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 ...
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  • 693
8 votes
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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 ...
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7 votes
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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{...
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  • 2,836
6 votes
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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 ...
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  • 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}]...
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  • 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{...
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  • 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 ...
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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 ...
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4 votes
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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 + ...
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  • 13.2k
4 votes
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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)...
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  • 20.4k
4 votes
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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 ...
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4 votes
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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\...
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  • 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 ...
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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)...
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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{\...
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4 votes
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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 ...
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  • 183
4 votes
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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\...
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  • 20.4k
3 votes
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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 ...
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  • 20.4k
3 votes
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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*}
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  • 20.4k
3 votes
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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 ...
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  • 4,217
3 votes
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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}{\...
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  • 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 ...
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  • 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)....
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  • 20.4k
3 votes
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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 ...
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  • 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 ...
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  • 13.8k
3 votes
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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 ...
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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 ...
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2 votes
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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*...
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  • 20.4k
2 votes
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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_{...
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  • 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} =...
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