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5
votes
1answer
84 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 ...
0
votes
0answers
48 views

Analytical solution to the Black-Scholes equation with time-dependent volatility

I am stuck with the following exercise and I would appreciate any help with it. I have to calculate the analytical function for the price of a call option given the following process for the ...
1
vote
1answer
91 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 ...
1
vote
1answer
80 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 ...
0
votes
0answers
48 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
17 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 ...
0
votes
0answers
19 views

CIR model, realistic parameters and usage

I'm currently working on SDE's, in particular with mean-reversion processes like CIR and Vasicek. The definition of the CIR model is \begin{equation} dX_t = \kappa(\theta-X_t)dt + \sigma ...
1
vote
0answers
17 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
132 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
278 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
144 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: ...
3
votes
1answer
54 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
109 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; ...
3
votes
2answers
135 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
82 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
125 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 ...
10
votes
3answers
1k 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 ...
1
vote
1answer
494 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: ...
1
vote
0answers
63 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
158 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
221 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 ...
2
votes
2answers
294 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 ...
0
votes
0answers
101 views

Is there a strong solution to $\frac{dS}{S}=\sigma(S)dw$?

Does someone know if there is a strong solution for this SDE : $$\frac{dS_t}{S_t}=\sigma(S_t)dW_t$$ where $$\sigma(S)=\begin{cases} 1\;\;\;S>1\\2\;\;\;S\leq 1 \end{cases} $$ $S_0=1$ and $W_t$ is ...
1
vote
1answer
117 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 ...
3
votes
2answers
225 views

Geometric Brownian Motion with non-negative random increments

I am attempting to model a cumulative time-series of a positive integer variable across independent entities. The cumulative series appears to follow a process of Geometric Brownian Motion (GBM) based ...