Questions tagged [sde]
The sde tag has no usage guidance.
3
votes
0answers
52 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
91 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
131 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:
$$dV = V_tdt+V_S\mu(S,t)dt+\frac{1}{2}V_{SS}\...
0
votes
0answers
38 views
Calibration of financial models when only pricing PIDE is known
This is a rather general question but I have not found much on it on the internet. Suppose I describe the stock price dynamics under the risk neutral measure by some SDE. Suppose that I don't have any ...
0
votes
0answers
81 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
1
vote
1answer
127 views
Two papers - two different solutions of the Ornstein-Uhlenbeck process
Bernal 2016 says that the solution of
$$ 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
74 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 ...
0
votes
0answers
45 views
Solve Poisson-SDE
for an assignment I have to solve stochastic differential equations involving Poisson processes. However, in class we only went from $F_t \rightarrow dF_t$ using Ito formula for Poisson processes, ...
1
vote
1answer
118 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
$$
\mathrm{d}L(t) = L(t)\left(\mu(t)\...
1
vote
0answers
269 views
Time integral of geometric brownian motion
Suppose $S_t$ is a geometric brownian motion. Then how to understand its time integral, i.e., $Y_t=\int_0^{t}S_udu$?
Is $Y_t$ still a stochastic process?
How to compute the expectation of $Y_t$?
...
3
votes
2answers
321 views
Hawkes process intensity solution
Hail to all,
I am struggling to solve the following SDE for intensity:
$d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t $
I know to expect the solution in the form of
$\lambda_t = c(0)e^{-...
2
votes
1answer
159 views
How to adjust Geometric Brownian Motion to be monotone?
I want to use stochastic process to model subscriber's mobile data consumption as time going in a month. So I think about Geometric Brownian Motion.
However, people's cumulative data consumption ...
0
votes
0answers
60 views
Why do we have to use discretization methods for SDE?
I haven't found the answer for the question above in google. Why can't we just discretize the equation instead of using methods like euler or milstein for the discretization.
0
votes
1answer
190 views
Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]
Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value:
\begin{align*}
\mathbb{E}(S_T\log(S_T)).
\end{align*}
Using a change of measure we can write
...
0
votes
2answers
275 views
CIR discretization Milstein scheme
The CIR model for spot rate $r_t$ is:
$$dr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$
where $\eta, \gamma, \alpha$ are constants.
How to express this SDE in discrete form using Milstein scheme?
...
1
vote
0answers
126 views
Characteristic function of SDE with coefficients depending upon second coupled SDE
Say we have the following two SDEs driven by the same single Brownian:
$$ 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$$
where $...
3
votes
1answer
306 views
How do you find variance of a sde?
I know how to find the mean of an SDE: write it on integral form, take derivative, solve a simple ODE.
But what to do when we want a variance?
In my case, $$X_{T + \delta t} = X_T + \int_T^{T + \...
2
votes
0answers
75 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):
$$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
4
votes
2answers
161 views
Moment Ito's Process Proof
I have a following stochastic integral - related problem that I have difficulty to solve:
Given
\begin{equation}
dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t
\end{equation}
and the second moment of $...
3
votes
2answers
109 views
How is this SDE interpreted?
I saw this model
$$\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
202 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
248 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
72 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
67 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
165 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
180 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
655 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
506 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
612 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.
6
votes
1answer
315 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
233 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
379 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
232 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
26 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
150 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
$$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
4
votes
1answer
964 views
How to do a Brownian Bridge with quasi-random numbers in the Heston model?
I'm required to use the Euler Monte Carlo method to compute the option price under Heston model settings.
I know from some paper that the convergence is volatile for the Heston model with a plain ...
2
votes
2answers
193 views
Transformation into Martingale
If $f$ is some function of BV on $\mathbb{R}$ and $dZ_t = f(W_t)dW_t + \mu_t dt$ ($W_t$ is a $1$-dimensional standard Brownian Motion), then what choice of real valued function $F$ makes:
\begin{...
3
votes
1answer
73 views
Swapping expectation operator with differential operator
Suppose I have a general SDE
$dx_{t} = \mu dt + \sigma dz_{t}$
Then I can put $E[]$ on both sides to get
$E[dx_{t}] = E[\mu dt] + E[\sigma dz_{t}]$
Now comes the question: I've seen some formulas ...
1
vote
1answer
162 views
Question about the stochastic differential equation in the Merton model
in the following stochastic differential equation merton model we have $$\frac{ds}{s}=(\alpha-\lambda k)dt+\sigma dW+dq$$
where $\alpha$ is the instantaneous expected return on the stock; $\sigma^2$...
3
votes
2answers
168 views
Why can't I multiply two SDE Solutions?
SDE 1 is
S1 = S10 exp( (r1-sigma^2/2) * dt + sigma dW1 )
S2 = S20 exp( (r2-sigma2^2/2) * dt + sigma2 dW2 )
E[dW1 dW2] = rho
I want to price an option on S1 x S2
I know I need to use the SDE's to ...
2
votes
1answer
97 views
Meaning of w in SDE
I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both
$w$ is one of events that could happen before ...
4
votes
1answer
165 views
Explicit solution SDE
I have the following SDE:
$$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$
where $W_{t}^{...
24
votes
4answers
5k views
Strictly local martingales: what is the intuition behind them?
A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{\tau_k,k=1,2,...\}$ the stopped process is a martingale. All true martingales are local martingales, but the ...
2
votes
1answer
2k views
How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?
The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$
Then the solution is given: $$S_t=S_0\,e^{\...
1
vote
0answers
95 views
Order 1.5 strong SDE integration methods for systems with diagonal additive noise
I'm looking into simple-to-implement and efficient order 1.5 strong SDE integration schemes for my system. My noise is diagonal and additive (possibly time-varying). Thus methods designed for either ...
3
votes
2answers
195 views
What information about the stochastic process is available from path-dependent options?
Assume the stock follows a process, which is defined by the following stochastic differential equation
$$\frac{dS}{S}=r(t)dt+\sigma(S,t)dW,$$
so that the stock price process has local volatility.
...
5
votes
2answers
315 views
Itô diffusion processes in finance with unknown distribution at a terminal value
In several papers it is argued that for many Itô diffusion processes,
$$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$
in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
4
votes
2answers
646 views
Shortcomings of generalized Brownian motion for asset price modelling
I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$
as a model for asset prices.
I know ...
1
vote
1answer
130 views
Bracket-Notation in SDEs
I often come across the following notation in my script, and I have not found it anywhere else. While our lecturer insists it is of utmost importance to write this way in his exams, he yet failed to ...
