Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
584
questions
37
votes
5answers
30k views
Integral of Brownian motion w.r.t. time
Let
$$X_t = \int_0^t W_s \,\mathrm d s$$
where $W_s$ is our usual Brownian motion. My questions are the following:
Expectation?
Variance?
Is it a martingale?
Is it an Ito process or a Riemann ...
30
votes
1answer
2k views
Law of an integrated CIR Process as sum of Independent Random Variables
It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :
$$dY_t= \kappa(\theta -Y_t)...
28
votes
4answers
9k views
What is a stationary process?
How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?
28
votes
1answer
7k views
What is the role of stochastic calculus in day-to-day trading?
I work with practical, day-to-day trading: just making money. One of my small clients recently hired a smart, new MFE. We discussed potential trading strategies for a long time. Finally, he expressed ...
23
votes
2answers
33k views
Worked examples of applying Ito's lemma
In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
23
votes
1answer
802 views
What is the trickiest thing to get right in Rates Quant recently (2019)?
What are the biggest challenges for Rates Quants in 2019? Most quants have been through a lot over the past years-shifting their SABR models in JPY swaptions, fixing the FVA models for negative rates, ...
17
votes
1answer
1k views
Probability distribution of maximum value of binary option?
A binary option with payout \$0/\$100 is trading at \$30 with 12 hours to
expiration.
Assuming the underlying follows a geometric Brownian motion (hence volatility remains constant), what ...
16
votes
8answers
6k views
Geometric Brownian motion - Volatility Interpretation (in the drift term)
A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution
$$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$
Recently ...
16
votes
2answers
2k views
Real world application of stochastic portfolio theory
There is a branche of stochastic portfolio theory (see also this question).
Fernholz and Karatzas have published research in this field (e.g. "Diversity and relative arbitrage in equity
markets") and ...
15
votes
2answers
4k views
Why is Brownian motion useful in finance?
The following is an interview question from Mark Joshi et al. Quant Job Interview.
Question: Why is Brownian motion useful in finance?
I am from a Pure Maths PhD background (functional analysis, ...
15
votes
2answers
796 views
Missing step in stock price movement equations
Assuming a naive stochastic process for modelling movements in stock prices we have:
$dS = \mu S dt + \sigma S \sqrt{dt}$
where S = Stock Price, t = time, mu is a drift constant and sigma is a ...
14
votes
9answers
7k views
Why the expected return rate of a stock has nothing to do with its option price?
OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
14
votes
6answers
5k views
Self-financing and Black-Scholes-Merton formula
Self-financing is an important concept in financial product replicating, normally used in pricing.
I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
13
votes
2answers
10k views
Why is this stochastic integral a martingale?
Suppose that:
$W^*_t$ is a Wiener process under probability measure
$\mathbb{P}^*$ and;
$\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$.
In my lecture notes, it says that $\tilde{S}_t$ is a ...
13
votes
2answers
2k views
Why Ito calculus?
Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
13
votes
3answers
1k views
Deterministic interpretation of stochastic differential equation
In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$...
12
votes
1answer
2k views
Girsanov Theorem and Quadratic Variation
Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation.
Below is the ...
12
votes
2answers
731 views
Filtration and measure change
I asked this question in math stackexchange but to no avail. So i'm trying the luck here.
I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand ...
12
votes
2answers
6k views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
12
votes
4answers
1k views
Solving Path Integral Problem in Quantitative Finance using Computer
I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again.
We know that for option value calculation, path ...
12
votes
1answer
278 views
Probability density function of simple equation, compound Poisson noise
I would like to find the probability density function (at stationarity) of the random variable $X_t$, where:
\begin{equation*}
dX_t = -aX_t dt + d N_t,
\end{equation*}
$a$ is a constant and $N_t$ is a ...
12
votes
2answers
470 views
Deriving the definition of stochastic integrals with respect to Ito processes from first principles
When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
12
votes
1answer
577 views
Transformation of Volatility - BS
I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev
\begin{equation}
\sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}}
\end{...
11
votes
2answers
400 views
Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$
Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
11
votes
2answers
2k views
FX forward with stochastic interest rates pricing
I would like to extend the following question about FX Forward rates in stochastic interest rate setup: "Expectation" of a FX Forward
We consider a FX process $X_t = X_0 \exp( \int_0^t(r^...
11
votes
2answers
4k views
How to use the stock as a numeraire to price a derivative with payoff of the form $(S_T f(S_T))^+$?
I have $\frac{dS_t}{S_t} = rdt + \sigma dW_t$ as usual under the money-market numéraire and I need to price options with payoffs
$$(S_T f(S_T))^+$$
How do I express the stock dynamics using the ...
11
votes
5answers
2k views
Free or Relatively Less Pricey Quant Finance courses online
I am trying to figure out what all online Quant Finance courses are out there which are free or relatively less pricey?
CQF is not less pricey
Financial Engineering course on Coursera - Not so great ...
11
votes
3answers
7k views
How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?
I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
11
votes
2answers
3k views
Distribution of stochastic integral
Suppose that $f(t)$ is a deterministic square integrable function.
I want to show $$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$.
I want to know if the following approach is ...
11
votes
1answer
533 views
Distribution of hitting time of the integrated CIR process
If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be
$$
\tau = \inf\{ u > 0 : X_u \geq B \}.
$$
Can ...
9
votes
4answers
12k views
How to use Itô's formula to deduce that a stochastic process is a martingale?
I'm working through different books about financial mathematics and solving some problems I get stuck.
Suppose you define an arbitrary stochastic process, for example
$ X_t := W_t^8-8t $ where $ W_t ...
9
votes
1answer
735 views
Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
9
votes
1answer
5k views
Multidimensional Ito's Lemma for Vector-Valued functions
Consider the vector of $n$ Ito processes
$$
d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t
$$
where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{...
9
votes
2answers
2k views
Ito, Stochastic Exponential and Girsanov
This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential.
Whilst reading notes relating to Girsanov it is stated that the ...
9
votes
2answers
477 views
Why does Black-Scholes equation hold on continuation region of American Option?
Explanation for Put Option:
$$ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0, $$
where
$\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\...
9
votes
3answers
414 views
Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?
I'm trying to derive the valuation equation under a general stochastic volatility model.
What one can read in the literature is the following reasoning:
One considers a replicating self-financing ...
9
votes
2answers
751 views
Change of measure discrete time
Suppose I have a random walk $X_{n+1} = X_n+A_n$ where $A_n$ is an iid sequence, $\mathsf EA_n = A>0$. How to construct a martingale measure for this case?
8
votes
2answers
4k views
Derivation of Ito's Lemma
My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
8
votes
3answers
5k views
Variance of time integral of squared Brownian motion
I want to calculate the variance of
$$I = \int_0^t W_s^2 ds$$
I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
8
votes
1answer
614 views
How were these SDE derived?
Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)?
\begin{align*}
\frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\
\ln(F_{...
8
votes
1answer
1k views
Ho and lee derivation for short rates model
A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
8
votes
2answers
1k views
Malliavin Calculus
From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing.
...
8
votes
3answers
1k views
Is the average of independent Brownian Motions still a Brownian Motion?
If $W$ and $B$ are independent Brownian Motions (BM thereafter), then the average of $W$ and $B$ is $X_t=\frac{1}{2}(W_t+B_t)$.
Where do I begin to show that indeed it is still a BM?
Also, if both ...
8
votes
1answer
1k views
Multi Fractals Models
From a quant point of view, how would you explain Multi Fractals Models in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing....
8
votes
2answers
416 views
Obtaining characteristics of stochastic model solution
I want to use the following stochastic model
$$\frac{\mathrm{d}S_{t}}{ S_{t}} = k(\theta - \ln S_{t}) \mathrm{d}t + \sigma\mathrm{d}W_{t}\quad (1)$$
using the change in variable $Z_t=ln(S_t)$
we ...
8
votes
2answers
742 views
how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?
I'm working in double exponential Jump-diffusion model (we know as kou model)
with following form , under the physical probability measure $P$:
\begin{equation}
\frac{dS(t)}{S(t-)}=\mu dt+\sigma ...
8
votes
1answer
1k views
How to perform basic integrations with the Ito integral?
From the text book Quantitative Finance for Physicists: An Introduction (Academic Press Advanced Finance) I have this excercise:
Prove that
$$
\int_{t_1}^{t_2}W(s)^ndW(s)=\frac{1}{n+1}[W(t_2)^{n+1}-...
8
votes
2answers
2k views
How do practitioners use the Malliavin calculus (if at all)?
This question is inspired by the remark due to Vladimir Piterbarg made in a related thread on Wilmott back in 2004:
Not to be a party-pooper, but Malliavin calculus is essentially useless in ...
8
votes
2answers
407 views
Anticipating stochastic integral $\int_0^T W_T dW_t$
Using basic techniques from Malliavin calculus it can be shown that
$$
\int_0^T W_T dW_t = W_T^2 - T
$$
As can be seen the above integral is a non-adapted stochastic integral.
We also know using Ito ...
8
votes
1answer
904 views
Girsanov's Theorem - Change of Measure
I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by:
$$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$
...
