61
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. ...
- 21k
18
votes
Accepted
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-...
- 15.1k
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 ...
- 481
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 ...
- 14.5k
13
votes
Accepted
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
...
- 14.5k
13
votes
Accepted
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 ...
- 2,402
13
votes
Accepted
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 ...
- 21k
13
votes
Accepted
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 ...
- 15.1k
12
votes
Accepted
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}{...
- 2,402
12
votes
Accepted
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$. ...
- 10.7k
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&...
- 15.1k
11
votes
How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
Let $P$ be the risk-neutral measure. We define the measure $P_S$ such that
\begin{align*}
\frac{dP_S}{dP}\big|_t &=\frac{S_t}{e^{rt}S_0}\\
&=e^{-\frac{1}{2}\sigma^2 t+\sigma W_t}.
\end{align*}
...
- 21k
11
votes
Accepted
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^...
- 21k
11
votes
Accepted
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 ...
- 6,864
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, \...
- 5,949
11
votes
Accepted
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{...
- 15.1k
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 ...
- 2,916
11
votes
Accepted
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 ...
- 6,260
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 ...
- 9,272
10
votes
Accepted
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 ...
- 15.1k
10
votes
Accepted
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^{...
- 216
10
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-...
- 1,644
9
votes
Accepted
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 ...
- 21k
9
votes
Accepted
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 ...
- 14.5k
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 ...
- 111
9
votes
Accepted
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 ...
- 11.7k
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