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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 ...
KarolisR's user avatar
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10 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 ...
Steinwolfe's user avatar
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{...
StackG's user avatar
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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{...
Gordon's user avatar
  • 21.2k
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}]...
Freelunch's user avatar
  • 1,086
5 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)...
Magic is in the chain's user avatar
5 votes
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4th Order Brownian Motion Martingale

Those are the expansion of $$ \exp(\sigma B_t - \sigma^2t/2) $$ in the power of $\sigma$. The general $n$-th order martingale is expressed by the probabilist's Hermite polynomials. The 4th order is ...
jChoi's user avatar
  • 1,174
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 ...
Negative Zero's user avatar
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\...
Gordon's user avatar
  • 21.2k
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 ...
Daneel Olivaw's user avatar
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 ...
Daneel Olivaw's user avatar
4 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)....
Gordon's user avatar
  • 21.2k
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{\...
Gabriele Pompa's user avatar
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 ...
Philipp's user avatar
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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\...
Gordon's user avatar
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4 votes
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How does the inclusion of stochastic volatility in option pricing models impact the valuation of exotic options?

Check out https://www.sciencedirect.com/science/article/pii/S0898122112003215 for barriers, think some searching could yield similar papers for Asian options. In practice this kind of stuff is mostly ...
QuantNero's user avatar
  • 233
4 votes

Can Heston volatility model be used to calculate VaR or CVaR?

Certainly! The Heston model is a well-known model in quantitative finance that describes the evolution of the volatility of an asset. It's a stochastic volatility model, meaning it assumes that the ...
Amit Kumar Jha's user avatar
3 votes

Expected Value of Stochastic Process

If you write the SDE in the integral form everything should be straightforward: $$ X_t = X_0 + \int_0^t a(X_s, s) ds + \int_0^t b(X_s, s) dz_s $$ If you now take the expected value the third term ...
clarkmaio's user avatar
  • 455
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 ...
Emma's user avatar
  • 460
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*}
Gordon's user avatar
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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 ...
Kiwiakos's user avatar
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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 ...
Gordon's user avatar
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3 votes
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Stochastic Calculus problem with three processes? (Itô calculus)

You have $$dZ_t = df\left(S_t, B_t, X_t\right) = \frac{\partial f}{\partial s}dS_t + \frac{\partial f}{\partial b}dB_t + \frac{\partial f}{\partial x}dX_t + \frac{1}{2}\left[\frac{\partial^2 f}{\...
siou0107's user avatar
  • 2,680
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 ...
Kevin's user avatar
  • 16k
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 ...
Question Anxiety's user avatar
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 ...
Jesper Tidblom's user avatar
2 votes

Piecewise Ito formula

Ito for diffusion is local so it holds locally if conditions are local. Let B a open ball of $\mathbb{R}^d $ Let I be a open time interval. Let f be $C^{1,2}(I,B) $. Let $A\subset B $ strictly in $B $...
M. Jeunesse's user avatar
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2 votes
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Merton portfolio allocation problem proportions/weights >1 or <0?

Your statement should be correct, the weights into the risky asset are not bounded between $0$ and $1$. Essentially, by setting $r=0$ you omit the term which shows that your weights always sum up to ...
Stefan Voigt's user avatar
  • 1,466
2 votes

What is meant by innovations in volatility?

Without context, I am not 100% sure but this is my interpretation: Just from wikipedia: In time series analysis (or forecasting) — as conducted in statistics, signal processing, and many other ...
vanguard2k's user avatar
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