# Tag Info

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A stationary process is one where the mean and variance don't change over time. This is technically "second order stationarity" or "weak stationarity", but it is also commonly the meaning when seen in literature. In first order stationarity, the distribution of $(X_{t+1}, ..., X_{t+k})$ is the same as $(X_{1}, ..., X_{k})$ for all values of $(t, k)$. ...

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This is pure speculation: MFE's are really tailored toward valuation models (how can we develop a model to price x swap, etc.). You don't entirely have to worry about those details in order to trade them: you're just quoted a price based on these models. But if you go in-house at a bank and are working as a product quant (structured products, etc.), then ...

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There are many numerical approaches to solving stochastic integrals such as the above. Assuming that there is no closed form slight-of-hand, the easiest approach is the Monte Carlo approach. I would recommend referring to Glasserman's excellent "Monte Carlo Methods in Financial Engineering" If you are not familiar with MC, think of it as evaluating ...

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Baxter and Rennie say it better than me, so I will summarize them. Suppose that $N_t$ is not stochastic and $f(.)$ is a smooth function then the Taylor expansion is $$df(N_t) = f'(N_t)dN_t + \frac{1}{2}f''(N_t)(dN_t)^2 + \frac{1}{3!} f'''(N_t)(dN_t)^3 + \ldots$$ and the term $(dN_T)^2$ and higher terms are zero. Ito showed that this is not the case in the ...

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(1) You analytically solve a stochastic differential equation (SDE) using Ito's lemma. Your second equation (the discretized one) is how you could model one path over one step. To find the solution, you would model many of these paths over many steps and then take the expectation (i.e., Monte Carlo methods). The solution to the SDE models all of these paths ...

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A process is defined here and is simply a collection of random variables indexed (in general) by time. Otherwise I know the concept stated by Shane under the name of "weak stationarity", strong stationary processes are those that have probability laws that do not evolve through time. More formally let $X_t$ be a given process, then let's call $P_X$ the ...

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This is the separable differential equation for simple continuous compounding! See this very accessible article for a step-by-step derivation (esp. under continuous compounding): http://plus.maths.org/content/have-we-caught-your-interest

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If $\alpha(t)$ is of finite variation, then the product rule is the same as in ordinary calculus: $$d(\alpha(t)X_t) = \alpha(t) dX_t + X_t d\alpha(t).$$ If you had $X_t$ and $Y_t$ as processes, you would get $$d(X_t Y_t) = X_t dY_t + Y_t dX_t + d [X,Y]_t.$$ If $Y$ has finite variation, the last quadratic covariation term is zero. The second equation ...

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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} ... 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 $$Z(T) = \frac{1}{T} \int_0^T S(0) + \sigma W(t) dt = S(0) + ... 6 A very excellent discussion of stationarity as it relates to trading can be found in Sherry's (Sherrys'?) Mathematics of Technical Analysis (poorly organized, but very useful book). As he puts it, if the price changes of a stock, etc., are stationary over a time period, the underlying rules generating the price changes are effectively unchanged. The ... 6 The classic argument using risk-neutral pricing is to assume that discounted stock prices are \tilde{P}-martingales where \tilde{P} is the risk-neutral probability measure. Then, you know that$$\frac{S_t}{(1+r)^t}=\tilde{E}[\frac{S_T}{(1+r)^T} | \mathcal{F}_t]$$by definition of a martingale process. As the discounts are non-stochastic, you can ... 6 I think this question has no easy answer but I'll give it a shot anyway (beware: oversimplification ahead!). The main idea of the Malliavin calculus is to be able to differentiate stochastic processes like Brownian motion (or more general martingales with bounded quadratic variation), which are not differentiable in the traditional sense (because of their ... 6 The part where you say that$$\frac{dS_t}{S_t} = d\ln(S_t)$$is wrong, because S is a stochastic variable. This is exactly what Itô tells you with his formula that you apply right do compute your dZ. The difference comes from the quadratic variation of the process S which you express as (dS)^2. If you don't add this term when the variable are ... 6 If you consider X_1 a random variable which is normally distributed with mean \mu and variance \sigma^2 them S_1 = \exp(X_1) is log-normally distributed with mean \exp(\mu + \sigma^2/2) and variance (\exp(\sigma^2)-1)\exp(2\mu+\sigma^2). This follows from the definitions of the normal distribution and the log-normal distribution and deriving the ... 5 In general, if you have a process that you can write under the form F(B_t,t) where F is \mathcal{C}^{2,1} then Itô's lemma gives you the drift term and diffusion term of dF. Then if the resulting SDE has a null drift (that's where Black Scholes PDE comes from), and you get a only local martingale. For it to be a proper martingale you can look at ... 5 I think you should see the hint as follows:$$d(W_t^{n+1})=d(f(W_t))$$with$$f(x)=x^{n+1}$$Apply Ito:$$d(W_t^{n+1}) = f'(W_t)dW_t + \frac{1}{2} f''(W_t) d<W>_td(W_t^{n+1}) = (n+1) W_t^n dW_t + \frac{1}{2} n (n+1) W_t^{n-1} dt$$If you integrate, you get:$$W_{t_2}^{n+1}-W_{t_1}^{n+1}=(n+1) \int_{t_1}^{t_2} W_t^n dW_t+ ... 5 Multi-fractal models can be applied to the modeling and forecasting of volatility. I read the following book with much interest and actually setup couple models in order to compare performance vs Garch family models and the application of multi-fractals much better captures discontinuous regime-changes than traditional volatility models. ... 4 Well the problems where Malliavin Calculus is applicable are mostly regarding greeks of exotic derivatives where some non smoothness in the payoff function creates trouble when trying to get this by finite difference methods. The thing is in my opinion that Malliavin Calculus is only an opening as it gives you basically an infinite number of ways to get ... 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: $$S_t = S_0 ... 4 This is not a finance concept. Augmented data is related to Bayesian inference. It's essentially a way to improve maximum likelihood estimation from incomplete data. For details see the article "The calculation of posterior distributions by data augmentation" by Martin A. Tanner and Wing Hung Wong (it's referred in the paper you are reading). 4 The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is$$ V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right] $$where the maximum is taken over all stopping times (exercise strategies \tau>0 permissible in the contract). With a PDE operator such as you have, the instantaneous ... 3 The standard method to manage your kind of problem (i.e. dealing with stochastic processes that are note presented or built thanks to a Brownian motion) is to use a measure change. The power of Brownian motion is that you have a lot of representation theorems (Doob-Meyer theorem, Wold theorem, etc) that allows to (thanks to a change of measure or a ... 3 Just following Musiela Rutkowski (the link redirects to Amazon). The risk neutral measure is derived form imposing that the present value of a self financed portfolio (i.e.; no infusion or withdraw of money) is a martingale. A portfolio can be seen as a stochastic process where its value at time t is given by$$ V_t = \phi^0_tP_t + \phi^1_tS_t\ , $$... 3 Do his first step first; integrate both sides:$$\displaystyle \ \ \int_0^T \frac{dS(t)}{S(t)} = \mu T - 0 \,\,\,\,\,\,\,\,\,\,\,(1)$$With zero diffusion, we know that \langle S_.\rangle_t = 0. Therefore, by applying Ito's lemma (or actually normal calculus):$$d\ln{S(t)} = \frac{1}{S(t)}dS(t)\,\,\,\,\,\,\,\,\,\,\,(2)$$Sub this into (1): ... 3 1 ) A first-order Taylor expansion gives \ln \left(\frac{S_{t+\Delta_t}}{S_t}\right)\approx \frac{S_{t+\Delta_t}-S_{t}}{S_t}+o(\Delta_t) , thus unless \Delta_t is not small you can drop the residual term and consider Z_t\overset{law}{=}\frac{S_{t+\Delta_t}-S_{t}}{S_t}. 2 ) Calculation of the moments: we can proceed by using the classical Dynkin way ... 3 Hi the forward rate equation is not dependent on the model it is calculated upon the prices of zero coupon bonds by the following equation :$$ P(t,T)=exp{-\int_t^T f_t(u).du} $$If you have a continuum of zero coupon bond prices which are sufficiently smooth then you can deduce from it that :$$f_0(T)=-\frac{\partial Ln(P(0,T))}{\partial T}$$Anyway, ... 3 (1) You can easily solve it in the case of constant coefficients. The answer will be \infty. In fact, this equation has no solution on any interval. The intuition is the following. For the SDE like$$ dS_t = \mu(t,S_t)dt+ \sigma(t,S_t)dw_t $$you can mention that dw_t = \xi_t\sqrt{t}, where \xi_t\sim\mathcal{N}(0,1) are standard gaussian i.i.d. random ... 3 "Like" Ito:$$d (B^2) = B dB + B dB + dB dB$$That is$$B dB = \frac{1}{2} d (B^2) - \frac{1}{2} dB dB$$Integrate. Last term is 1/2 the quadratic variation. I understand the questions as follows: In iii) one has to define what$dB dB$stands for and one has to "proof" the first line in my answer. In ii) one may use Ito to "know" that$dB dB = dt$. 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.

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