All Questions
Tagged with stochastic-processes stochastic-calculus
312
questions
0
votes
0
answers
77
views
Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion
Now, I'm given an SDE $$dS_t = 2S_t\,dt + 4 S_t\,dW_t$$ which I need to find the solution of. I have the solution on paper, but I want to know if there's any way I can solve this directly in python. ...
- 101
1
vote
0
answers
94
views
Do we model stock prices using non-Markovian processes in continuous setting?
In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
- 183
2
votes
0
answers
346
views
Testing the fit of an Ornstein-Uhlenbeck process
I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE:
$$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$
where
$\lambda > 0$ is the mean-reversion ...
- 135
3
votes
0
answers
49
views
Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale
I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that:
The existence of an equivalent martingale measure $Q$ on the discounted price ...
- 459
2
votes
1
answer
157
views
COS Method and existence of density
Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I ...
- 71
3
votes
0
answers
117
views
Explicit form for forwards Feynman-Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
- 53
-1
votes
1
answer
619
views
How can I learn stochastic process & stochastic calculus in two weeks? [closed]
I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
- 11
0
votes
0
answers
128
views
SDF derivation by a stochastic process
I have a stochastic process to model the stochastic discount factor (SDF) with M:
\begin{equation}
dM_t = aM_tdt + bM_t d Z_t
\end{equation}
where, $Z_t$ is a standard brownian motion. How do I show ...
1
vote
1
answer
232
views
What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$
Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
- 459
0
votes
1
answer
523
views
Calibrating the Ornstein-Uhlenbeck process with an additional parameter
Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
- 135
0
votes
0
answers
118
views
What is the difference between the geometric brownian motion and cumulative product of percentage returns?
I wonder why the following code: one using GBM and the other using cumulative product of normally distributed percentage returns slightly different values.
...
- 17
4
votes
0
answers
185
views
Summary of Stochastic Derivatives, Integrals, Expectations, and Variances
I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
- 113
2
votes
0
answers
141
views
Is it possibile to use Ito Formula here?
I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$.
Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
- 33
6
votes
3
answers
1k
views
Expectation of exponential of 3 correlated Brownian Motion
Consider,
are correlated Brownian motions with a given
I want to calculate the,
,
I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
- 73
0
votes
1
answer
190
views
Pairs trading by transforming two cointegrated series into a mean-reverting process?
I am slightly confused about the following.
Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they ...
- 135
4
votes
1
answer
250
views
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
- 347
1
vote
0
answers
54
views
Moments of a SDE: a detail on the information set
Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let
$$dS_t = \mu S_t dt + \sigma S_t dz_t.$$
When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
- 2,386
5
votes
2
answers
301
views
Can a Process with a Stochastic Drift be a Martingale?
I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts?
Suppose I have a process with a stochastic drift:
$$...
- 5,513
1
vote
2
answers
890
views
Integral of the square of Brownian motion using definition of variance
Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
- 121
2
votes
1
answer
333
views
Discounted price process - martingale
I have a process $S_{t}=S_{0}e^{\left(r-q\right)t+mt+X_{t}}$, where $X_t$ is a Levy process and I want to check for which $m$ the process $e^{-(r-q)t}S_t$ is a martingale. The third condition of a ...
- 423
1
vote
0
answers
282
views
First Hitting Time and Monte Carlo simulation
I am interested in implementing a Monte Carlo simulation in Python of a first hitting time (first passage time) of an Ornstein-Uhlenbeck process (or similar). Specifically interested in fatter tails ...
- 135
3
votes
1
answer
410
views
Correlated Stochastic Processes
Let say, I have 2 stochastic processes:
$$\begin{align}
dS_1 &= \left( r - q_1 \right)S_1 dt + \sigma_1 S_1 dW_1
\\
dS_2 &= \left( r - q_2 \right)S_2 dt + \sigma_2 S_2 dW_2
\end{align}$$
The ...
- 245
0
votes
0
answers
49
views
Why can't we ignore the second term in Taylor Expansion in Ito's lemma? [duplicate]
Why can't we neglect the $dt$ there?
$$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
- 1
1
vote
1
answer
150
views
How can the increments of a CIR process be derived?
For a CIR process, which has SDE
$$
dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t
$$
how can I derive the increments over the discrete time-interval from $r_t$ to $r_{t+1}$?
- 175
3
votes
0
answers
50
views
Characteristic function of time-changed Levy processes
Let $X_t$ be a Levy process, and $Y_t$ be a subordinator i.e. process with nondecreasing trajectories. I have to find characteristic function of $X_{Y_t}$. I know that I have to calculate:
$$E[e^{iuX_{...
- 43
4
votes
1
answer
609
views
Mean Reverting Heston Model?
Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity ...
- 143
1
vote
1
answer
239
views
Covariance of mean-reverting Vasicek process?
I am dealing with a mean-reverting Vasicek process defined as:
\begin{equation}
S_t = S_0 e^{-at} + b(1-e^{(-at)}) + \sigma e^{(-at)} \int_{0}^{t} e^{(-as)} \ W_t
\end{equation}
I want to ...
- 25
1
vote
1
answer
252
views
Transition density of geometric Brownian motion with time-dependent drift and volatility
Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE
...
- 113
2
votes
0
answers
262
views
Correct application of Feynman Kac formula
I have a question on Feynman-Kac formula but can I ask the community if I have done it correctly? If no, may you point out to where I went wrong? Thanks!
The original FK formula states: Assume $f(t,x)$...
- 184
0
votes
1
answer
108
views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
- 1
1
vote
0
answers
101
views
Option that never expires
I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
- 127
2
votes
3
answers
1k
views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
- 5,513
4
votes
0
answers
128
views
Why is the Schöbel-Zhu model affine?
In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.
The characteristic function of the stock process can be found by arguing that the model is ...
- 41
2
votes
0
answers
52
views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ S_t := S_0 + \int_{h=t_0}^{h=t}\alpha(\mu -S_h)dh + \int_{h=t_0}^{h=t}\sigma S_h dW(h) $$.
I actually don't know how ...
- 5,513
4
votes
0
answers
70
views
Confused about discretization
I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf
The following is my confusion.
The paper has the following continuous time model for the price ...
- 479
2
votes
0
answers
31
views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
- 499
1
vote
1
answer
99
views
Covariation of Ito semimartingales
If we have two Ito semimartingales over $[0,T]$:
$$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$
What is the relationship between
$$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \...
- 288
2
votes
2
answers
182
views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
- 342
1
vote
1
answer
125
views
Taking Expectation of Stopping Time and Integral Manipulation
Consider a stopping time $\tau$ that represents the point in time when the first credit event (e.g. default) occurs on a compact interval $[0,T]$.
Consider the expectation of the indicator function, $\...
- 313
4
votes
0
answers
104
views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
- 288
3
votes
1
answer
302
views
The most general conditions under which Ito lemma holds
Prompted by a question that came up in the comments here, namely why we can apply the Ito lemma to a function of the form $f(x)=(x-K)^{+}$, I would be interested in knowing what are the least ...
- 576
3
votes
0
answers
291
views
Rigorous proof of Dupire formula (e.g. using Gyöngy's theorem)
Where can I find a rigorous proof of the Dupire formula (for example, using using Gyöngy's theorem)? I imagine this would be covered by a paper or by a standard financial math text, but I could not ...
- 576
3
votes
1
answer
134
views
Under which conditions the given random process is martingale and under which submartingale?
Let $a_t $ be adapted to the filtration random process $a_t: P\{\int _0^T|a_t|dt < \infty \} = 1 $ and $ b_t \in M_T^2. \quad$ Under which conditions the random process $$X_t = exp\{\int _0^ta_sds+\...
- 133
2
votes
0
answers
37
views
Differentiation of value function in perpetual american option
I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.
$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t $
where $...
- 51
2
votes
1
answer
270
views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
- 21
1
vote
1
answer
89
views
Properties of integrated GBM
(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
- 157
0
votes
1
answer
269
views
Stochastic Interest Rates in Option pricing
My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
- 411
2
votes
1
answer
732
views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
- 385
3
votes
1
answer
440
views
Difference between $W_t$ and $X_t= \sqrt{t}Z$
$W_t$ is a brownian motion and $X_t:= \sqrt{t}Z$, where: $Z\sim N(0,1)$.
How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
- 63
1
vote
1
answer
99
views
Derivation of stock price formula John C. Hull 9th Ed p309
It says assuming a no-uncertainty Weiner process that models stock price:
$$
\Delta S = \mu S\Delta t
$$
Can be rearranged to (after taking the limit of $\Delta t \to 0$...
$$
\frac{dS}{S}=\mu dt
$$
...
- 113