16
votes
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
16
votes
Accepted
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 ...
- 16.2k
8
votes
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
8
votes
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
6
votes
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
6
votes
Accepted
Probability of an Option maturing In-the-money vs. Volatility
Call option:
$$\mathbb{P}\left(S_t\geq K\right)=\mathbb{P}\left(S_0e^{(rt-0.5\sigma^2t+\sigma W_t)}\geq K\right)=\\=\mathbb{P}\left(W_t\geq \frac{ln\left(\frac{K}{S_0}\right)-rt+0.5\sigma^2t}{\sigma}\...
- 6,308
6
votes
Accepted
Pricing an Option with payoff $\left(1-\frac{K}{S_t}\right)^{+}$
$\frac{1}{S_t}$ is log-normal
If $S_t$ is a geometric Brownian motion, so is $\frac{1}{S_t}$ and indeed any power $S_t^\alpha$. Simply use Itô's Lemma and set $f(t,x)=\frac{1}{x}$,
\begin{align*}
\...
- 16.2k
6
votes
Accepted
Geometric Brownian Motion as the limit of a Binomial Tree?
We can show that the moments of the Binomial tree agree with the moments of the continuous model for the case where we pick symmetrical probability value $p=0.5$.
I will change the notation slightly (...
- 6,308
5
votes
What is the stock price expectation?
In Hull's textbook, the stock price dynamics is lognormal: $S_T = S_0 \exp(\mu T - \frac{1}{2}\sigma^2T + \sigma W_T)$, where $W_t$ is a standard brownian motion. And so the mean of this is the mean ...
- 666
5
votes
Accepted
Normality or Log-Normality of Regular Returns
You're right but a GBM doesn't assume that percentage returns are normally distributed. It's about log-returns.
If the log-return $r_t=\ln\left(\frac{S_{t+dt}}{S_t}\right)$ is normally distributed (...
- 16.2k
5
votes
Probability of an Option maturing In-the-money vs. Volatility
This is quite a brain teaser, at least it was for me. The way I thought about this initially was based on statistics. This lead me to believe that higher IVOL should always decrease the probability of ...
- 9,349
5
votes
Accepted
Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM
I think the easiest way to derive the solution to the GBM is via Ito's Lemma.
The GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a short hand for:
$$ S_t = S_0 + \int_{h=0}^{h=t}\left(\mu S_h\right)dh ...
- 6,308
5
votes
Accepted
Default intensity in Black-Cox model
As shown in Credit Risk Modeling Notes (Bielecki, Jeanblanc, Rutkowski), Corollary 1.3.1, for $t < s$, we have:
$$ P(\tau \leq s | {\cal F}_t) = N\left( -Y_t \sigma^{-1}(s-t)^{-1/2}- \nu(s-t)^{1/2}\...
- 5,123
4
votes
Accepted
How to Understand Lognormal Distribution in the Following Case
The drift of $\mathrm{d}\ln(S_t)$ is indeed $r-\frac{1}{2}\sigma^2$ which is always negative if $r=0$. The extra $-\frac{1}{2}\sigma^2$ has many explanations. You could see it as a convexity ...
- 16.2k
4
votes
Accepted
Why do I get this difference when simulating geometric Brownian motion?
Here are the things you need to correct in your code:
Although you are setting a seed, you are generating the random numbers twice, and therefore they are not identical. Try this:
...
- 5,835
4
votes
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
It's just Girsanov's theorem.
I suppose that under the risk neutral measure Q
$$dS_{t}= r S_{t} dt + \sigma S_{t}dW_{t},$$ $$S_{t} = S_{0}\exp\left((r-\frac{\sigma^{2}}{2})T + \sigma W_{T}\right)$$
By ...
- 98
4
votes
Normality or Log-Normality of Regular Returns
The return $R_i$ as expressed in
$$R_{i+1,i}=\frac{S_{i+1}-S_i}{S_i}=\mu \Delta t + \sigma \Delta W(t_{i+1},t_i)$$
is not possible.
To see this, let's get the returns over two small time steps of $\...
- 241
4
votes
Accepted
Geometric Brownian Motion simulation in Python: strange results
Great question!
Abstract: Your code and math are correct, but you use too high vol and drift to be real world realistic. Your simulations decay to zero due to high vol and LogNormality.
Basically, I ...
- 1,272
4
votes
Accepted
How to compute the Present Value of this path-dependent option?
We have
$$
\begin{align}
V(t) &= \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)] \\
&= \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B}\mathbb{1}_{S(T_2)>K} (S(T_2)-K))] \\
&= \mathbb{E}^{...
- 1,033
3
votes
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
3
votes
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
3
votes
Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world
Part 1: deriving the drift of the stock price process under the stock Numeraire.
Under the risk-neutral measure, the process for $S_t$ is as follows:
$$ S_t = S_0 + \int_{h=t_0}^{h=t}rS_h dh + \int_{h=...
- 6,308
3
votes
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Question 1 is answered in parts 1 through to 6: the idea is that each part slowly builds the tools required to derive the process equation for $S_t$ under the $S_t$ Numeraire.
Question 2 & ...
- 6,308
3
votes
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Black scholes formula based on $S_t$ measure , theory, and formulas you mention are derived in detail in "Steven Shreve: Stochastic Calculus and Finance" draft pdf from 1997 , page 328 "...
- 861
3
votes
Accepted
Covariance of logarithms of geometric Brownian motion
Let $Y = \log X$, then:
$$\begin{align}
Y &= Y_0 + (\mu-\frac{\sigma^2}{2})t + \sigma W_t
\\
EY_t &=Y_0 + (\mu-\frac{\sigma^2}{2})t
\\
EY_tEY_s &= Y_0^2 + Y_0 (\mu-\frac{\sigma^2}{2}) (t+...
- 211
3
votes
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
3
votes
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
3
votes
Accepted
Drawing values from a lognormal distribution of a GBM
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^{\...
- 5,815
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