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### Libor Market Model: numeraire change

I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
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### 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 ...
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### PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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### How to express the Black Derman & Toy Model in a $dr=A\,dt+B\, dW$ form?

The Black Derman & Toy (BDT) model is given by $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t))}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ How can one rewrite the BDT model as $dr=A\,dt+B\, dW$, ...
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### 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 ...
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### FX Rate dynamics

Let's suppose USD/EUR price in USD follows a GBM with $$dS_t = rS_tdt + \sigma S_tdW_t$$ What process does EUR/USD follow in EUR?
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### Differenced Brownian Motion covariance

I am having some difficult showing what the following equals, where $x$ and $y$, $x>y$, distinct times: $\mathbb{E}[\Delta W_x \Delta W_y]$ where each $\Delta W_t = W_t - W_{t-1}$. I have ...
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I am having trouble taking the following limit of CVaR/VaR for a normal distribution as alpha approaches 1: $\lim_{\alpha \to 1} \frac{\mu + \sigma \frac{\phi^{-1}(\alpha)}{1-\alpha}}{\mu + \sigma ... 1answer 70 views ### backward Kolmogorov equations - Markov properties I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ... 3answers 263 views ### Geometric Brownian motion - Volatility Interpretation (in the drift term) A Geometric Brownian motion satisfying the SDE$dS_t = rS_t dt+\sigma S_t dW_t$has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ... 1answer 97 views ### The distribution of jump gaps for Levy processes Assume$X_{t}$is a Levy process with triplet$(\sigma^{2}, \lambda, \nu)$, here$\nu$is the Levy measure of$X_{t}$. Define$\tau_{1},\tau_{2},\dots$be the time gap between the successive jumps ... 1answer 128 views ### Derivation of the Stochastic Vol PDE I'm trying to follow the derivation of the stochastic vol pde for an option price - as given in Gatheral (The vol surface), Wilmott on Quant Finance and many other places. As usual one starts off with ... 2answers 221 views ### How to compute$\mathbb{E} \left[ (W_s + W_t - 2W_0)^2 \right]$? The solution to the SDE $$dx_t= -kx_t dt + cx_t dW_t$$ is $$x_t = x_0 e^{\left(c - \frac{k^2}{2} \right)t}e^{-k W_t}$$ with mean $$\mathbb{E} \left[ x_t \right] = x_0 e^{\left(c - ... 1answer 135 views ### unique equivalent martingale measure in incomplete markets Do you have any idea about how we can prove, and under which conditions, that an equivalent martingale measure (EMM) in an incomplete market is unique? The assumptions we have made are: 1) that the ... 1answer 63 views ### Problems to understand a stochastic DGL equality currently I am reading a paper called "Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model" for self-study reasons. The paper can be found here: ... 2answers 189 views ### Itô diffusion processes in finance with unknown distribution at a terminal value In several papers it is argued that for many Itô diffusion processes,$$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$in mathematical finance the distribution of X_T for fixed T>0 is unknown, which makes ... 1answer 138 views ### What is the stochastic differential of a general semimartingale? By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3:$$H = B + H^c + h(x) ... 2answers 403 views ### Differential equation for log-returns I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian ... 0answers 218 views ### Test for stationarity and make use of non-stationary points in financial market? I have two questions to ask: What are the best methods to determine stationarity in a financial market (such as stocks) using MATLAB? What methods would you recommend to use in order to change from ... 2answers 152 views ### A question on Ito If we know the dynamics of$S$, then we can estimate the value of$S$at a time point,$t$. Here, I have a question concerning how to solve for$S_t$by Itô because I obtained different results by ... 0answers 105 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 ... 1answer 514 views ### Multi Fractals Models From a quant point of view, how would you explain Multi Fractals Models in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am ... 1answer 375 views ### Malliavin Calculus From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing. ... 2answers 289 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 ...
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Here I have this question (i) state Ito's formula (ii) hence or otherwise show that $\int^t_0B_s dB_s = \dfrac{1}{2}B^2_t -\dfrac{1}{2} t$ (iii) define the quadratic variation $Q(t)$ of Brownian ...
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### 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 $1 d$- ...
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### 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 ...
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 $$\nu \left( dx\right) = A ... 1answer 617 views ### Derivation of Ito's Lemma My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ... 1answer 483 views ### Regime switching in mean reverting stochastic process Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an ARMA(p,q). This time series could be ... 2answers 2k views ### What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process? I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.$$d X(t) = a ( L - X_t ) dt + V X_t dW_t Is anyone able to calculate the mean ...
What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let $X_t$ be a process and $\alpha(t)$ a real function. What would be $d(\alpha(t)X_t)$?