# Tag Info

Accepted

### Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

I provide a solution in three steps. The first step carefully outlines how to split up the expectation and what new measures are used. This first step does not require any special model assumption ...
• 16.2k
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### From VG and NIG processes to GBM

Intuition Yes it is possible. Both, the NIG and the VG process are exponential Lévy processes, i.e. they model the stock price via $S_t=S_0e^{X_t}$, where $X_t$ is a Lévy process. Here's a recent ...
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### What the expectation of S^2 is from GBM?

As Sanjay said, you can apply Itô's Lemma to $f(t,x)=x^2$ and obtain \begin{align*} \mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t. \end{...
• 16.2k
Accepted

### Dynamics of FX rate

I am answering now instead of commenting. The rate of change in FX is naturally forward looking in this case. What you confuse is what happened to Spot due to changes in interest rate environments ...
• 9,349
Accepted

### Simulation of Geometric Brownian Motion in R

The issue is that you do not plot one sample path but for each time point $t$, you simply plot one possible realisation of the random variable $S_t(\omega)$. Thus, you don't get a connected path. (...
• 16.2k
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• 6,308
Accepted

• 1,033

### What is the meaning that Geometric Brownian motion is leptokurtic?

Heavier tails, or a higher probability of extreme outlier values, meaning the investor is more likely to experience extreme events (e.g. tail losses). EDIT: @noob2, valid point, see this article on ...
• 370
Accepted

### Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world

Following this answer, let $\mathbb Q$ be the probability measure associated to the risk-free bank account as numeraire and $\mathbb Q^1$ the probability measure associated to the stock as numeraire. ...
• 16.2k

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Accepted

### How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion

Because $\mathbb{E}\left(e^{\sigma W_t}\right) = e^{\frac{1}{2}\sigma^2T} > 1$, you need that correction to ensure that your asset grows on average at rate $\mu$ (or $r$ in the risk-neutral measure)...
• 2,680
Accepted

### Probability of a stock price using implied volatility

I assume you want to real-world probability, because the risk-neutral probability is not a probability in the 'likelihood' sense. Under the real-world measure, we model the stock under the B-S model ...
• 6,308
Assuming that your GBM is given by $$S_{T}=S_{0}e^{(r -{\frac {\sigma ^{2}}{2}})T+\sigma W_{T}}$$ then its mean and variance are: $${Mean=S_{0}e^{r T},}$$  {Variance=S_{0}^{2}e^{2r T}\left(e^{\...