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1
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0answers
37 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 ...
3
votes
0answers
156 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 ...
1
vote
1answer
186 views

Examples of non-increasing variance of a time homogeneous Markovian process

This is an edit to the previous question, on stationary process, which was answered by Richard below. Let $x_t$ be a zero mean, time homogeneous Markovian process over time $t$ starting from ...
4
votes
2answers
191 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
1answer
223 views

Closed form european option prices for a variance gamma process with a randomly distributed drift, volatility, and variance rate

Does an option pricing model with a closed form European option price exist that takes into account randomly distributed drift, volatility, and variance rate? I prefer a modification to the variance ...
7
votes
1answer
415 views

How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
5
votes
1answer
184 views

From $AR(p)$ to SDE

Let the Vasicek model to be $$\Delta r_{t}=k(\theta - r_{t-1})\Delta t+\sigma\Delta z_{t}$$ Due to the fact that $$\Delta r_{t}=r_{t}-r_{t-1}$$ if you let $\Delta t=1$, it is easy to see by ...
4
votes
6answers
5k views

Why non-stationary data cannot be analyzed?

Searching online, i found out that non-stationary cannot be analyzed with traditional econometric techniques as in case of non-stationarity some basic model assupmtions are not met and correct ...
1
vote
2answers
483 views

Brownian motion - first passage time

Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters. I am ...
8
votes
1answer
143 views

How to test that a distribution has infinite mean?

I observe a sample from a distribution that I expect to be the hitting time $$\tau = \inf\{t>0| X(t)>a\}$$ where $X(t)$ is a Lévy process with $X(0)=0$ and $a$ is some constant. $X$ is not a ...
1
vote
2answers
132 views

how to extend lognormal model so that $\sigma$ is correlated to $\mu$?

Consider a log-normal model, $dx / x = \mu dt + \sigma dW$, where $W(t)$ is a Wiener process. Let's say $\mu$ and $\sigma$ change with time, slowly, so we note them by $\mu(t)$ and $\sigma(t)$. ...
0
votes
0answers
111 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
2answers
242 views

Change option B&S pricing

Consider a market composed by two stocks whose prices $X$ and $Y$ are given by B&S diffusion $$dX_t= \mu X_t dt+ \sigma X_tdW_t$$ $$dY_t= \mu Y_t dt+ \sigma Y_tdB_t$$ Supposing the market is ...
7
votes
2answers
296 views

Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option: $ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0 $, where $\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S ...
3
votes
1answer
570 views

Monte Carlo simulating Cox-Ingersoll-Ross process

The CIR process is given by the SDE $$ \mathrm dr_t = \theta(\mu-r_t)\mathrm dt + \sigma\sqrt{r_t}\mathrm dW_t $$ where $W_t$ is a Brownian motion. I am interested in finite-difference schemes of ...
0
votes
0answers
125 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 ...
3
votes
0answers
138 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 $f(\cdot)$ ...
14
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2answers
860 views

Parameter estimation of Ornstein–Uhlenbeck and CIR processes

I would like to estimate Ornstein–Uhlenbeck process' parameters via Kalman filter. My process is the following one: $\text{d}x_{t}=\alpha(\theta-x_{t})\text{d}t+\sigma\text{d}W_{t}$ I'm interested ...
3
votes
2answers
278 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
172 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 $- ...
4
votes
1answer
225 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 $- ...
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?
2
votes
0answers
457 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 ...
11
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1answer
309 views

Are BSDE's used in practice?

In the academic applied probability/math finance community, Backwards Stochastic Differential Equations (BSDE's) are extremely popular, and they provide a single framework for several different ...
1
vote
0answers
287 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 ...
4
votes
3answers
508 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 $\phi(T/N)$ ...
2
votes
1answer
131 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. ...
6
votes
1answer
261 views

Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
0
votes
1answer
141 views

how to quantify non-fundamental risk if variance is 100% discounted?

If there's better vocabulary, forgive me. If you were required to ignore variance as risk, how would you quantify non-fundamental risk? Many thanks in advance!
2
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1answer
222 views

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

What are the formulae for d1 & d2 using a Laplace distribution?
4
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0answers
123 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 ...
6
votes
1answer
264 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 ...
5
votes
2answers
777 views

Simulation of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
3
votes
1answer
495 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 ...
7
votes
2answers
2k views

What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?

I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process. $$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$ Is anyone able to calculate the mean ...
6
votes
2answers
784 views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
4
votes
5answers
844 views

How to improve the Black-Scholes framework?

Since the distribution of daily returns are obviously not lognormal, my bottom line question is has BS been reworked for a better fitting distribution? Google searches give me nada. The best dist ...
3
votes
1answer
2k 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 ...
8
votes
3answers
815 views

How to simulate stock prices using variance gamma process?

I want to simulate stock prices with the variance gamma process. The model is given by: $S_T=S_0 e^{ {[}(r-1)T + \omega + z{]}} $ where $S_0= $ starting value $T= $ Time ...
6
votes
2answers
10k views

How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: ...
2
votes
0answers
155 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 ...
3
votes
1answer
295 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)?
2
votes
0answers
102 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) ...
4
votes
1answer
247 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 ...
6
votes
2answers
213 views

How to get an analytic result for option price based on this model?

I defined such a model for stock price (1).... $$dS = \mu\ S\ dt + \sigma\ S\ dW + \rho\ S(dH - \mu) $$ , where $H$ is a so-called "resettable poisson process" defined as (2).... $$dH(t) = ...
5
votes
2answers
2k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
8
votes
1answer
362 views

Is there a closed-form solution for the partial autocorrelation function of a Markov regime-switching process?

Consider a Markov Regime-switching process $X_{t}$ with $k$ regimes represented by $s_{t}$ such that $$X_{t}=\mu\left(s_{t}\right)+\epsilon_{t}$$ and $$\epsilon_{t}\sim ...
3
votes
2answers
145 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 ...
8
votes
3answers
500 views

How to test for and how to simulate price rise/fall asymmetry in the stock market

One of the stylized facts of financial time series seems to be a fundamental asymmetry between smooth upward movements over longer periods of time followed by abrupt declines over relatively shorter ...
8
votes
1answer
131 views

Simulating property price index

I am trying to write a Monte Carlo simulation to calculate risk associated with some property based products. What is the most reasonable stochastic process to model property price index? Do people ...