# Tag Info

### 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{...
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### Proof that no trading system always wins

At the first glance, what you are asking for is a model admitting arbitrage, so there is a zero chance of losing money and positive chance of yielding profits. Well, many equilibrium models start with ...
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### Book recommendation: math toolkit for quantitative finance and statistics

All the topics you've mentioned are wonderful and shouldn't be eschewed by reading some finance-oriented review book. I recommend these instead. Linear algebra: Hoffman and Kunze and Halmos Set ...
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### How can I go about applying machine learning algorithms to stock markets?

Sorry, but despite being used as a popular example in machine learning, no one has ever achieved a stock market prediction. It does not work for several reasons (check random walk by Fama and quite a ...
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### Preparation for interview: influx of power of the moon

Here's how i'd have at it; * I happen to know these are okay guesses. ** Let's assume it's just the potential energy, and that ...
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Let define $\mathbb{Q}$ and $\mathbb{P}$ two equivalent probabilities on a filtered space $(\Omega,(\mathcal{F}_t)_{t\geq 0})$ Let define $Z_T=\frac{d\mathbb{Q}}{d\mathbb{P}}$ restricted to $\mathcal{... • 2,382 5 votes Accepted ### Do we have a Brownian motion Aside from the independence requirement for the increments, that is, the independence of$X_{s+t}-X_s$and$\mathcal{F}_s$, you can check whether the increment$X_{s+t}-X_s$has the distribution of$N(...
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It is actually rather simple. Lets start with the fixed rate market. A can borrow at 5% while B can borrow at 7%. Simply said, A has a comparative advantage of 2% in the fixed rate market. In the ...
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### Abstract algebra in economics and finance

Yes, I've seen some interesting papers that improve one's insight into how things work, even if it is not clearly applicable to practice. Belal Ehsan Baaquie published several books on applications ...
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### How do you derive this Carr-Madan-like equation?

Equation (11) in Kammeyer and Kienitz' paper is a very well-known and popular option pricing formula. It goes back to the work from Lewis (2001), see Theorem 3.2 in Lewis' paper. Original Formula ...
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### Using crude Monte Carlo

Here's some pseudo code to generate your valuations: ...
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### Intuition behind log return of portfolio = weighted sum of log returns

The above relation really only approximately. If you consider arithmetic retunrs then it is exact. For the approximation you just need to look at the Taylor series of the exponential:  e^x = 1 + x +...
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### Mark Joshi, Quant Interview Question problem 2.34; replicating a digital option on a 4-step symmetric binomial tree

The answer above is only confusing because it is missing the the bet amounts. You have a series of events, you are only allowed to bet on single events. You want to construct something such that the ...
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It is a complete solution. Bearing in mind the SDE verified by $(X_t)_{t \geq 0}$, applying Itô's lemma to compute the (stochastic) differential of $f(X_t)$ yields \begin{align} df(X_t) &= \...