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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} ... 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 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 ... 4 The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is $$V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right]$$ where the maximum is taken over all stopping times (exercise strategies$\tau>0$permissible in the contract). With a PDE operator such as you have, the instantaneous ... 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: $$S_t = S_0 ... 3 If you allow X_t to be two dimensional then a model with a stock price X_t^1 and its variance process X_t^2 (stochastic volatility) would fit your definition. In such cases to my knowledge we often don't have a closed form of the density of X_T^1 but in some cases we have a closed form of the Laplace transform. An example is the Heston model. 3 It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved. Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be ... 3 Note: There is a typo in your third equations. Instead of S(u) it should be S(t_{i}) and in place of S(t) there should be S(t_{i+1}). In fact, given S(t_{i}) we have that$$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)= 0 M(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 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 There are a lot of methods for simulating such a process, the real problem here is to preserve positivity of the next simulated step as the Gaussian increment might result in negative value and then a non definite value for the next "square-root" step. An approach that might be suitable to your more general needs is the following where a ... 2 Note that you can understand the \Delta as an "operator" acting on r. So just act on r twice:$$\Delta^2 r_t = r_t - 2 r_{t-1} + r_{t-2}.$$In fact if you write the$r$as a vector,$r = (r_1, r_2, \ldots, r_N)$, then$\Delta$is an$N\times N$matrix with elements$\Delta_{i,j} = \delta_{i,j} - \delta_{i-1,j}$. The AR(2) model can be written as ... 1 Any of a wide variety of local vol models, where (from your equation)$b(\cdot,\cdot)$is some fitted surface, are unlikely to have closed-form solutions for the terminal distribution. Indeed it's well-known that these models tend to have very unusual forward term structures of volatility. As a specific example, take$b(\cdot,\cdot)$to be an approximation ... 1 Saying that you can't analyze something as is does not make it garbage. You can't eat flour "as-is", but that doesn't mean you throw it out. In order to use "standard" analysis tools, you must first transform the series into something compatible. Some examples of such a transformation include k-th order differences or a log transformation. These ... 1 In log space the explicit solution for the density of the first passage time is the Inverse Gaussian Distribution. See, e.g., http://www.springerreference.com/docs/html/chapterdbid/205395.html or the Wikipedia page for the distribution. The only thing that should matter is the interval from the initial state to the threshold, and that is the parameter "a" ... 1 Just work in log-space to get rid of the starting point, then by invariance of BM you only need threshold-X(0) and X(0)=0 is enough to work with at first. In the no-drift case the solution is also invariant if you scale diffusion and threshold simultaneously (Levy dist), therefore you can effectively get rid of one parameter (that is you can reduce to the ... 1 As @Rustam notes, "correlation" of deterministic functions in the sense you describe is a special case of allowing$\mu$and$\sigma$to have a term structure of arbitrary shape. Since the latter is easy to treat, no one bothers with restricted forms of it. Now, there quite a few people who deal with models that let$\sigma$change with$S$. I am thinking ... 1 That's a great question and it is what I always wanted to try to do. I guess I found a solution using PDE approach. Change of numeraire would be more intuitive indeed, but I am not very good in stochastic calculus. The idea is as follows: 1) Let's consider portfolio$\Pi = V(X,Y,t) - \Delta_X X - \Delta_Y Y$. I will found$\Delta_X$and$\Delta_Y$such ... 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 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 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 In addition to local volatility and stochastic volatility models, discontinuous jumps are also an important component of stock price movements that are cannot be properly explained by diffusion models. This article: "Which model for equity derivatives?", gives an overview of the rationale behind discontinuous jumps and their importance for modeling the ... 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 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. ...