Skip to main content

Questions tagged [sde]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
98 views

Deep calibration in the Heston Model

I am doing my master thesis on deep calibration in the Heston Model, and after reading a few academic paper (eg. Horvath et al. 2019) on the subject I understand pretty well the procedure and the ...
sxminho's user avatar
  • 33
0 votes
0 answers
63 views

How to simulate a conditional expectation given a filtration

I had a question regarding how to simulate a certain conditional expectation. I am given two processes $X_1(t), X_2(t)$ which both follow their own SDE, but both are of the form \begin{equation*} dX_i(...
Tipeg's user avatar
  • 1
3 votes
0 answers
94 views

Option pricing boundary condition

I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
Valentin's user avatar
  • 135
2 votes
1 answer
151 views

Pricing PDE of Asian option by Shreve

I am currently working on "Stochastic Calculus for finance II, continuous time model" from Shreve. In chapter 7.5 Theo 7.5.1 he derives a pricing PDE with boundary conditions for an Asian ...
Valentin's user avatar
  • 135
0 votes
0 answers
94 views

Volatility of a stochastic Process given by an SDE

I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and ...
Valentin's user avatar
  • 135
0 votes
0 answers
135 views

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 $dW_{1t}dW_{2t}=\rho dt$....
Mr. Ivan's user avatar
0 votes
0 answers
51 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 ...
lukas kiss's user avatar
0 votes
0 answers
30 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 ...
Valentin's user avatar
  • 135
0 votes
1 answer
161 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 $\alpha_t$ ...
Alessandro's user avatar
0 votes
1 answer
112 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 -{\...
Alessandro's user avatar
2 votes
0 answers
55 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 ...
Rudy S's user avatar
  • 21
0 votes
0 answers
86 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 ...
Marc Allan's user avatar
1 vote
0 answers
129 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 ...
CasMath's user avatar
  • 59
1 vote
1 answer
236 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 ...
JMNQC's user avatar
  • 53
2 votes
0 answers
128 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 ...
UNOwen's user avatar
  • 128
1 vote
0 answers
49 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-...
Nap D. Lover's user avatar
0 votes
3 answers
221 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 ...
Nic's user avatar
  • 1
2 votes
1 answer
119 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 $$\...
Nap D. Lover's user avatar
3 votes
0 answers
77 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}{...
Pefok's user avatar
  • 163
1 vote
0 answers
103 views

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{\...
Pefok's user avatar
  • 163
1 vote
1 answer
59 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{...
John Stevens's user avatar
2 votes
0 answers
59 views

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} \...
John Stevens's user avatar
1 vote
0 answers
100 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 ...
John Stevens's user avatar
4 votes
0 answers
177 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 ...
Khalil Belghouat's user avatar
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 ...
Landscape's user avatar
  • 548
1 vote
1 answer
74 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(...
Siron's user avatar
  • 21
2 votes
0 answers
222 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 ...
Mr Frog's user avatar
  • 253
1 vote
0 answers
61 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 ...
Carl's user avatar
  • 123
3 votes
1 answer
206 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\...
BS.'s user avatar
  • 165
0 votes
0 answers
193 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_{...
Alejandro Andrade's user avatar
1 vote
1 answer
121 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{...
Alejandro Andrade's user avatar
2 votes
2 answers
1k 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 ...
Bohdan_'s user avatar
  • 21
2 votes
0 answers
36 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 $k$...
Thomas's user avatar
  • 21
3 votes
2 answers
2k 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 ...
Aian 's user avatar
  • 31
2 votes
1 answer
116 views

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 \...
robsmith11's user avatar
2 votes
1 answer
496 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 ...
curious123456789's user avatar
1 vote
0 answers
216 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: ...
StefanH's user avatar
  • 201
4 votes
1 answer
469 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 ...
Stochastichelp's user avatar
3 votes
0 answers
84 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. ...
Leguan3000's user avatar
0 votes
0 answers
91 views

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 ...
Ratanna's user avatar
3 votes
1 answer
291 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)=...
Martin_Gale's user avatar
0 votes
2 answers
264 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 $df = f_tdt+f_{B_t}dB_t + \frac{1}{2}f_{B_tB_t}dt$) ...
user13232877's user avatar
1 vote
0 answers
83 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 ...
user13232877's user avatar
0 votes
1 answer
190 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 ...
user144410's user avatar
2 votes
0 answers
144 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 ...
RedLapm's user avatar
  • 33
5 votes
2 answers
360 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: $$...
Jan Stuller's user avatar
  • 6,118
4 votes
0 answers
345 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{\...
fwd_T's user avatar
  • 747
0 votes
0 answers
103 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 ...
Question Anxiety's user avatar
2 votes
0 answers
54 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 ...
Jan Stuller's user avatar
  • 6,118
3 votes
2 answers
812 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 ...
RedZoro's user avatar
  • 33