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4
votes
1answer
3k views

How to simulate a Merton Jump Diffusion process?

I am talking about the Merton Jump Diffusion model, on this page, where they give the following formula: $$ dS_t = \mu S_t dt + \sigma S_t dW_t + (\eta-1) dq$$ where $W_t$ is a standard brownian ...
4
votes
1answer
276 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
1answer
226 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
258 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
1answer
119 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
80 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 dt + d N_t, \end{equation*} $a$ is a constant and $N_t$ is a ...
4
votes
0answers
57 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
129 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
151 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
135 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
511 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
3answers
240 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
114 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 ...
3
votes
2answers
113 views

Fractional Brownian motion

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ ...
3
votes
2answers
179 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
6answers
346 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
3
votes
1answer
93 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
93 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.
3
votes
1answer
130 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 ...
3
votes
1answer
146 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
307 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
3answers
171 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[ ...
3
votes
1answer
275 views

Monte Carlo for MultiFactor Ornstein Uhlenbeck

I'm following loosely the exposition given in "Monte Carlo Methods in Financial Engineering by Glasserman. For a multifactor OU process: $dX(t)=C(b-X(t))dt+DdW(t)$ Where C and D are d*d matrices ...
3
votes
2answers
157 views

What mathematical characteristics are required from the asset price process in order to stay within the RNP framework?

I'm currently doing a course in derivatives pricing and I'm having some trouble wrapping my head around the sweet spot where theory meets reality in terms of Risk Neutral Pricing. I know that the ...
3
votes
1answer
598 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 ...
3
votes
1answer
331 views

How to calculate probability of touching a take-profit without touching a stop-loss?

How to calculate probability of touching a take-profit without touching a stop-loss (no-dividend stock, infinite time)?
3
votes
1answer
81 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) ...
3
votes
1answer
160 views

Normalized price process $Z(t)=\frac{\Pi(t)}{B(t)}$

If an interest rate model with the following $P$-dynamics for the short rate. $$dr(t)=\mu(t,r(t))dt+\sigma(t,r(t))d\bar{W}(t)$$ Now consider a $T$-claim of the form $\chi = \Phi(r(T))$ with ...
3
votes
1answer
119 views

Solving the Jamshidian Zhu (1997) PCA short rate model

This is my first time posting a question. I have very limited experience in the field of stochastic calculus and interest rate modelling. I have been tasked with implementing the short rate model ...
3
votes
0answers
107 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 ...
3
votes
2answers
255 views

Arbitrage and dominant strategies

If there is no arbitrage there is no dominant trading strategy, but there may be arbitrage opportunities even if there are no dominant trading strategies. Could you explain this statement and bring ...
3
votes
0answers
178 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 ...
3
votes
1answer
184 views

Foward-start option pricing

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 is generated by $1 d $- ...
2
votes
2answers
131 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? ...
2
votes
2answers
63 views

Binomial representation of stochastic processes

It is common knowledge that a random walk can be represented in the form of a binomial process. Is it possible to represent any generic stochastic process (including non-linear) of the form ...
2
votes
3answers
166 views

How to estimate parameters of geometric brownian motion with time-varying mean?

Does anyone know how to estimate $A$, $\sigma_1$,$\sigma_2$ from the following system? $$dx = \mu_t x dt + \sigma_1 x dB_x$$ $$d\mu = A(\bar\mu - \mu) dt + \sigma_2 dB_\mu$$ Variation in $x$ could ...
2
votes
1answer
205 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 ...
2
votes
1answer
99 views

Distribution of minimum of hazard functions

Suppose I have two random variables, $X_1$ and $X_2$, that are independent (but not identically distributed) and assume both have hazard functions $\lambda_1(s)$ and $\lambda_2(s)$, for $s > 0$. ...
2
votes
1answer
79 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 ...
2
votes
1answer
43 views

Underlying Sample Space in Continuous Market Model

E.g., a model for $N$ stocks might have each follow a GBM $dS_i = \mu_i S_i dt + \sigma_i S_i dW_i$, where each $W_i$ is independent of the others. Letting $(\Omega, \mathcal{F}, P)$ be the ...
2
votes
1answer
80 views

Martingale Measure for Vasicek process

First, under Black-Scholes we have the usual method to transform the discounted asset price into a martingle: Let the asset price $S_t$ be goverend by $$ dS_t = \mu S_t dt + \sigma S_t dW_t, $$ so ...
2
votes
2answers
111 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
2
votes
2answers
60 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 ...
2
votes
1answer
99 views

CIR model: is the short rate really non-central $\chi^2$ distributed?

Probably simple question. Consider the CIR (1985) model for interest rates $$ dr = k(\theta - r)dt + \sigma \sqrt{r}dz $$ Then it is known in closed form the conditional pdf $f(r(s),s|r(t),t)$ ($s ...
2
votes
1answer
136 views

Help with integrating stochastic calculus expression from yield curve model

I am very rusty on stochastic calculus, and I am having trouble integrating the following simple term from a yield curve model: $$z(t)=\int_0^t\exp(-k(t-s))dW(s)$$ Any suggestions appreciated.
2
votes
1answer
115 views

Ito integral approximation by Euler?

I was wondering how to find the solution of the following stochastic integral: $$dY_{t}=a(W_{t},Y_{t})dW_{t}+b(W_{t},Y_{t})dZ_{t}$$ or in integral notation ...
2
votes
1answer
228 views

Variance of Multi-Dimensional OU process

I'm trying to implement this model shown here: http://www.sciencedirect.com/science/article/pii/S0304407611000388 As part of the modelling process I have to calculate the unconditional variance of X ...
2
votes
1answer
138 views

American Option price formula assuming a logLaplace distribution?

What are $d_1$ and $d_2$ for Laplace? may be running before walking. When I tried to use the equations provided, the pricing became extremely lopsided, with the calls being routinely double puts. ...
2
votes
1answer
232 views

What are $d_1$ and $d_2$ for Laplace?

What are the formulae for d1 & d2 using a Laplace distribution?
2
votes
1answer
47 views

martingale decomposition problem

Let $G_{t}$ be a filtration and $M_{t}$ a $G_{t}$-martingale. Why do we have this decomposition: $H_{t}=\mathbb{E}[H|G_t]=\int_{0}^{t}h_{s}dM_{s}+R_{t}$ where $R_{t}$ is a martingale orthogonal with M ...