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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{...
Kevin's user avatar
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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}...
Gordon's user avatar
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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}...
Freelunch's user avatar
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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 ...
will's user avatar
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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 ...
Richard Hardy's user avatar
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{...
M. Jeunesse's user avatar
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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(...
Gordon's user avatar
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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 ...
MH.Q's user avatar
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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}{...
Raskolnikov's user avatar
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Examples of discrete math and graph theory within quantitative finance

There is a huge strand of literature on graph theory in finance which analyzes networks summarized here: Allen, F., and A. Babus (2009): “Networks in Finance,” in Network-based Strategies and ...
skoestlmeier's user avatar
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5 votes

Do quants need to know bloomberg terminal and VBA?

(these are just my random opinions. Sorry, I expect many people to disagree with some of them.) Bloomberg makes no effort to make its terminals available to students - quite the opposite. Hence lots ...
Dimitri Vulis's user avatar
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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}\...
Canardini's user avatar
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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 ...
Dimitri Vulis's user avatar
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 ...
Kevin's user avatar
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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 &...
ir7's user avatar
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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 ...
user9403's user avatar
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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 ...
Quantuple's user avatar
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4 votes
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taylor expansion of PnL

$\require{cancel}$ $$\text{PnL} = -[P(t+\delta t,S+\delta S)-P(t,S)] + rP(t,S)\delta t + \Delta(\delta S - rS \delta t + q S\delta t)$$ Assuming a pure diffusion, at the order 1 as $\delta t \to 0$ $$...
Quantuple's user avatar
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4 votes
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Using crude Monte Carlo

Here's some pseudo code to generate your valuations: ...
will's user avatar
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4 votes
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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 +...
Richi Wa's user avatar
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4 votes
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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 ...
will's user avatar
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4 votes
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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) &= \...
Quantuple's user avatar
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4 votes
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Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 6

I don't have Joshi's book with me. But I guess you can use Feynman-Kac, right? It says if X(t) follows the stochastic differential equation: $dX(u) = \beta(u, X(u))du + \gamma(u, X(u))dW(u)$ and $g(...
Lipton's user avatar
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4 votes
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Approximation of Forward Rates in discrete time

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\...
Gordon's user avatar
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4 votes

Examples of discrete math and graph theory within quantitative finance

A famous example of using graph theory in finance is the detection of triangle arbitrage by finding a negative cycle in a graph. More on this problem and the solution on Math.SE.
Bob Jansen's user avatar
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4 votes
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How to derive Black-Scholes equation with dividend?

We assume that the stock price process $\{S_t,\,t>0\}$ satisfies, under the real-world probability measure $P$, an SDE of the form \begin{align*} dS_t=S_t\big((\mu-q)dt+\sigma dW_t\big), \end{align*...
Gordon's user avatar
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4 votes
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Question on the use of a limit in a proof

It makes no sense to write $C^h \to e^{\delta C}$ as $T \to \infty$ when $C = I_K +\Lambda/T$ since $e^{\delta C}$ on the right-hand side depends on $T$. What can be confirmed is $(C^h)_{k,k} \to e^{\...
RRL's user avatar
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3 votes
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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)...
Gordon's user avatar
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3 votes
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Implied Expected Stock Return from European Option Prices

Note that \begin{align*} \frac{S_T-S_t}{S_t} &= \frac{S_T-K +K-S_t}{S_t}\\ &=\frac{(S_T-K)^+-(K-S_T)^+ +K-S_t}{S_t}. \end{align*} Then, \begin{align*} E\left(\frac{S_T-S_t}{S_t} \mid \mathcal{...
Gordon's user avatar
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