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63 votes

Integral of Brownian motion w.r.t. time

This type of integral has appeared so many times and in so many places; for example, here, here and here. Basically, for each sample $\omega$, we can treat $\int_0^t W_s ds$ as a Riemann integral. ...
Gordon's user avatar
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18 votes
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Why is Brownian motion useful in finance?

Brownian motion is simply the limit of a scaled (discrete-time) random walk and thus a natural candidate to use. It is very intuitive and arguably one of the simplest and best understood time-...
Kevin's user avatar
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15 votes

What is the trickiest thing to get right in Rates Quant recently (2019)?

Of course making money is always the key issue. That (not completely facetious) comment aside: On the practical side, in many firms IT is struggling with being clear, transparent, and intuitive in ...
Patrick S Hagan's user avatar
14 votes

Integral of Brownian motion w.r.t. time

Just to add to the already nice answers, the result can also be obtained using the (stochastic) Fubini theorem. \begin{align} \int_0^t W_s ds &= \int_0^t \int_0^s dW_u\, ds \tag{$W_s=\int_0^s ...
Quantuple's user avatar
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13 votes
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Girsanov Theorem for Quanto/Compo adjustment

Assume deterministic and constant interest rates. For an investor in the foreign economy i.e. a market participant that can only trade assets delivering a payout in the foreign currency, let us define ...
Quantuple's user avatar
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13 votes
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Can I always use quadratic variation to calculate variance?

Quadratic variation and variance are two different concepts. Let $X $ be an Ito process and $t\geq 0$. Variance of $X_t$ is a deterministic quantity where as quadratic variation at time $t $ that ...
M. Jeunesse's user avatar
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13 votes
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Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let \begin{align*} Y_t = e^{(a+\frac{c^2}{2})t-cW_t}. \end{align*} Then \begin{align*} dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right]. \end{align*} Moreover, \begin{align*} d(X_tY_t) &= Y_t ...
Gordon's user avatar
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13 votes
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Intuition for Martingale Representation Theorem

Let me give my intuition as a former Electrical Engineer. This is going to be very sloppy. Suppose you have a Brownian Motion with increments (or "noise term" in EE language) $dB_t$. ...
nbbo2's user avatar
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13 votes
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Clarification on Deriving Ito's Lemma

Just a few notes How to make sense of $\text dW_t$ is the entire point of stochastic calculus. It's far beyond the scope of any answer here. You should read some introductory lecture notes/books on ...
Kevin's user avatar
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12 votes
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Correlation coeffitiont between two stochastic processes

if you talk about correlation then: compute expectation: $$\mathbb{E}(W_t)=0\text{ and }\mathbb{E}(\int_0^tW_d ds)=0$$ variance: $$\text{Var}(W_t)=t\text{ and }\text{Var}(\int_0^tW_s ds)=\frac{t^3}{...
M. Jeunesse's user avatar
  • 2,422
12 votes

Heston stochastic volatility, Girsanov theorem

Consider the Heston (1993) model under the real world measure ($\mathbb{P}$) \begin{align*} \mathrm{d}S_t&=\mu^\mathbb{P} S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}^\mathbb{P}, \\ \mathrm{d}v_t&...
Kevin's user avatar
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11 votes
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How were these SDE derived?

From $(2)$, \begin{align*} \ln S_t &=\ln F_{t, t} \\ &= \ln F_{0, t}-\frac{1}{2}\int_0^t\sigma^2 e^{-2\lambda (t-s)}ds+\int_0^t \sigma e^{-\lambda(t-s)} dB_s\\ &=\ln F_{0, t}-\frac{\sigma^...
Gordon's user avatar
  • 21.1k
11 votes
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What is an adapted process

Let $\{X_t\}$ be a stochastic process and $\mathcal{F}$ be a filtration. The intuitive idea is that for $\{X_t\}$ to be adapted, it can't reveal what's unknowable (according to the filtration). By ...
Matthew Gunn's user avatar
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11 votes

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Note that the Ito integral of a deterministic integrand $f: \mathbb{R}_+ \rightarrow \mathbb{R}$ is normally distributed \begin{equation} \int_0^t f(u) \mathrm{d}W_u \sim \mathcal{N} \left( 0, \...
LocalVolatility's user avatar
11 votes
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Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$

By construction, the Itô integral, $I_t=\int_0^t X_s\text{d}W_s$, is a martingale if $\int_0^t \mathbb{E}[X_s^2]\text{d}s<\infty$. The martingale property, $\mathbb{E}_s[I_t]=I_s$ implies $\mathbb{...
Kevin's user avatar
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11 votes

Expectation of exponential of 3 correlated Brownian Motion

You need to rotate them so we can find some orthogonal axes. A simple way to think about this is by remembering that we can decompose the second of two brownian motions into a sum of the first ...
StackG's user avatar
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11 votes
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Expectation of exponential of 3 correlated Brownian Motion

Besides @StackG's splendid answer, I would like to offer an answer that is based on the notion that the multivariate Brownian motion is of course multivariate normally distributed, and on its moment ...
Kermittfrog's user avatar
  • 6,628
10 votes
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FX forward with stochastic interest rates pricing

The formula $F^X(t,T) = E_t^d\left(X_T \right)$, under the domestic risk-neutral measure, is problematic. Note that, at time $t$, the forward exchange rate $F^X(t,T)$, for maturity $T$, is the ...
Gordon's user avatar
  • 21.1k
10 votes

Why is Brownian motion useful in finance?

Physical objects move according to simple smooth curves that can be represented by low order polynomials: a straight line, a parabola, an ellipse, etc. Financial market prices move in a completely ...
Alex C's user avatar
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10 votes
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Periodic functions when determining No Arbitrage price

It is, of course, possible to price such a contract in a no-arbitrage market. Indeed, if $f$ is a sufficiently smooth function, then you can price all contracts paying $f(S_T)$. Note that your ...
Kevin's user avatar
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10 votes
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Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

First, let's check if these models are abritrage free. The first fundamental theorem of asset pricing says that if there exists an equivalent probability measure under which $\frac{S_t}{\beta_t} = e^{...
user6247850's user avatar
9 votes
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Girsanov Theorem, Radon-Nikodym Derivative backward

The result you're looking for is $$ \left. \frac{d\Bbb{P}}{d\Bbb{Q}}\right\vert_{\mathcal{F}_t} = \left( \left. \frac{d\Bbb{Q}}{d\Bbb{P}}\right\vert_{\mathcal{F}_t} \right)^{-1} $$ This is a result ...
Quantuple's user avatar
  • 14.6k
9 votes

How to compute $E[W(T)\exp(W(T)]$

Hereunder is how I would solve that. I would say this is some sort recurring exercice in probability classes at university. Solution based on the derivation of the characteristic function $e^{\lambda ...
Thomasunny's user avatar
9 votes
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Book/reference to practice stochastic calculus and PDE for interviews

You may like: Probability and Stochastic Calculus Quant Interview Questions by Ivan Matić, Radoš Radoičić, Dan Stefanica 150 Most Frequently Asked Questions on Quant Interviews, Second Edition by Dan ...
Dimitri Vulis's user avatar
9 votes

Three mathematical mistakes in Black-Scholes-Merton option pricing?

Don't take this as an answer per se, but as mentioned in my comment more a summary of imo Bjork's clear explanation that hopefully can convince you there is nothing wrong with the BS PDE and self-...
Frido's user avatar
  • 1,854
8 votes
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Ito Formula for Stochastic Integral

A stochastic differential equation is nothing more than a short-hand notation for a corresponding integral equation. So the initial SDE you provided actually means $$ \int_0^t d S_u = \int_0^t \mu(...
Olaf's user avatar
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