Questions tagged [sde]
The sde tag has no usage guidance.
118
questions
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121
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Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$
Two stocks are modelled as follows:
$$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$
$$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$
with $dW_{1t}dW_{2t}=\rho dt$....
- 33
0
votes
0
answers
36
views
Modelling support and resistance using sde
This initiative was sparked by the identification of cointegrated pairs, fitting them to an OU process, and devising an optimal strategy based on the OU process—areas that have already been well ...
- 129
0
votes
0
answers
23
views
State space equation of CARMA(p,q) processes
Thanks for visting my question:)
I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by:
for $p>q$
the ...
- 43
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votes
1
answer
95
views
Maximizing the expected log utility
Let's assume that we have a self-financing portfolio made by $\delta_t$ shares and $M_t$ cash, so that its infinitesimal variation is:
$$
dW_t = rM_t \, dt + \delta_t \, dS_t
$$
We define $\alpha_t$ ...
0
votes
1
answer
67
views
Solving the SDE for GBM [closed]
Let's assume that we have the following stochastic differential equation:
$dX_t = \mu X_t dt + \sigma X_tdW_t$
and that we have to prove that this is its solution:
$X_t = X_0 \exp\left(\left(\mu -{\...
2
votes
0
answers
49
views
Benchmark Model for Path-Dependant Monte Carlo Simulations?
As part of my research for my masters thesis, I'm testing out the effectiveness of some different models in Monte Carlo simulations for path dependant options.
I will be attempting a model-free ...
- 21
0
votes
0
answers
72
views
Affine Jump Diffusion
I'm currently looking into affine jump-diffusions. I would like to get to know the literature better and I know the paper by Duffie, Pan, and Singleton (2000) is a very celebrated paper. Although I ...
- 107
1
vote
0
answers
108
views
Stochastic volatility with jumps [closed]
I'm reading the Duffie, Pan, and Singleton (2000) paper now and I've stumbled upon something that seems to me as an inconsistency. Whenever I look up the SVJJ model, I see that its log-transform is ...
- 59
0
votes
1
answer
164
views
How calculate expectation and variation of stochastic integral Based on Heston model?
I was calculated Heston volatility model. But I think it is wrong.
$dS_t = \mu dt + \sqrt V_t dW_t^s$
$dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$.
$dW^s_t dW^v_t = \rho dt$
take integral to ...
- 43
2
votes
0
answers
121
views
Is the time derivative of asset returns expressible as an SDE?
Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$,
\begin{equation}
\mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right),
\end{equation}
which (in Langevin form) may ...
- 128
1
vote
0
answers
48
views
References for path-dependent GBMs or continuous time analog of discrete time filters
Consider a path-dependent GBM model for a stock price:
$$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$
where $\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$ are previsible path-...
- 225
0
votes
3
answers
194
views
Simulate from a SDE where drift and diffusion terms are matrices using Yuima in R
I'm trying to implement an SDE in R using Yuima. Most of examples and literature show how to implement and how the math works for SDE where drift and diffusion terms are scalar. What if I want to ...
- 1
2
votes
1
answer
92
views
Filtering SDE for Heston Volatility
Consider a GBM model with Heston volatility:
$$dS_t = \mu S_t dt + \sqrt{V_t} S_t dB_t^1$$
$$dV_t = \kappa(\theta-V_t)dt+\xi \sqrt{V_t}dB_t^2,$$
where $(B_t^1, B_t^2)$ is a correlated BM. Let
$$\...
- 225
3
votes
0
answers
67
views
Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
- 163
1
vote
0
answers
95
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Feymann Kac for multidimensional pde
I Have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
- 163
1
vote
1
answer
58
views
Dynamics of discounted prices (multi-dimensional)
My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
- 61
2
votes
0
answers
59
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Munk (2011) exercise 3.6
I'm trying to solve the exercise in Munk (2011). The exercise reads:
"Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
- 61
1
vote
0
answers
97
views
Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
- 61
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162
views
optimal stopping time problem
I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
5
votes
2
answers
1k
views
Dynamics of FX rate
I've see a couple of places where a FX rate, denoted $X$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ...
- 548
1
vote
1
answer
68
views
SDE linear combination of stock price
Assume that $X_t$ is a process with dynamics $dX_t = \sigma X_t dW_t$ is where $W_t$ is a standard Brownian motion. Given two deterministic functions $p(t)$ and $q(t)$, compute $\mathbb{E}[p(t)X(t)+q(...
- 21
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votes
0
answers
178
views
How did Bachelier characterize the Brownian motion?
The model for a stock price
$$
dS_t=\mu dt + \sigma dB_t
$$
where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
- 221
1
vote
0
answers
59
views
Is there a closed form solution to the following system of SDEs?
Suppose we have the system
\begin{align}
dr_t=\alpha_r(x_t-r_t)dt+\sigma_rdW_t^r\\
dx_t=\alpha_x(\bar{x}-x_t)dt+\sigma_xdW_t^x\\
\end{align}
As this system is affine, I believe there should be an easy ...
- 123
3
votes
1
answer
203
views
Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
Assume $(X_t)_{t\geq 0}$ follows an SDE of the form:
$$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$
where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
- 165
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votes
0
answers
180
views
Simulating sum of squared brownian motions process
I'm trying to simulate the following stochastic process:
\begin{equation}
R_t = \sum_{i=1}^nB_{i,t}^2
\end{equation}
which has the following dynamics:
\begin{equation}
\begin{aligned}
dR_t = \sum_{...
1
vote
1
answer
114
views
Show that the solution to a SDE is strong
I have the following SDE
\begin{equation}
dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t
\end{equation}
that has the solution:
\begin{equation}
\begin{aligned}
X_t = \frac{X_0 + B_t}{1+t} = \frac{...
2
votes
2
answers
991
views
Solving SDE using integration factor and Ito's lemma [closed]
I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
- 21
2
votes
0
answers
31
views
Existence of the solution for SDE with Gaussian Process
I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by
$\dot{x}=f(x),\quad f(x)\sim GP(0,k(x,x')),$
where $f$ is a Gaussian Process with kernel $k$...
- 21
3
votes
2
answers
1k
views
How to incorporate momentum in Ornstein Uhlenbeck to capture overshooting in financial markets?
In modelling asset prices, it is a good idea to model it using a fair value or target price concept. Recently Carr & Prado explored this idea to find optimal stop loss/take profit levels when the ...
- 31
2
votes
1
answer
112
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Confused by derivation of variance swap payoff
I'm trying to follow
https://en.wikipedia.org/wiki/Variance_swap#Pricing_and_valuation
where it seems to me that they're just subtracting a simple return:
$$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
- 71
2
votes
1
answer
387
views
Finding Option Probability Density Using Local Volatility from Dupire Model
This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price?
I also asked this on Math ...
1
vote
0
answers
175
views
Programming the Milstein method and computing the increments
In the wikipedia article on the Milstein method, the following python code to simulate a geometric Brownian motion is presented:
...
- 201
4
votes
1
answer
387
views
Weak solution of a SDE
$\text { Consider the } \operatorname{SDE} d X_{t}=\operatorname{sign}\left(X_{t}\right) d t+d B_{t} \text { on } 0 \leq t \leq T, \text { where } \operatorname{sign}(x)=1\\
\text { for } x>0 \text ...
3
votes
0
answers
79
views
Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
- 161
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0
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86
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What is the difference between "stochastic" heat equation and just heat equation?
I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
- 1
3
votes
1
answer
266
views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$dX(t)=rdt+aX(t)dW(t),\ t>0$$
$$X(0)=x$$
where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor
$$F(t)=...
- 129
0
votes
2
answers
224
views
SDE Exmaple (no drift) [closed]
Assume, $X_t := ∫^t_0e^{μs}dB_s$ ($B_s$ is Brownian motion)
My Reference Said $dX_t = e^{μt}dB_t$.
I tried to Ito's formula to solve this (that is $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$)
...
- 303
1
vote
0
answers
82
views
Geometric Brownian Motion SDE
I recently saw the clip : GBM which quantpie made.
here is the link https://www.youtube.com/watch?v=98xF6b0PZpo
In the time 1:33 of the clip, it naturally said $dX^2 = σ^2X_t^2dt$
To proof this it ...
- 303
0
votes
1
answer
184
views
Differentiability of solutions of a stochastic differential equation
I would like to clarify a confusion I have.
It is well known that a Wiener process (Brownian motion) is nowhere differentiable. I have no difficulty in understanding that. But I am wondering about the ...
- 101
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0
answers
143
views
Is it possibile to use Ito Formula here?
I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$.
Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
- 33
5
votes
2
answers
342
views
Can a Process with a Stochastic Drift be a Martingale?
I have repeatedly come across the statement that "a process with a drift cannot be a martingale". Is this true also for stochastic drifts?
Suppose I have a process with a stochastic drift:
$$...
- 5,998
4
votes
0
answers
279
views
Angular bracket notation (physics)
In a few papers I have seen the following notation:
$$
\langle X_t \rangle
$$
Also, in Bergomi's book, at page 8, we have the following equality:
$$
\biggr\langle \int_0^T e^{-rt}s^2 \frac{d^2P_{\hat{\...
- 747
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votes
0
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93
views
Alternatives to Lognormality for negative Prices
If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
- 342
2
votes
0
answers
52
views
Solution to Stock Price SDE with mean reversion [duplicate]
Suppose $S_t$ follows the process (notice the $S_t$ term in the diffusion part):
$$ 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) $$.
I actually don't know how ...
- 5,998
3
votes
2
answers
699
views
SDE Jump-Diffusion
If you combine the compound Poisson process with the Brownian motion you obtain the simplest case
of a Jump-diffusion. Let’s define
$$X_t = \mu t + \sigma W_t + J_t$$
where $W_t$ is a Wiener process ...
- 33
2
votes
0
answers
32
views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
- 509
3
votes
1
answer
385
views
Application of Ito's Lemma in expected utility theory
An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests
A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
0
votes
0
answers
228
views
Dynamic programming and Bellman equation to obtain the maximum
This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM"
Suppose an endowment economy where the representative ...
- 193
2
votes
1
answer
882
views
Idea of using logarithm for solving SDE in Black-Scholes model
In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$ dS = \mu S dt + \sigma S\ dW $$
I was wondering, was it common at the time they work on this ...
- 385
1
vote
3
answers
719
views
How can I prove that the solution to the Heston SDE is a Markov process?
Consider the Heston model expressed as
\begin{align}
dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\
dV_t &= \kappa(\theta - V_t)dt + \...
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