Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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Motivation: Stochastic Interest rate model

what is a reason that someone might be interested in a stochastic-interest model such as the Chen model? Also can you provide me with a link to an easy to read motivational paper/part of a paper on ...
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34 views

Figuring out parameters for a geometric brownian motion [closed]

So i have this problem: And this is my solution so far: Isolating the μ make sense to me. According to wikipedia, the formula is: E(S(t)) = S(0) * exp(μ*t). But what I am not sure about, is can you ...
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35 views

Why can't we ignore the second term in Taylor Expansion in Ito's lemma? [duplicate]

Why can't we neglect the $dt$ there? $$df = f'(B_t) dB_t + \frac{1}{2} f''(B_t) dt$$
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Time-changed Levy processes

in different articles the authors use the CIR process to change the time in different processes. They mostly use the CGMY, VG, NIG etc process, but I haven't noticed anybody using the Kou process. ...
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Price of a Forward Contract

I have the following, Let ${F_t,t\geq0}$ be the price process of the forward contract on the risky asset with maturity $T' > 0$. Since interest rates are deterministic, we have $$F_t=S_t\ e^{r(T^\...
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59 views

Arbitrage free pricing of option to trade stocks

Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
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24 views

Ito's formula with a random jump measure

Suppose all processes and functions defined are nice enough such that all the following definitions make sense. On a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a filtration $\...
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19 views

Statistical test for comparing two different speed of mean reversion parameters for CIR model

I am trying to compare two different values of speed of mean reversion parameter for CIR model. I would like to know if there exists a statistical test for comparing these two parameters. the estimate ...
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36 views

Stochastic differential equation of Avellaneda model

I was reading this paper and page 14 the model is given. I'm trying to find the steps to get to the SDE given. OI is the open interest, E is the elasticity of demand. $$ \frac{\Delta S}{S} ∼ E·Q^p \...
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98 views

Stochastic Interest Rates in Option pricing

My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
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44 views

Compo/Quanto Adjustment & Multivariate Ito

Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below: By exploring StackExchange, I noticed the ...
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Hedging a short position in the Lookback Option

SOLUTION I got the correct answer using this formula $X_2(HH)=(1+r)*[X_1(H)-\Delta_1(H)*S_1(H)]+\Delta_1(H)*S_2(HH)$ $(1+0.25)[2.24-(.06667*8)]+0.06667*16=3.20$
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Deriving coupling equation(s) for Heston Stochastic Volatility Model

In Bergomi Smile Dynamics (2003) Section 2.1 we are given the following coupled equations for the mean and for the variance of the hedger's portfolio: $ \begin{align*} \frac{dm}{dt} + \mathcal{L}m - ...
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23 views

Order of expectation versus expectation of order (error terms in Taylor expansion)

Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as $$ E[F(X)] = F(a) + E[ O((X-a))] $$ Under what ...
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55 views

Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework

Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model $$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
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Call Option on the Square of a Log-Normal: Process of Underlying under Stock Measure and Risk Neutral Measure

I'm working on some quant interview questions from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). Here are the questions from the bookd, and the answers ...
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67 views

Not clear on an SDE solution example on YouTube [closed]

This video, from about 6 to 12 minutes: https://youtu.be/qdbkvD4N-us I feel like I’m following him ok, but then at the end his f(t,B(t)) has become an f(t,x) and there is no B(t) in his result, so it ...
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453 views

Trouble understanding jump part in Kou double exponential jump diffusion model

I am trying to work with Kou's double exponential Jump-diffusion model and simulate a price path in a programming language. So the dynamics of the asset price in Kou's model follow: \begin{equation} ...
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37 views

Variance Equations is missing definition

here: https://www.nrc.gov/docs/ML1208/ML12088A329.pdf Campbell, Lo, Mackinlay: The Econometrics of Financial Markets on page 159 i am looking at equation 4.4.9 in the last line, = $I\sigma_{\...
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98 views

Approximating an SDE for Volatility Estimation

Consider the SDE $$ dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t) $$ where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so ...
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145 views

When $E[f(\alpha,X)] = f(\alpha, E[X])$

When $E[f(\alpha,X)] = f(\alpha,E[X])$, where $f$ is some convex function of the first and second variables, except when the first variable takes the value $\alpha$ in which case the equality holds, ...
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How is the Wiener integral $\int{WdW}$ calculated?

I want to calculate $\int ^t _0 W_tdW_t$ I know that the reasoning is the following: Let $x(t)=W(t)$ with $a=0$ and $b=1$ in the definition of an Ito Process, and $f(t,x)=x^2$. Then, applying Ito'...
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73 views

Stochastic Vol Mathematical derivation [closed]

I want to understand the mathematical steps done. Can someone please simplify the derivation of d(pi) from Pi? Thanks in advance.
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227 views

Why does the partial derivative, $X_t$, of an ABM $X(t)$ not involve standard Brownian motion $Z(t)$, even though $Z(t)$ varies with $t$?

Consider the arithmetic Brownian motion $X(t) = \alpha t + \sigma Z(t)$ and evaluating $dX(t)$ using Ito's lemma. We have $\frac{\partial X}{\partial t} = \alpha$, which does not involve $Z(t)$, even ...
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2answers
485 views

Close form solution for Geometric Brownian Motion

I have a very fundamental problem, please help me out. I am little confused with the derivation for the close form solution for the Geometric Brownian Motion, from the very fundamental stock model: $$\...
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1answer
2k views

On the application of Itos lemma to Geometric Brownian motion [closed]

I recently read this from a book: The canonical SDE in financial math, the geometric Brownian motion, ${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution $${S_t} = {S_0}{e^{(\mu -...
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1answer
111 views

squaring stochastic calculus and other solutions [closed]

It is well-known that the solution to the stochastic SDE $$ dS = S_0(\mu dt + \sigma dWt) $$ is $$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$ Were $\sigma=0$, this is simply the formula for ...

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