5

As stated here, for $f = f(t, x) ∈ C^{1,2}(\mathbb{R}^2)$ a deterministic function and Ito process $$X_t = W_t^2,$$ the stochastic process $$Y_t = f(t,X_t)$$ is an Ito process and we have $$df (t,X_t) = \partial_tf(t,X_t)\,dt + \partial_xf(t,X_t)\,dX_t + \frac{1}{2} \partial_{xx}^2f(t,X_t)(dX_t)^2. $$ Since $$ dX_t = 2W_t dW_t + dt $$ and $$ (dX_t)^2 = ...


4

Answering the title question: Let $f(t,W_t)=W_t^2-t$, then it is easier to derive the dynamics using the "general formula" for Itô's lemma (reference, see eq. 10): $$df(t,W_t)=\frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial W_t} dW_t + \frac{1}{2}\frac{\partial^2f}{\partial W_t^2} dW_t^2$$ where, $$\frac{\partial f}{\partial t} = -1, \...


3

$$ X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$ $$ [X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$ (the author probably intended to use $0.5$ as jump size too)


2

One can use the Euler-Maruyama discretization scheme for CIR, 'fixed' for $v$ positivity, to get: $$ v(t+\epsilon) -v(t)\approx \kappa (\bar{v} -v(t)^+)\epsilon + \omega \sqrt{v(t)^+} (W_v(t+\epsilon) - W_v(t)). $$ So, one approximation of the Brownian increment, when $v(t)$ and $v(t+\epsilon)$ are given, is: $$ W_v(t+\epsilon) - W_v(t) \approx \frac{v(t+\...


1

Given your code, the following will yield what you are after: t,S = generateGBM(TIME_HORIZON/365, DRIFT, ANNUALIZED_VOL, INITIAL_PRICE, 1/365/24/3) As all inputs are annualized, you must also think in units of year fractions: The time horizon is 30 days over 365 days, and the time step size, being 20 minutes, is one year over 365 * 24 * 3 (there are three ...


1

The examples provided by Sin in their article Complications with Stochastic Volatility Models might help to answer your questions. I'm transcribing the abstract below: We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what it is usually ...


1

Edit: This is probably incorrect. The Quadratic Exponential scheme is the best one I have seen as it converges in distribution and is pretty fast, so nice choice there! When $\eta$ is constant you can simplify the integral $$ \int_t^{t+\varepsilon}\eta dW(u)=\eta\int_t^{t+\varepsilon}dW(u)=\eta\left(W(t+\varepsilon)-W(t)\right) $$ In the QE scheme you either ...


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