# 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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### 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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### Is a linear combination of GARCH processes also a GARCH process?

No, a sum of two GARCH processes is generally not a GARCH process. (I am not even sure whether there exists a nontrivial special case where the opposite holds.) By GARCH I mean the classic ...
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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 $\mathcal{... • 2,432 5 votes Accepted ### Do we have a Brownian motion Aside from the independence requirement for the increments, that is, the independence of$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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### Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance

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) &= \...
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### Value of a perfect hedge

Using the values for $\phi$ and $\psi$ that you have derived, \begin{align*} V_0(X) &= \phi S_0 + \psi B_0\\ &= \frac{X(u) - X(d)}{S_1(u) - S_1(d)} S_0 + B_1^{-1}\left(X(u) - \frac{X(u) - X(d)...
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