Questions tagged [sde]
The sde tag has no usage guidance.
62
questions
0
votes
1answer
55 views
Baxter and Rennie: A question on Notation
On page 56 of Baxter and Rennie (Financial Calculus), we have
The definition of a continuous stochastic process, in terms of the drift $\mu_s$ and volatality $\sigma_s$. Its important to keep in ...
2
votes
1answer
60 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
2
votes
1answer
65 views
Expectation of Stochastic Differential
First of all, I am a mathematician, so I apologize for my ignorance regarding stochastic calculus. What exactly does an expression like:
$$
\mathbb{E}[dX_tdY_t]
$$
here $X_t,Y_t$ are stochastic ...
0
votes
1answer
70 views
Vasicek model and spot interest rate parametrised by reversion rate
By solving an SDE I want to derive the analytical results for mean and variance of the process of extended Vasicek model.
$$
dr(t) = \left(\eta - \gamma r(t) \right)dt + c dX(t)
$$
where $\gamma$ ...
1
vote
0answers
70 views
How to solve these SDE Problems
Quuestion1.
I make a solution $r(t)$ used by Ito's lemma
$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$
Is this right?
and I try to make ...
2
votes
0answers
37 views
How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?
I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112:
I am unable to understand:
How does instability arise when we use the Euler scheme on X(t)?
What change does ...
3
votes
1answer
128 views
How to determine the order of convergence of the Euler-Maruyama method?
To make this simple let us consider the Geometric Brownian Motions.
My questions:
1. How can I show that the Euler-Maruyama Method is convergent using GBM?
2. How can I determine the order of ...
2
votes
1answer
84 views
Expected value of stochastic optimization
I have a optimization problem where the SDE is:
$$
dX(t) = [X(t)(u(t)-\beta(t))+\theta(t)]dt+X(t)u(t)\sigma dW(t), t \in [0,T], X(0) = X_0
$$
where $\beta(t)$ and $\theta(t)$ are deterministic ...
3
votes
1answer
123 views
Expectation in a stochastic differential equation
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
\begin{align}
d(X_t) = \frac{1}{2}X_t dt ...
1
vote
0answers
60 views
Proving Flow Property of Stochastic Differential Equation
I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE:
\begin{equation*}
d X(u)=b(X(u))d ...
1
vote
1answer
143 views
Correlated stock prices and geometric Brownian motion
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
3
votes
0answers
37 views
Bounded solution for a SDE
I have this SDE
$$
dX(t) = [X(t)(u(t)(\delta-r)+r-\beta(t))+\theta(t)(1-\alpha(t))]dt+X(t)u(t)\sigma dW(t), t \in [0,T] \\
X(0) = X_0(1-\alpha(0))
$$
I've checked some books and I find the solution ...
1
vote
1answer
159 views
How to calculate mean and volatility parameters for Geometric Brownian motion?
Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
4
votes
0answers
408 views
Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?
I found this great post deriving the solution to the Merton Jump-Diffusion SDE
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{2}\right)t + \sigma W_t\right)\prod_{j=0}^{N_t}V_j$$
The first part of ...
3
votes
0answers
68 views
Solving BSDE in R
I was wondering how to implement a BSDE approximation in R. For example, if I have the toy BSDE
$$
dX_t = \mu dt + \sigma dW_t ; X_T\sim N(\mu_1,\sigma_1),
$$
for fixed real numbers $\mu,\mu_1,\sigma,...
2
votes
1answer
183 views
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$
dr_t = a(b-r_t)dt + \sigma dW_t
$$
Every solution is proceeding to multiply both sides of the equation by the ...
3
votes
2answers
193 views
SDE for option value
Given an SDE for an underlying:
$$dS(t) = \mu(S,t)dt+\sigma(S,t)dW(t)$$
the SDE for the value of the option $V=V(S,t)$ is given via Ito's lemma as:
$$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
1
vote
1answer
150 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$
equals
$$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
3
votes
0answers
118 views
What is the purpose of short rate models?
Just venturing into quantitative finance and studying short rate models (Vasicek, CIR, Hull-White etc.). Wanted to ask a very simple intuitive question. How would a practitioner use these models? I ...
1
vote
1answer
181 views
Dynamics of LIBOR foward rate under T-forward measure
Assume that under the physical measure $\mathbb{P}$ we have for the LIBOR forward rate $L(t):=L(t;S,T) = \frac{1}{T-S}\left(\frac{P(t,S)}{P(t,T)}-1\right)$ that
$$
\mathrm{d}L(t) = L(t)\left(\mu(t)\...
1
vote
0answers
359 views
Time integral of geometric brownian motion
Suppose $S_t$ is a geometric brownian motion. Then how to understand its time integral, i.e., $Y_t=\int_0^{t}S_udu$?
Is $Y_t$ still a stochastic process?
How to compute the expectation of $Y_t$?
...
2
votes
2answers
412 views
Hawkes process intensity solution
Hail to all,
I am struggling to solve the following SDE for intensity:
$d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t $
I know to expect the solution in the form of
$\lambda_t = c(0)e^{-...
2
votes
1answer
190 views
How to adjust Geometric Brownian Motion to be monotone?
I want to use stochastic process to model subscriber's mobile data consumption as time going in a month. So I think about Geometric Brownian Motion.
However, people's cumulative data consumption ...
0
votes
0answers
90 views
Why do we have to use discretization methods for SDE?
I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
0
votes
1answer
295 views
Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]
Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value:
\begin{align*}
\mathbb{E}(S_T\log(S_T)).
\end{align*}
Using a change of measure we can write
...
0
votes
2answers
453 views
CIR discretization Milstein scheme
The CIR model for spot rate $r_t$ is:
$$dr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$
where $\eta, \gamma, \alpha$ are constants.
How to express this SDE in discrete form using Milstein scheme?
...
1
vote
0answers
138 views
Characteristic function of SDE with coefficients depending upon second coupled SDE
Say we have the following two SDEs driven by the same single Brownian:
$$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$
where $...
2
votes
1answer
384 views
How do you find variance of a sde?
I know how to find the mean of an SDE: write it on integral form, take derivative, solve a simple ODE.
But what to do when we want a variance?
In my case, $$X_{T + \delta t} = X_T + \int_T^{T + \...
1
vote
0answers
80 views
Transformation of coupled forward-backward stochastic differential equations in 3 dimensions with Ito formula
Maybe this is the right place for my question:
I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates):
$$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
3
votes
2answers
164 views
Moment Ito's Process Proof
I have a following stochastic integral - related problem that I have difficulty to solve:
Given
\begin{equation}
dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t
\end{equation}
and the second moment of $...
2
votes
2answers
112 views
How is this SDE interpreted?
I saw this model
$$\frac{dF(t,T)}{F(t,T)} = \sigma(t,T) dW_t + (\exp(e^{-a(T-t)}dJ_t)-1) + \mu_J(t,T)dt$$
to model the forward curve. Rewriting
$$dF(t,T) = \sigma(t,T)F(t,T) dW_t + F(t,T)(\exp(e^{-...
-2
votes
1answer
229 views
How to solve $dX_t = X_t(\sigma_t dW_t + \mu_t dt)$?
Solve the SDE $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$ where $\sigma_t$,$\mu_t$ are deterministic.
Attempted solution
We have $$dX_t = X_t(\sigma_t dW_t + \mu_t dt)$$
Let $f(x) = \log X$, applying ...
1
vote
1answer
301 views
Stochastic differential equation of a Brownian Motion
I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.
Take $W_t$ as a standard Brownian motion and $g(s)$ as some ...
0
votes
0answers
81 views
Approximating an SDE for Volatility Estimation
Consider the SDE
$$
dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t)
$$
where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...
0
votes
0answers
69 views
How are Levy driven SDE simulated?
Do you just use an Euler scheme as before?
E.g. take this process, OU process with a Levy driver.
\begin{equation}
\text{d}V_t = -\lambda V_t\text{d}t + dZ_t
\end{equation}
Do you just have
$V_{...
4
votes
1answer
173 views
Interpreting Units of Short Rate Parameters
I've estimated the parameters for the Vasicek model
$$
dr(t) = a(b - r(t))dt + \sigma dW(t)
$$
and the CIR model
$$
dr(t) = a(b - r(t))dt + \sigma\sqrt{r(t)} dW(t)
$$
to one-year Treasury yield data ...
2
votes
3answers
249 views
Understanding the HJM drift condition's dimensions
In an HJM model the forward rate dynamics follow
$$
df_t(T) =a_t(f_t(T))dt+b_t(f_t(T))dW_t
$$
where $W_t$ is a $d$-dimensional brownian motion, $b_t$ takes values in $\mathbb{R}^{d\times d}$ and $a_t$ ...
1
vote
4answers
843 views
Exploding Libor Rates in Libor Market Model
I have implemented the Libor Market Model in Matlab. When I generate a number of paths, I notice that some of them explode. Does anybody have an idea what could cause this?
I already tried solving ...
1
vote
1answer
624 views
Integration in the Hull-White SDE
I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble ...
11
votes
1answer
687 views
Processes used in quant finance
What are the main stochastic processes (and their SDE) used in quant finance?
For example to model currency prices, stock prices, etc.
7
votes
1answer
365 views
Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?
I'm trying to simulate a process as close as possible to EUR/USD of the ten past years.
I've used a Ornstein-Uhlenbeck process:
$$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$
with the ...
1
vote
1answer
271 views
Geometric Brownian Motion: d(S) vs. d(ln(S))
I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel].
I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (...
2
votes
1answer
440 views
Methods of SDE Calibration
There is somewhere summary of methods that can be used to estimate parameters of SDE?
I currently using MLE and regression due to linear dependence between samples. I searching for something ...
1
vote
1answer
267 views
Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$
Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
1
vote
0answers
33 views
jump-resetted diffusion process
I'm working on a model in which there are two processes, $H$ and $L$, and the final variable to model starts as $H$ and then whenever a jump occurs, an instance of the $L$ processes starts and ...
1
vote
0answers
29 views
Stiffness of numerical methods for SDE
What can I do with stiffness of numerical methods for SDE?
I want to use numerical approach for solving SDE in market's scenarios generation.
Is there any general approach to handle it?
4
votes
1answer
158 views
Computation of Expectation
This question has so long preoccupied my mind.Please help me to solve it.
Question: Assume $X_t$ described by the following stochastic differential equation
$$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
4
votes
1answer
1k views
How to do a Brownian Bridge with quasi-random numbers in the Heston model?
I'm required to use the Euler Monte Carlo method to compute the option price under Heston model settings.
I know from some paper that the convergence is volatile for the Heston model with a plain ...
2
votes
2answers
205 views
Transformation into Martingale
If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes:
\begin{...
3
votes
1answer
78 views
Swapping expectation operator with differential operator
Suppose I have a general SDE
$dx_{t} = \mu dt + \sigma dz_{t}$
Then I can put $E[]$ on both sides to get
$E[dx_{t}] = E[\mu dt] + E[\sigma dz_{t}]$
Now comes the question: I've seen some formulas ...
