stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

learn more… | top users | synonyms

2
votes
3answers
132 views

How to understand nonrandom/random process in Shreve book?

I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given ...
2
votes
2answers
163 views

How to price a stock under Q and stochastic interest rates?

I am interested in pricing a stock under $\mathbb{Q}$ when I assume that $$dS(t) = \mu(S(t))dt + \sigma(S(t))dW(t)$$ where $W(t)$ is a Wiener process under $\mathbb{P}$ and $$dr(t) = a(b-r(t))dt +...
3
votes
1answer
79 views

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

The probability that an arithmetic Brownian motion process $dt = \mu dt + \sigma dW$ hits an upper Barrier $U$ before it hits a lower barrier $L$ is given by $$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
2
votes
1answer
100 views

Mean Crossing for Ornstein-Uhlenbeck

Suppose we have classic Ornstein-Uhlenbeck process. How can we calculate expected number (and variance too) of crossing mean value over the certain period of time? Say, if we have discrete OU process ...
9
votes
0answers
124 views

2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
6
votes
0answers
87 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 $n-...
6
votes
0answers
122 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).
6
votes
0answers
99 views

Transition densities in the Heson model

Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
6
votes
0answers
69 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
0answers
108 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
159 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
162 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 $f(\cdot)$ ...
3
votes
0answers
54 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq 0},\...
3
votes
0answers
126 views

Is there a countably infinite Sigma-Algebra? Why?

Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and if $A_1,A_2,...\in\...
3
votes
0answers
378 views

Law of a geometric brownian motion first hitting time (formula dont match Monte Carlo Simulation)

I posted this question before on MSE I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all ...
3
votes
0answers
194 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
2
votes
0answers
76 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
2
votes
0answers
28 views

Heston Model Maximum Return Distribution

What is the joint probability distribution of the maximum of the return between time $0$ and $t$ and the return at $t$, for the Heston model, when the return drift is $0$ and the correlation between ...
2
votes
0answers
75 views

Risk Neutral Variance Gamma

In the risk neutral version of the Variance Gamma model the stock dynamics are $S_T=S_0 e^{ (r-q+\omega)t + X(t;\sigma,\nu,\theta)}$ with $\omega=\frac{1}{\nu}ln(1-\theta \nu - \frac{\sigma^2 \nu ...
2
votes
0answers
133 views

Multivariate Itô's lemma

Hey guys I'm looking for worked examples who show how to apply Itô's lemma in several variables, starting from the very basics. Thank you in advance!
2
votes
0answers
296 views

How to price zero coupon bonds with the Monte Carlo method?

Im trying to calculate monthly ZCB bond prices with a fixed maturity T, over a period of months via Monte Carlo methods. Here is my attempt: For the first month, the price is $P_{t_0}(0,T) = E[exp(-...
2
votes
0answers
685 views

Does the geometric Ornstein-Uhlenbeck process have stationary variance?

I know that the long run variance of the standard OU process is $\lim_{s\rightarrow \infty}\mbox{Var}(P_{t+s}|P_t) = \frac{\sigma^2}{2\theta}$ I'm using the geometric version of the process. I ...
2
votes
0answers
190 views

Measure change in a bond option problem

This is not a homework or assignment exercise. I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
1
vote
0answers
65 views

Estimate Volatility process

How can I estimate the process $\sigma_{t}$ given in the following paper: Spot volatility estimation for high frequency data. J. Fan, Y. Wang. Does anyone have an idea? Free source Edit: Iam very ...
1
vote
0answers
33 views

The best process for foreign exchange rate

I have a simple research project and I need to explain a behavior of a foreign exchange rate. Could you propose a stochastic process without jumps so that it could be estimated with QMLE? Is GBM ...
1
vote
0answers
47 views

How can we observe volatility smile from the market. Drawbacks of Heston Stochastic Volatility Model

Here are two questions related to implied volatilities. a) The set up here is for an European option. We can get its implied volatility smile from calibration, the question is why could we also ...
1
vote
0answers
25 views

Arbitrage and completeness in multiperiod model?

Given a 2-period market with above stock price process along with a riskfree stock with a return of 5%, how do I determine whether the market is arbitrage-free and complete when I only have knowledge ...
1
vote
0answers
78 views

SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{...
1
vote
0answers
43 views

On the construction of a Brownian motion from a Gaussian process

Let $X$ a Gaussian process defined by $$ X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] $...
1
vote
0answers
33 views

How to perofrm a simple GARCH simulation example?

How is it possible to simulate one million of tick data for, say EUR-USD price, using a GARCH model? For example, how do I simulate $X_i$ for $i = 1 \dots 1000000$, with $\text{mean}(X)=X_0 \...
1
vote
0answers
42 views

Avellaneda/Cont model Order Book Model

The model given in the following paper by Avellaneda et al http://people.stern.nyu.edu/jreed/Papers/limitorder.pdf On page 7 he explains that the initial Bid and Ask size should be normalised by ...
1
vote
0answers
26 views

Soft: Interpretation Fractional BM in finance

Suppose we are in the BS framework. If we replace the Brownian Motion with a more general fractional Brownian motion therein, how can it be interpreted? That is what is a financial interpretation of ...
1
vote
0answers
74 views

State of Art - Nelson Siegel Modeling

My idea is to work with dynamic Nelson Siegel models(DNS) on my master's thesis. As I am finishing undergraduation this year I started researching on the subject. I wonder what is being discussed in ...
1
vote
0answers
14 views

Is there any theoretical work to find an optimum size for the size of horizon in finite-horizon optimization or control?

we learn a lot about finite and infinite horizon control in dynamic programming. but I was wondering if we want to minimize the cost per time(discrete time) is there any work to find the optimum size ...
1
vote
0answers
122 views

In what kind of stochastic process Ito's lemma is adopted?

I have been told that Ito's lemma serves as the stochastic calculus counterpart of the chain rule. And yet again my tutor mentioned it is not used for all stochastic processes. Is this statement true?...
1
vote
0answers
51 views

Intensity Function of Stochastic Processes

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
1
vote
0answers
235 views

one-step-ahead Stochastic Volatility for 5-minute VWAP prices

I'm trying to run an SV model against prices of Euro/USD. For those not familiar with SV, its a volatility model in which each point gets its own volatility parameter $h_t$ with 3 main parameters that ...
1
vote
0answers
83 views

Max Likelihood via Marquardt Optimisation

I asked a related question here: How to apply Levenberg Marquardt to Max Likelihood Estimation I tried the approach suggested it works for some of the parameters but not the variances. I spoke to ...
1
vote
0answers
47 views

Weak convergence of Lookback payoff with correction term

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
1
vote
0answers
392 views

Call options portfolio: what would the underlyings' moments to be maximized?

Let you have only three underlyings, like SPY, TLT and GLD, and you want to buy $n_{1}$ Call options on SPY, $n_{2}$ Call options on TLT and $n_{3}$ Call options on GLD... with a limited budget, that ...
0
votes
0answers
13 views

Replicant portfolio with commissions (Jarrow Rudd)

I have created a Jarrow Rudd three for a call option that I know how to replicate with a portfolio. A replicating portfolio of a option works this way: At time 0 we form a replicating portfolio ...
0
votes
0answers
51 views

Spread Return and Mean Reversion Model

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2399915 The above paper proposes an interesting method for modeling credit spreads. I have tried to implement it in R but keep obtaining unrealistic ...
0
votes
0answers
63 views

Stochastic Integration

I have the following derivation question: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and ...
0
votes
0answers
50 views

For a square-root process (CIR), how to verify the characteristic function of the transition density?

I am trying to solve a financial mathematical question. I derived PDE (a) for the characteristic function as follows. But, I don't know how to verify the following characteristic function of the ...
0
votes
0answers
100 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 ...
0
votes
0answers
63 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 ...
0
votes
0answers
53 views

Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...
0
votes
0answers
42 views

Law of one price in continuous time

The law of one price (i.e. for assets $S^{(i)}$ and $S^{(j)}$, $S^{(i)}_T = S^{(j)}_T $ almost surely implies that $S^{(i)}_t = S^{(j)}_t $ almost surely for all $ 0 \leq t \leq T$) is known to hold ...
0
votes
0answers
152 views

Exact value of mean reversion rate knowing terminal value of the process

Let you have the following mean reverting process: $\text{d}x_{t}=a(\theta-x_{t})\text{d}t$, where the diffusion term is absent, that is this process is not stochastic. Let you know the value of $\...