38
votes
Explaining the Risk Neutral Measure
Life Without a Risk-Neutral Measure
How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$...
- 16.2k
24
votes
Accepted
Which process is the most commonly used for modeling stock prices?
I give you a brief outline about some key properties of Lévy processes.
Lévy processes have stationary and independent increments but do not necessarily have continuous sample paths. In fact, ...
- 16.2k
22
votes
Accepted
Explaining the Risk Neutral Measure
Intro:
Great answer given by Kevin. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage&...
- 6,308
20
votes
Accepted
Finding distribution of $\int_0 ^T uW_u du$
Using the Ito Formula
The general approach that often works for these kinds of question is to search for functions such that their Ito differential contains the terms that we are interested in. In ...
- 6,074
14
votes
Accepted
Can I always use quadratic variation to calculate variance?
Quadratic variation and variance are two different concepts.
Let $X $ be an Ito process and $t\geq 0$.
Variance of $X_t$ is a deterministic quantity where as quadratic variation at time $t $ that ...
- 2,442
13
votes
Accepted
How to find the formula for the half-life of an AR(1) process (using the Ornstein–Uhlenbeck process)
Convenient rewriting
Let $$X_t = c + \phi_1 X_{t-1} + \epsilon_t, \quad \vert \phi_1 \vert < 1 \tag{1} $$
denote a weakly stationary AR(1) process. Weak stationarity notably implies that $$\Bbb{E}[...
- 14.7k
13
votes
Accepted
Does numeraire have to be a tradable asset
This is an interesting question that I have asked myself. Below is my take.
Let us consider an economy $(\Omega,\mathcal{F},P)$ equipped with a filtration $(\mathcal{F})_{t \geq 0}$ consisting on a ...
- 8,169
13
votes
Accepted
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let
\begin{align*}
Y_t = e^{(a+\frac{c^2}{2})t-cW_t}.
\end{align*}
Then
\begin{align*}
dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right].
\end{align*}
Moreover,
\begin{align*}
d(X_tY_t) &= Y_t ...
- 21.2k
12
votes
Accepted
Correlation coeffitiont between two stochastic processes
if you talk about correlation then:
compute expectation:
$$\mathbb{E}(W_t)=0\text{ and }\mathbb{E}(\int_0^tW_d ds)=0$$
variance:
$$\text{Var}(W_t)=t\text{ and }\text{Var}(\int_0^tW_s ds)=\frac{t^3}{...
- 2,442
12
votes
Finding distribution of $\int_0 ^T uW_u du$
Another approach consists in using the Fubini theorem to write that
\begin{align}
\int_0^T u W_u du &= \int_0^T \int_0^u u\, dW_v\, du \tag{$W_u = \int_0^u dW_v$} \\
&= \...
- 14.7k
12
votes
Heston stochastic volatility, Girsanov theorem
Consider the Heston (1993) model under the real world measure ($\mathbb{P}$)
\begin{align*}
\mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\
\mathrm{d}v_t&...
- 16.2k
11
votes
Accepted
What is an adapted process
Let $\{X_t\}$ be a stochastic process and $\mathcal{F}$ be a filtration.
The intuitive idea is that for $\{X_t\}$ to be adapted, it can't reveal what's unknowable (according to the filtration). By ...
- 7,004
11
votes
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed
\begin{equation}
\int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
- 6,074
11
votes
Accepted
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$
By construction, the Itô integral, $I_t=\int_0^t X_s\text{d}W_s$, is a martingale if $\int_0^t \mathbb{E}[X_s^2]\text{d}s<\infty$.
The martingale property, $\mathbb{E}_s[I_t]=I_s$ implies $\mathbb{...
- 16.2k
11
votes
Expectation of exponential of 3 correlated Brownian Motion
You need to rotate them so we can find some orthogonal axes.
A simple way to think about this is by remembering that we can decompose the second of two brownian motions into a sum of the first ...
- 3,056
11
votes
Accepted
Expectation of exponential of 3 correlated Brownian Motion
Besides @StackG's splendid answer, I would like to offer an answer that is based on the notion that the multivariate Brownian motion is of course multivariate normally distributed, and on its moment ...
- 7,035
10
votes
Accepted
List: Behavioural characteristics of key Ito processes used in finance
I will provide some references such that you can see where the different processes are used. These papers typically motivate their models and show which effect the single paramaters have and what ...
10
votes
Accepted
How to simulate Levy processes
You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and ...
- 16.2k
10
votes
Accepted
Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{...
- 216
10
votes
Accepted
Expected value and Variance of a stopped random process
Although Math SE might be a bit more suited for this one, I wanted to give it a try.
The answer relies on the law of total expectation, the law of total variance, and the relationship between Euler's ...
- 7,035
10
votes
If the spread between two assets is an OU process, what processes do the two assets follow?
If we allow the mean reversion speeds to be identical, we could assume OU processes for the two components:
Let
$$
\begin{align}
dx_1&=\kappa_1(\theta_1-x_1)dt+\sigma_1dW_1\\
dx_2&=\kappa_2(\...
- 7,035
9
votes
Accepted
SABR Model Pricing Engine in Python QuantLib
Here is a simple example that might be useful. Basically finding parameters for a given section. Some of the parameters might be assumed at start instead of calibrated.
...
- 5,835
8
votes
Accepted
Integral of Wiener process w.r.t. time
@Ivan's comment regarding the covariances is the key.
Consider an equally spaced partition $\Pi_n = \left\{ t_0 = 0, t_1 = \Delta_n, \ldots, t_n = t \right\}$ of the interval $[0, t]$, where $t_i = i ...
- 6,074
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
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
Let $(W_t)$ be a standard Brownian motion and $a>0$. We define $X_t=e^{aW_t-\frac{1}{2}a^2t}$. Then, the process $(X_t)$ is adapted and integrable which are the first two conditions of being a ...
- 16.2k
8
votes
Difference between $W_t$ and $X_t= \sqrt{t}Z$
The means are equal
Suppose $f$ is analytic so that we can give it a Taylor series that works everywhere such that $f(x) = \sum a_n x^n$, and then let us let this be bounded too. To show that the ...
- 1,399
8
votes
How can I learn stochastic process & stochastic calculus in two weeks?
This is impossible unless you are very intelligent with good memory-retention skills and already mathemathically proficient in the field of analysis and statistics (and no, a single course in basic ...
- 101
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