is a martingale. But in …
- 2,339
2
votes
How to simulate correlated stock prices (not returns)
For two GMBs
$$
S_i(t)=S_i(0)e^{(r-\delta_i)t+\sigma_i W_i(t)-\sigma_i^2 t/2}\,,\quad i=1,2
$$
with ${\rm Corr}[W_1(t),W_2(t)]=\rho\,t$
we have
\begin{align}
\mathbb E\left[S_i(t)\right]&=S_i(0)e^{(r- …
- 2,339
2
votes
Accepted
Change of numéraire for two risky assets without bank account (Margrabe’s formula?)
Under the risk-neutral measure both stocks follow the GBMs
\begin{align}
S^{(i)}_t=S^{(i)}_0\exp\left((r-q_i)t+\sigma_iW^{(i)}_t-\frac{\sigma_i^2t}{2}\right)\,,\quad i=1,2\,,
\end{align}
where the con …
- 2,339
1
vote
American option under Ornstein-Uhlenbeck stock price
If one insists on first principles then OU processes should be limited to variables that do not describe prices of trades assets. This limits OU to -as you say- interest rates for example.
A process t …
- 2,339
1
vote
Estimate of GBM Return Variance
It is correct that the definition of realized variance that you gave depends on the drift of $S_n$ which seems unwanted at first glance. This is however the official definition that is used in derivat …
- 2,339