12
votes
Is it really possible to create a robust algorithmic trading strategy for intraday trading?
Here's my favorite example of an intraday strategy on S&P500 futures that at least used to work:
Intraday Share Price Volatility and Leveraged ETF Rebalancing
I pull it out whenever people start ...
- 693
8
votes
Accepted
Is it really possible to create a robust algorithmic trading strategy for intraday trading?
Such a complex question...
Geometric Brownian Motion (GBM) will not typically work to aid one finding strategies based on technicals, as the pursuit of the technical trader is to find market ...
- 371
7
votes
Accepted
conditional expectation of stochastic integral
What a great question! I've had a go at it below, I'd say I'm about 75% sure of the result I've got to but I'd love feedback from others.
I'm going to use the definition of the Ito integral,
\begin{...
- 2,836
6
votes
Accepted
How to differentiate a brownian motion?
In order to apply Ito's lemma, your function needs to be a twice-differentiable function. There is no issue with the non-differentiability of the Wiener process. $\frac{dF}{dX}$ involves ...
- 520
5
votes
What is meant by innovations in volatility?
A volatility innovation is the difference between our best prediction of future volatility and what is actually observed. Say that we predict the volatility at the next time step as $E_t[ \sigma_{t+1}]...
- 1,048
5
votes
Invariance Scaling of Brownian Motion
Note that
\begin{align*}
\int_0^t e^{B_s}ds &= t\int_0^1 e^{B_{tu}}du\\
&=t\int_0^1 e^{\sqrt{t}\frac{1}{\sqrt{t}}B_{tu}}du\\
&=t\int_0^1 e^{\sqrt{t}W_u}du,
\end{align*}
where $\{W_u=\frac{...
- 20.4k
4
votes
Is it really possible to create a robust algorithmic trading strategy for intraday trading?
I have been through your confusion myself for the last five years. Until recently, my account started to get some consistent performance.
First, I started with Technicals, Spent $$$ on a automated ...
- 141
4
votes
Intergral of Brownian motion w.r.t. Brownian motion
While Richard's answer is technically correct, just saying the result can be obtained using Ito's formula doesn't make the issue much clearer. So let me go into the microscopics of the issue.
The Ito ...
- 402
4
votes
Accepted
Intergral of Brownian motion w.r.t. Brownian motion
Apply Ito's lemma to $f(W_t) = W_t^2$ then
$$
f(W_T) = f(W_0) + \int_0^T f'(W_t) dW_t + \frac{1}{2} \int_0^T f''(W_t) dt.
$$
Thus
$$
W_T^2 = 2 \int_0^T W_tdW_t + \frac12 2 T = 2 \int_0^T W_tdW_t + ...
- 13.2k
4
votes
Accepted
Why $W_{t}^3$ is not a martigale?(by Definition)
Note that, for $0 \leq s < t$,
\begin{align*}
W_t^3 &= (W_t-W_s+W_s)^3\\
&= (W_t-W_s)^3 + 3(W_t-W_s)^2 W_s + 3 (W_t-W_s) W_s^2 + W_s^3.
\end{align*}
Moreover,
\begin{align*}
E\big( (W_t-W_s)...
- 20.4k
4
votes
Accepted
Hawkes process intensity solution
Let us define the auxiliary process $\Lambda_t=e^{\kappa t}\lambda_t$. Note that:
$$ \Lambda_t = \kappa e^{\kappa t} \int_0^t(\rho_s-\lambda_s)ds+\delta e^{\kappa t}\int_0^tdN_t$$
Hence after a jump ...
- 7,091
4
votes
Accepted
Expected value of exponential of hitting time of GBM
Let $W_t= -B_t$. Moreover, let $a= - \frac{r-\frac{1}{2}\sigma^2}{\sigma}$ and $b= -\frac{1}{\sigma}\ln \frac{S^*}{S_0}$. Then, as in this question,
\begin{align*}
\mathbb{P}\left(\tau \ge T \mid W_T\...
- 20.4k
4
votes
Evaluating the SDE $dX_t = t\,dS_t$
Using Itô's Lemma, notice that:
$$d(tS_t)=tdS_t+S_tdt=dX_t+S_tdt$$
Hence:
$$X_t=tS_t-\int S_udu$$
Using independence of Brownian increments, $E(S_udW_u)=E(S_u)E(dW_u)=0$, and the chain rule for the ...
- 7,091
4
votes
Stochastic Calculus problem with three processes? (Itô calculus)
If you are happy to try the brute force approach, then here are the relevant formulae:
In ordinary calculus, you have the product rule for the differential of two variables:
$$d \left( x_1 x_2\right)...
- 6,044
4
votes
conditional expectation of stochastic integral
Just wanted to add to @StackG's great answer using a different approach. Please, double-check my solution as well because I'm not 100% sure.
Let $\sigma_t$ be sufficiently regular such that $\dot{\...
- 680
4
votes
Accepted
Taleb's Black-Swan: interpretation of the exponent
I finally got the idea behind the example. To illustrate it in a more general setting I will present a rigorous proof:
Let $x_k$ denote the salary and $b_k$ the number of persons that earn $x_k$ or ...
- 183
4
votes
Accepted
Change of measure for a stochastic process to be a martingale
Let $Y_t= e^{B_t}$ and $Z_t = B_{t}-t / 2$. Then,
\begin{align*}
dX_t &= Z_t dY_t + Y_t dZ_t + d\langle Y, Z\rangle_t\\
&=(B_{t}-t / 2)e^{B_t}\big( dB_t + 1/2\,dt \big) + e^{B_t}\big(dB_t -1/2\...
- 20.4k
3
votes
Accepted
forward option, stochastic calculus
For the last question. We assume that
\begin{align*}
S_t = S_0 e^{(r-q-\frac{1}{2}\sigma^2)t + \sigma W_t},
\end{align*}
where $W$ is a standard Brownian motion, $r$ is the interest rate, $q$ is the ...
- 20.4k
3
votes
Accepted
Problem with derivating integral
We assume that $\gamma(s, t)$ is differentiable with respect to $t$. Then,
\begin{align*}
dx_t = \left(\int_0^t \frac{\partial\gamma(s, t)}{\partial t} dW_s \right)dt + \gamma(t, t) dW_t.
\end{align*}
- 20.4k
3
votes
Accepted
existence of implied volatility
You can show that "the implied variance of an ATM short maturity option is equal to the expectation under the risk neutral measure of the integrated variance over the life of the option." As you move ...
- 4,217
3
votes
Accepted
stochastic calculus - Itô formula?
For the first question, since by definition,
\begin{align*}
\varepsilon_t^{if} = e^{i \int_0^{t}f\big(\frac{1}{\xi}\langle M\rangle_s\big)\frac{dM_s}{\sqrt{\xi}} + \frac{1}{2}\int_0^t f\big(\frac{1}{\...
- 20.4k
3
votes
Detect trend of an index
Questions:
1=> Does anyone have a suggestion to determine a trend correctly.
My answer is in general and an opinion. Hong Kong Stock Exchange is third largest market behind Tokyo and Shanghai and ...
- 460
3
votes
Problem of stochastic differential equation (SDE)
We assume that the price at time $t$ of a zero-coupon bond, with maturity $u$ and unit face value, is of the form
\begin{align*}
f(u-t, r_t, x_t) = E\left(e^{-\int_t^u r_s ds}\mid \mathcal{F}_t\right)....
- 20.4k
3
votes
Accepted
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Note that, for $t>s>0$,
\begin{align*}
X_t-X_s &= \frac{1}{t}\int_0^t udW_u - \frac{1}{s}\int_0^s udW_u\\
&=\frac{1}{t}\bigg(\int_s^t u dW_u + \int_0^s udW_u \bigg)- \frac{1}{s}\int_0^s ...
- 20.4k
3
votes
How can I prove that the solution to the Heston SDE is a Markov process?
I am not providing a full proof but a reference for you to read up the details. The key step is mentioned below.
Most models used in finance are Markovian which is kind of in line with the efficient ...
- 13.8k
3
votes
Accepted
What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
The SDE you are describing is called the Geometric Brownian Motion. In the end its just a model, which underlies certain assumptions, which are usually not met in the real world scenarios. There are ...
- 342
3
votes
What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
Take the analogy of equations modelling something in physics.
Just because you write down an equation, it does not mean it has to be connected to anything in reality. It only do so to the extent you ...
- 561
2
votes
Accepted
stochastic calculus - brownian motion
Note that $X$ is a continuous martingale. Moreover, the quadratic variation is given by
\begin{align*}
\langle X_t, \, X_t\rangle = \int_0^t |\sigma_u|^2 du = c^2 t.
\end{align*}
That is,
\begin{align*...
- 20.4k
2
votes
Accepted
equality in distribution
It appears that we need only to observe the following:
\begin{align*}
\lim_{\lambda\rightarrow 0}\frac{1}{\lambda}\int_0^{\lambda t}\sigma^2_u du &= \lim_{\lambda\rightarrow 0}\int_0^{ t}\sigma^2_{...
- 20.4k
2
votes
How to differentiate a brownian motion?
We write the differential form of Ito formula for simplification. Actually, the differential form for Ito formula
$$
dF(W(t)) = 2W(t)dW(t) + dt
$$
means the integral form for Ito formula,
$$
\int{dF} =...
- 43
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