11 votes

Why Ito calculus?

My understanding is because the Ito's integration definition keeps the martingale property. With Brownian motion $W(t, \omega)$ defined, to define stochastic integration in a Riemann–Stieltjes style:...
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9 votes
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Why Ito calculus?

In fact Ito and Stratonovich calculus are both mathematically equivalent. In the following paper you can e.g. see that both derivations lead to the same result, i.e. the Black-Scholes equation: Black-...
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8 votes

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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7 votes

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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  • 1,021
7 votes
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Understanding the solution of this integral

Let $\tau = T-t$. Then \begin{align*} S_T = S_t e^{(\mu - \frac{1}{2}\sigma^2) \tau + \sigma \sqrt{\tau}\, Z}, \end{align*} where $Z$ is a standard normal random variable, independent of $\mathcal{F}...
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  • 20.5k
6 votes

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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  • 61
6 votes
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Is linear programming important for quant?

To price financial instruments such as options, bonds and stocks must be priced so as to be "arbitrage free". The concept of arbitrage can be made precise by one of the fundamental ideas of ...
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6 votes

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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  • 5,062
6 votes
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Clarify a derivation in Pat Hagan's Convexity Conundrums

Swap Just to be clear, (3.4c) leads to (3.5a) when we assume lognormal $R(\tau)$. Lognormal $R(\tau)$ means we can write $$R(\tau) = R_0 e^{-\frac{1}{2}\sigma^2 \tau + \sigma \sqrt{\tau} Z}$$ with $...
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  • 1,849
6 votes
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Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 4

Under the risk-neutral measure the discounted (under some numéraire) price process is a martingale. If we have a bank account with dynamics $dB_t = r B_t dt$ then the discounted asset $X_t = \frac{S_t}...
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  • 1,048
5 votes

Is linear programming important for quant?

Optimization is definitely important in Quantitative Finance, especially for portfolio optimization where we maximize utility of the return of a portfolio as linear weighted vector of asset returns ...
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  • 5,649
5 votes
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derivation of heston pde in gatheral

1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want ...
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  • 3,836
5 votes

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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5 votes
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How is stock data objectively different to this random walk?

I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is ...
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5 votes
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Put-Call Parity Application

First, we have $P(t)+S(t)=C(t)+B(t,T)\cdot K$, Then, $\frac{\partial P(t)}{\partial S(t)} + \frac{\partial S(t)}{\partial S(t)} = \Delta^{\text{put}}_{t}+1$ and $\frac{\partial C(t)}{\partial S(t)} +...
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  • 106
5 votes
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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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  • 2,426
5 votes
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Formula for conditional expectation. Related to the Fundamental Theorems of Asset Pricing

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{...
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  • 2,362
5 votes
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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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5 votes
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Swap contract comparative advantage

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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  • 173
5 votes
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Finding the process of $X/Y$

You are right about the dropped $\sim$, it's probably just a typo. Furthermore, remember that in stochastic calculus, you have to take into account second order derivatives, i.e. $$d\left(\frac{1}{...
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  • 1,487
5 votes
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How can I express this sum in a easier way?

As you mentioned, we have $$\sum_{l=0}^{k}{p}=\frac{k(k+1)}{2}$$ You want to know $$\sum_{k=0}^{n}{\sum_{l=0}^{k}{p}}=\sum_{k=0}^{n}{\frac{k(k+1)}{2}}=\frac{1}{2}\sum_{k=0}^{n}{k^2}+\frac{1}{2}\...
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  • 753
5 votes

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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5 votes
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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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5 votes

Reflection principle of the Brownian motion

I'll only show it for $M_T = \max_{u\leq T} B_u$ and $(x,h)$-domain $$ \{ h> 0, h > x \}. $$ By the reflection principle we have: $$ P\left( B_T < x, M_T > h \right) = P\left( 2h - B_T &...
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  • 5,038
4 votes
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Distribution of minimum of hazard functions

Note that $$P(X_i >s)= \exp\Big(-\int_0^s \lambda_i(u) du \Big),$$ for $i=1, 2$. Then, $$P(\min(X_1, X_2) >s) = P((X_1>s)\cap (X_2>s)) = P(X_1>s)P(X_2>s) = \exp\Big(-\int_0^s (\...
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  • 20.5k
4 votes

Recommendations for books to understand the math in quantitative finance papers?

Paul Wilmott on Quantitative Finance.
4 votes

Book recommendation: math toolkit for quantitative finance and statistics

I would recommend the books from Steven Shreve. Here is a link to some one of his older online pdf's (1997 but nevertheless true) so you can check if that fits the bill. http://citeseerx.ist.psu.edu/...
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  • 226
4 votes

Risk-Neutral Probabilities, Trinomial Model

Trinomial trees give incomplete markets so there is a range of possible risk neutral prices. So you have to find the possible probabilities that make the tree risk-neutral and see what prices you get. ...
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  • 6,763
4 votes

Perpetual American options

Use Dynkin's formula to write the expectation: $\mathbb{E}[e^{-r\tau} \phi(S_\tau)]= g(S_0)+\mathbb{E}[\int_ 0 ^ \tau (A g -rg) dt]$ where $\phi$ is the payoff. Use the infinitismal generator $A$ to ...
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  • 1,329
4 votes

Understanding the solution of this integral

Another take on the question which uses stochastic calculus [Digression] Assume deterministic and constant rates without loss of generality. Also assume the absence of arbitrage opportunities and ...
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