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The way you do it in the first place is a discretization of the Geometric Brownian Motion (GBM) process. This method is most useful when you want to compute the path between $S_0$ and $S_t$, i.e. you want to know all the intermediary points $S_i$ for $0 \leq i \leq t$. The second equation is a closed form solution for the GBM given $S_0$. A simple ...
I like Richard's answer, but I think we can compute the mean and the variance of $\int_0^T W_t dt$ by ourselves using Ito's lemma. Let $f(W_t, t) = t W_t$. $$d( t W_t ) = W_t dt + t dW_t .$$ Integrating both sides, and re-arranging the terms, we get $$\int_0^T W_t dt = T W_T - \int_0^T t dW_t \, .$$ We'll be using Ito's isometry formula $\mathbb{E} ... 6 The model for the stock is the Bachelier model with the solution $$S(t) = S(0) + \sigma W(t)$$ Thus the law of the stock$S(t)$is Gaussian with mean$S(0)$and variance$\sigma^2 t$. For average process$Z(T)$is thus the average of linear Brownian motion, we can rewrite this as $$Z(T) = \frac{1}{T} \int_0^T S(0) + \sigma W(t) dt = S(0) + ... 5 For completeness, let's restate that the discrete case goes like this:$$\Delta S_t = S_{t+\Delta t}- S_t = \mu S_t \Delta t + \sigma \sqrt{\Delta t} Z_t $$with Z_t \sim \mathcal{N}(0,1). What you are doing in your case (although there is a typo in your formula) is to use the exact solution of the SDE to model the move between two points of S. ... 4 I believe your problem is that you're assuming all Lévy processes are stable with exponent 2. Here is what happens if we try to use your argument: Let X be a Lévy process (that is a martingale, for simplicity). At time t, for any N, we have$$ X_t \sim\sum_{i=1}^N X^i \left(\frac{t}{N}\right), $$with each X^i \left(\frac{t}{N}\right) i.i.d. and ... 4 Okay so I'll take Jase answer and format it properly so that it answers your question and it will be useful for users in the future. For clarity, let me restate the dynamics of the Modified Ornstein-Uhlenbeck model using the more common notation:$$dS_t = \theta (\mu-S_t)dt + \sigma S_t dW_t$$This blog post provides a closed form solution:$$ S_t = S_0 ... 4 Check out these resources: The book Levy Processes in finance. This paper basically enabling you to use any distribution for asset prices: Option Valuation Using the Fast Fourier Transform 4 Hi here are my two cents, It is true that BSDE's framework represents a very powerful theoretical tool to attack abstract problems in mathematical finance. Nevertheless to my knowledge they are very rarely used in practice for at least three reasons. First they are very "unnatural" in their expression (integrating in the future in time and still being ... 3 It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved. Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be ... 3 I think a simple solution is to try to construct a Brownian motion$W_t$through known points (e.g.,$W_0 = W_1 = 0$); it is also known as a Brownian Bridge [ http://en.wikipedia.org/wiki/Brownian_bridge ]. See also question 3 in http://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2012/assignments/assignment4.pdf . 3 Note: There is a typo in your third equations. Instead of$S(u)$it should be$S(t_{i})$and in place of$S(t)$there should be$S(t_{i+1})$. In fact, given$S(t_{i})$we have that $$S(t_{i+1}) = S(t_{i}) \exp\left( (\mu - \frac{1}{2} \sigma^2) (t_{i+1} - t_{i}) + \sigma (W(t_{i+1}) - W(t_{i})) \right)$$ is the exact solution of the SDE. Hence, the ... 3 Here couple points that at least helped to formulate a daily guide for myself: Losses are just what they are, losses. You return tomorrow to play again. But bankruptcy means game over, you are done. Thus such event is to be avoided at all cost. Long-term, equities exhibit positive drift and have outperformed other competing asset classes. However, the ... 2 The Lévy theorem states that the conditions that have to be met for$M(t)$to be a Brownian motion (and hence be normally distributed):$M(t)$, for$t>0$, be a martingale relative to some filtration$F(t), t>0$.$M(0)= 0M(t)$has continuous paths$[M,M](t) = t$for all$t\geq0$; So, to test each condition you simply differentiate your Lévy ... 2 Have you looked at using Laplace in a Monte Carlo simulation? Here is how you price American style options within a MC framework: http://www2.math.uu.se/research/pub/Jia1.pdf and the Longstaff, Schwartz paper: http://escholarship.org/uc/item/43n1k4jb#page-1 Regarding the discretization of a process that draws its random variables from a Laplace ... 2 Your question is interesting because I thought that the only chance with Lévy-processes is to use Fourier-transform approaches (see e.g. Cont,Tankov). But in the paper Option Pricing for Log-Symmetric Distributions of Returns by Fima C. Klebaner· Zinoviy Landsman they consider models, where the log of the price has a symmetric distribution. In Corollary 3.2 ... 2 As far as I can tell, you've essentially written the model that you are concerned with. The only difference is that you would instead have$\theta_{i}$when$s_{t}=i$where$s_{t}$is a latent variable that reflects the probability of being in state$i$. You would also need to include the dynamics that drive the probability transitions as another part of ... 2 Martingale and Markov process are both stochastic processes where the sequences of random variables are not entirely independent, and their differences are: In martingale, the expectation of the next value IS the present value, so this property is sometimes called 'fair game'. In Markov process, the expectation of the next value only DEPENDS ON the present ... 2 a) you can run a Monte Carlo simulation in which you model stock price movements and then you can look at the future pay off as a function of path dependency into which you incorporate your stop losses and take profits. Done this over many iterations you will be able to derive your probabilities. Caveat here is your result will be strongly dependent on your ... 2 The standard method to manage your kind of problem (i.e. dealing with stochastic processes that are note presented or built thanks to a Brownian motion) is to use a measure change. The power of Brownian motion is that you have a lot of representation theorems (Doob-Meyer theorem, Wold theorem, etc) that allows to (thanks to a change of measure or a ... 2 For Itô Processes$dX(t) = \mu(t) \mathrm{d}t + \sigma(t) \mathrm{d}W(t)$you have the result that (under appropriate assumptions which ensure that the local martingale is a martingale, e.g.$E( (\int \sigma(t)^2 \mathrm{d}t )^{1/2} ) < \infty$, etc.):$X$is a martingale$\Leftrightarrow\mu(t) = 0$. So in order to check if a process$X$is a ... 2 I take it you want to do a Monte-Carlo simulation. You just need to decide of an unit of time$dt$and then start simulating the path.$dW_t$is simulated using a random normal value. In Excel$N\left(\mu, \sigma\right)$would be simulated by NORMINV(rand(), mu , sigma). For your Poisson process you just have to simulate random numbers between 0 and 1 and ... 1 As Quartz says it is possible to make non-linear transformations taking into account skew and kurtosis, but this is mostly is limited to univariate processes (one approach for a t distribution is to match moments). For multivariate processes, it is considerably more difficult. A more general solution is to rely on Entropy Pooling. You could take views on ... 1 For the first one absurd reasoning allows you to construct an arbitrage (as r=0) by investing (or short selling according to the sign of$\mu$) at the time where$\sigma$is null, or if you prefer as soon as$t$is in$B$(which is not a Lebesgue negligible set by hypothesis) which is absurd as no-arbitrage holds. The details that remain to be proved is that ... 1 There are many, which are mostly generalizations of the Black-Scholes model (Geometric Brownian Motion). For Equity stocks, the most widely used (IMHO) is the deterministic generalization of Black-Scholes model, the Local Volatility model. Followed by stochastic volatility models such as Heston or SABR, also there is a generalization of the Local volatility ... 1 A very good question. In other words you ask why the central limit theorem does not hold, right? A sum of iid should be somehow normal, right? Looking at the Levy-Kinchin representation we see the Gaussian part, which comes from increments of a continuous process, and the rest from the jumps. So one answer (which is not mathematically rigorous) is the ... 1 If you say stock prices are following GBM then you can say$dS_t = \mu S_tdt + \sigma S_t dW_t$solving which it brings where$\sigma$is volatility and$r$is risk free rate . **EDITED For a Variance Gamma process theta is the deterministic drift in subordinated Brownian motion and sigma standard deviation in subordinated Brownian motion. I ... 1 Let$(\Omega,\mathcal{F},\mathbb{F},\mathbb{\mu})$be a filtered probability space. Market efficiency implies that the stock price process is Markov with$\mathbb{E}[f(X_t)|\mathbb{F}_s] = g(X_s)$for$0 \leq s \leq t$where$f$and$g\$ are Borel measurable functions. It additionally implies that the discounted stock price process is a martingale w.r.t. ...