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5
votes
2answers
169 views

Ito, Stochastic Exponential and Girsanov

This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential. Whilst reading notes relating to Girsanov it is stated that the ...
5
votes
2answers
117 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
1
vote
2answers
98 views

Discounted Stock Price

I have the following Question : Prove that under the risk-neutral probability p the stock and the banjaccount have the same average rate of growth. In other words, if $ S_0 , S_N $ are the initial ...
8
votes
1answer
336 views

Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ...
7
votes
1answer
155 views
+100

Why do people always seek finite-variance models for option pricing

For the purpose of getting fatter tails than the Guassian, I have seen people for example use $\alpha$-stable processes to model the stock. But in that case they end up using 'tempered' versions of ...
2
votes
1answer
46 views

Negative Interest Rate & Basis Models

Since markets are showing negative interest rate, I'm forced to find a model that can catch this behaviour. Because of that, I have implemented and calibrated the G2++ (or the Hull-White 2 Factors) ...
0
votes
1answer
60 views

Derivation o HJB equation

I am trying to derive the HJB equation in a stochastic setting. Let me exemplify my problem with the simplest case where there is no control, just one state variable. Assume the payoff is given by $$ ...
5
votes
0answers
94 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 ...
4
votes
0answers
38 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 ...
4
votes
0answers
124 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
148 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
176 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
42 views

pdf of simple equation, compound Poisson noise

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where: \begin{equation*} dX_t = -aX_t + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
2
votes
0answers
37 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 ...
2
votes
0answers
169 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 ...
2
votes
0answers
144 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) = ...
2
votes
0answers
531 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
171 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 ...
2
votes
0answers
107 views

Difference between kappa and delta in mixed-effects model

(This question is a crosspost from Cross Validated) I have a following stochastic model describing evolution of a process (Y) in space and time. Ds and Dt are domain in space (2D with x and y axes) ...
1
vote
0answers
55 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!
1
vote
0answers
88 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
1
vote
0answers
162 views

Provide a bond pricing differential equation and invoke Feynman-Kac

Grateful for any assistance. Consider the process: $dZ=r(t)Z\,dt$ , where $r(t)$ is stochastic and $Z=Z(r,t;T)$ is a zero coupon bond. Provide a bond pricing differential equation and invoke ...
1
vote
0answers
108 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 ...
1
vote
0answers
45 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
167 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
73 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
41 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
315 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
7 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 ...
0
votes
0answers
69 views

Ornstein-Uhlenbeck / Vasicek and no-arbitrage

I'm working my way through a common question which asks to derive the solution, the mean and the variance to the following Ornstein-Uhlenbeck process: \begin{align} dS_t = (\theta(t) - \beta\,S_t)\,dt ...
0
votes
0answers
39 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
149 views

close form for stochastic integral

I am new to stochastic calculus. Can I know how to compute the close-form solution for $$\int_0^t \exp(\alpha s - \sigma W_s) \; ds$$ and $$\int_0^t \exp(\alpha s - \sigma W_s) \; dW_s.$$ I encounter ...
0
votes
0answers
141 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 ...