# Tag Info

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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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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### 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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### 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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### 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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### 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 ...
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### 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 ...

### 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 ...
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### 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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### 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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### 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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### 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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### 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}{\...
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### 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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### 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 ...

### 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 ...

### 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 $B$...
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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 ...
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