The tag has no usage guidance.

learn more… | top users | synonyms

5
votes
2answers
85 views

Constructing a Brownian motion from a Simple Random Walk

I'm trying to get my head around how a Brownian motion is formed from a simple random walk. I've seen two similar methods used: Why has one approach used $\frac{1}{\sqrt{k}}$ and the other ...
5
votes
1answer
149 views

justification of square root process

In finance, many stochastic processes $X(t)$ are defined via \begin{equation} dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t \end{equation} with $\gamma = 1/2$ (for instance the Heston model ...
5
votes
2answers
198 views

Reflection Principle

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{W_t ∶ t ≥ 0\}$ be a standard Wiener process. By setting $\tau$ as a stopping time and defining \begin{align} ...
5
votes
3answers
2k views

Stochastic modeling of stock price process

Apart from the model of Geometric Brownian motion is there any other "widely accepted" stochastic model to characterize the dynamics of a stock price process?
5
votes
0answers
82 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends ...
5
votes
0answers
115 views

Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

With respect to the answer by user AFK in Ideas about Stochastic volatility models. I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).
5
votes
0answers
66 views

What kind of errors arise when I fit ARMA(1,1) to data generated from ARMA(1,1)-GARCH(1,1) process?

As far as I know estimates of parameters of ARMA(1,1) are asymptotically optimal when fitted to data from ARMA(1,1)-GARCH(1,1) process, and only their variance increase, so when we assume large ...
5
votes
1answer
531 views

PDF Calculation by Fourier Inversion of Characteristic Function for Affine Intensity Process in Matlab

I'm trying to use the Fourier inversion formula to plot the PDF of an Affine Stochastic Intensity Reduced Form Credit Model, given its characteristic function. The characteristic function of an ...
4
votes
3answers
264 views

Determine $E[W_p W_q W_r]$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let 0 < p < q < r. Determine $E[W_p W_q W_r]$. ...
4
votes
2answers
144 views

What's the name of this nearly-brownian stochastic process?

1) Does the following algorithm (my question is math, not programming-related): ...
4
votes
2answers
174 views

Fractional Brownian motion

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ ...
4
votes
1answer
139 views

Stochastic Differential

Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because $$ \mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta. $$ But am I allowed to actually write ...
4
votes
3answers
667 views

Central Limit Theorem and Lévy processes

Lévy processes are self-decomposable and independent on any non-overlapping interval, so how come the distribution of the process at time T,$\phi(T)$, which is the sum of N i.i.d with law $\phi(T/N)$ ...
4
votes
1answer
148 views

How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
4
votes
1answer
90 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $ ...
4
votes
1answer
90 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
4
votes
1answer
182 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
4
votes
1answer
81 views

Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
4
votes
1answer
140 views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
4
votes
2answers
149 views

Stationary distribution for square root process

Consider the process, $$ dX_t=(-aX_t+b(1-X_t))dt + \sqrt{X_t(1-X_t)}dW_t $$ How do I show that the stationary distribution for the transition density is a beta distribution? I tried expanding the ...
4
votes
1answer
319 views

Quadratic exponential method (by Andersen) in Heston model

I am having trouble understanding the reasons that led Andersen to define his QE scheme to efficiently simulate Heston Stochastic volatility model (you may check the celebrated scheme here). The ...
4
votes
1answer
151 views

The distribution of jump gaps for Levy processes

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
4
votes
1answer
298 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
4
votes
2answers
281 views

Does No arbitrage(NA) imply efficient markets (EMH)?

The EMH states that stocks are traded at its fair values. This means there is no arbitrage strategy in efficient markets. However, if the market is no arbitrage, can we conclude the market is ...
4
votes
1answer
266 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
4
votes
2answers
778 views

Regime switching in mean reverting stochastic process

Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$. This time series could be ...
4
votes
1answer
36 views

Analytical Bond Price under Rendlemen-Bartter?

Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process $$ dr_t = ar_t dt + \sigma r_t dW_t, $$ does anyone know of an analytical bond price formula? We ...
4
votes
1answer
130 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 ...
4
votes
0answers
90 views

How to trade the Ornstein-Uhlenbeck process?

My question comes from this paper, which is a short version of Avellaneda's paper The picture bellow provides a summary of the equations. Do I understand correctly that in order to trade OU process ...
4
votes
0answers
151 views

Finding the dynamics of a dividend paying asset under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
4
votes
0answers
161 views

Is it random walk?

I would like to ask a question about random walk. Campbell, Lo & Mackinlay defined the random walk, in the following way (RW3): $$ cov[f(r_{t}),g(r_{t+k})]=0,\qquad k\neq0 $$ for all $f(\cdot)$ ...
4
votes
0answers
140 views

Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A ...
3
votes
3answers
526 views

How to create a Stochastic Process through pre specified points?

I want to create a random (quasi random) process which goes through pre determined points and constraints. E.g. I have a daily price series but want to generate intra-day prices with the same OHLC ...
3
votes
2answers
221 views

Relationships between white noise and random walk

I would like to ask 5 questions about relations between these processes. 1) Could white noise be also a random walk? 2) Could random walk be also a white noise? 3) Could white noise be stationary? ...
3
votes
3answers
280 views

Show that $E[B_t|\mathscr{F}_s] = B_s$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
3
votes
1answer
79 views

Pricing a log-contract using Monte Carlo

Having a payoff of log-contract defined as $$ \Pi_T = \ln \left(\frac{S_T}{S_0} \right) $$ How would you express the MC-estimator for the price of this contract? The stock price dynamics here is ...
3
votes
1answer
45 views

clarification to log-stock price formula

Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$ what is the log-stock price? Using Ito formula it is ...
3
votes
2answers
93 views

CIR model problem - deriving PDE, Feynman-Kac

I am reviewing a CIR model problem, where $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some constants $ab>\frac{\sigma^2}{2} \quad$ Letting T ...
3
votes
2answers
333 views

Ito's formula for Jump process

Let $\{N_t\,|\,0\leq t\leq T\}$ be a Poisson process with intensity $\lambda>0$ defined on the probability space $(\Omega,\mathcal{F}_t,P)$ with respect to the filtration $\mathcal{F}_t$ and ...
3
votes
1answer
303 views

How to apply Levenberg Marquardt to Max Likelihood Estimation

In this paper on p315: http://www.ssc.upenn.edu/~fdiebold/papers/paper55/DRAfinal.pdf They explain that they use Levenberg Marquardt (LM) (along with BHHH) to maximize the likelihood. However as I ...
3
votes
2answers
78 views

ARMA-GARCH model, bset model selection and confidence levels calculations

I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch. So, 2 main questions. 1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it ...
3
votes
1answer
378 views

open problems in mathematical finance

What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven. I have ...
3
votes
1answer
139 views

Extended CIR and discretization

Did someone know how to discretize this process efficiently : $dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$ I am looking for something more sophisticated than the trivial Euler ...
3
votes
1answer
152 views

What is wrong in this GBM simulation?

I am trying to generate a few samples of GBM using the following very simple MATLAB code: ...
3
votes
2answers
334 views

Transformation to reduce standard deviation without changing median

Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points. Suppose that I was to create an ...
3
votes
1answer
53 views

CIR model - nth moment generation $E^*[r_T^n]$

I am analyzing the nth moment generation process for $r_t$ with dynamics defined by CIR model $r_t$ has following dynamics $$dr_t=a(b-r_t)dt+\sigma \sqrt{r_t} dW_t^* \quad \quad (1)$$ for some ...
3
votes
2answers
31 views

The relation between exchange rate SDE and respective interest rates

The exchange rate between a domestic currency money market and a foreign currency money market can be expressed as $$ dQ(t) = (r_d - r_f)Q(t)dt + \sigma Q(t)d\tilde{W}(t) $$ where $r_d$ is the ...
3
votes
1answer
65 views

Expected value of log-GARCH process

Is there a way to analitycally compute expectation of log-GARCH process? The GARCH(1,1) process: $dU_t = \theta(\omega - U_t) dt + \xi U_t d W_t$ The log-GARCH(1,1) process: $e^{U_t}$ The ...
3
votes
2answers
75 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
3
votes
3answers
207 views

Convergence of GBM mean after simulation?

As a follow up of my previous question, I am now simulating the GBM step by step for $n$ steps. I am using the following implementation for the simulation: $$S_{t+1} = S_t \exp \left[ ...