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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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328 views

Different definitions of arbitrage

Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "...
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0answers
80 views

Transformation of coupled forward-backward stochastic differential equations in 3 dimensions with Ito formula

Maybe this is the right place for my question: I have a system of coupled FBSDEs in 3 dimensions as follows (in cartesian coordinates): $$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
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58 views

Using malliavin derivative to find the worst Delta-positive hedge?

Background: I've heard that Malliavin Calculus can be used to show the explicit form of a delta-neutral hedge (given an SDE driven market model). For example, here is a sketch here on page 21 on how ...
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490 views

SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{...
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1answer
413 views

Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
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0answers
64 views

Is there anyone tried to use simultaneous stochastic differential equations?

I am looking for some examples or attempts of using simultaneous stochastic differential equations for financial analysis but there has been none so far. Is it just so nasty to apply such thing in ...
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0answers
228 views

stochastic calculus and multidimentional itos lemma

I am considering a number of assets (N) in a portfolio. each asset follows a geometric Brownian motion process therefore the stochastic differential equation is dS(i) = S(i)μdt + S(i)σdX(i). The ...
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0answers
107 views

Term Structure and short rates

If I have a term structure/yield curve given by: $$f(t, T) = f(0, T) + σ^2t(T − \frac{t}{2}) + σB_t $$ and want to find the short/spot rate $r_t$, is this simply: $$f(t,t) = f(0,t) + \sigma^2t(t-\...
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1answer
64 views

Complete Multiperiod Binomial model

I have the following deifnition of a Complete multiperiod binomial model: A multi period binomial model can be called complete if every derivative security can be replicated by trading in the ...
1
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2answers
220 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it fully,...
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0answers
101 views

Intensity Function of Stochastic Processes

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
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2answers
388 views

How to find the mean and variance of this stochastic process?

$ I_t = \int_0^t e^{i W_s} dWs $ where $W_s$ is the standard brownian motion and $i$ is the complex number. Any help will be appreciated!
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2answers
217 views

How to compute the conditional variance of this jump process?

Let $N_t$ be a Poisson process with intensity $\lambda>0$ and $S_t$ follows a pure jump process $$dS_t=S_t(J_t-1)dN_t$$ where $J_t$ is the jump size variable if $N_t$ jumps at time $t$. Also, ...
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1answer
132 views

How to prove $\int_0^t W_s^2dWs = \frac{1}{3}W_s^3 - \int_0^t W_s ds$ using Ito's formula? [closed]

Please help me with this problem.
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1answer
3k views

Correlation coeffitiont between two stochastic processes

I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks
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1answer
319 views

Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]

Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write ...
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2answers
393 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 - \frac{k^2}{2}...
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2answers
83 views

Advantage of continuous time stochastic calculus over discrete version?

I'm new to the stochastic calculus, and I keep converting the continuous stochastic differential equation to its counterpart in discrete time, such as the autoregressive models. I wonder in practice, ...
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2answers
251 views

Show a process is Martingale

$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
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1answer
397 views

Change of numeraire from bank account to Zcb [closed]

Why is there no drift adjustment when numeraire is changed from bank account (risk neutral measure) to zero coupon bond who matures at time of payoff (fwd risk neutral measure) ?
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1answer
56 views

Notation clarity on continous proesses [closed]

Can someone clarify differences between $dX_t,\frac{\partial X_t}{\partial t},\int_0^t X_{t'}dt',\int_0^tdX_{t'}$? Does $\int_0^t\frac{\partial X_{t'}}{\partial{t'}}d{t'}=X_t$?
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2answers
199 views

Stochastic process and brownian motion

I just read the following and i am having some difficulty to interpret it: We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, ...
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1answer
2k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
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1answer
304 views

How to define the $f$ function to apply Ito's lemma?

\begin{equation} Z(t) = \exp (a W(t)) \end{equation} I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
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1answer
162 views

Why don't I get this right $\frac{d}{dt}\mathop{\mathbb{E}}\left[ e^{-\int_t^Tr(s)ds}|\mathscr{F}_t \right]$

Let $r$ a random process defined by : $$dr_t=\theta(t)dt + \sigma dW_t$$ $\theta$ is deterministic in $t$ and $W$ a brownian motion. I don't know where my calculation below is going wrong : Let $...
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1answer
182 views

Variance of the Cox-Ingersoll-Ross short rate

Shreve II page 151, the Cox-Ingersoll-Ross model is defined as $$dr_t=(\alpha-\beta r_t)dt+\sigma\sqrt{r_t}dW_t$$ By applying Ito's Lemma, we obtain \begin{align} r_t&=r_0e^{-\beta t}+\frac{\alpha}...
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1answer
113 views

Does the partition of time in a simple process depend on the omega in probability space?

In Steven Shreve's book "Stochastic Calculus for Finance 2", page 126, a simple process $\Delta(t)$ is a stochastic process such that there is a partition of time $0 < t_1 < ... < t_n \leq T$,...
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1answer
111 views

stochastic discount factor transformation

I have $$\frac{dM_t}{M_t}=-\frac{\mu}{\sigma} dW_t + \gamma_t dB_t, \tag{1}$$ where $B_t$ and $W_t$ are two independent Brownian Motions, which was further presented as $$ M_t=\exp \left( -\frac{\mu}{...
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1answer
410 views

Ho-Lee model - A and B derivation for $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$

I am analyzing the transition of the bond prices in the affine models in the form of $P(t,T)=e^{-A(t,T)-B(t,T)r_t}$ using the property that the diffusion and the drift of an affine model can be ...
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2answers
150 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
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1answer
60 views

integration of squared brownian motion w.r.t time

How to prove $\int_0^1 B_s^2ds$ is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion.
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1answer
54 views

Event Occurs Almost Surely

Consider an uncountably infinite space, an infinite coin-tossing. Let $(\Omega,\mathcal{F},\mathbb{P})$ be the probability space. If a set $A\in\mathcal{F}$ satisfies $\mathbb{P(A)=1},$ then we say ...
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2answers
182 views

Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$: $$d(\int W_t dt ) = W_t dt$$ I ...
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2answers
988 views

Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
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1answer
47 views

standard brownian vs brownian motion

We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
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1answer
142 views

Zero value of cash flow for future in Shreve's book

Here is the statements of future price in Shreve's book Stochastic Calculus for Finance II page 244 to proof the ...
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1answer
218 views

How to calibrate an SDE's by finite difference equation?

I would like a general framework for the calibration of the unknown parameters in an arbitrary stochastic differential equation. I have a proposed method that seems reasonable in theory, but is ...
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1answer
453 views

Mean Reverting to its own variance?

Good morning all, When trying to decipher some documentation I have come across this stochastic process which seems to me much like a Ornstein-Uhlenbeck (or Vasicek) process. $$dX_t=-\kappa(X_t-\...
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1answer
142 views

Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
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1answer
184 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...
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1answer
120 views

Lebesgue-Stieltjes integration and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
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1answer
129 views

approximating fBm stochastic integral

Suppose I have the following stochastic integral: $$\int_a^b f(t)dB_H(t)$$ with the term $dB_H(t)$ a fractional brownian motion with associated $H$ parameter. Is it true that for $H \in (1/2,1)$, ...
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1answer
55 views

trading strategy problem - initial capital x buys S over time [0,T] at the constant rate of x/T euros per unit of time

I am looking for clarification to the trading strategy problem where the number of stocks is depending on time. In the Market with zero safe rate and stock dynamics defined as $$\frac{dS_t}{S_t}=\...
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1answer
63 views

Please help me with this problem of double exponential distribution

please help me with this problem of double exponential distribution
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1answer
98 views

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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1answer
14 views

Arithmetic Asian Option

Assume the risk-free bond Bt and the stock St follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $μ$ and volatility $σ$). Let $A_T:=\frac{1}{T}...
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33 views

Change of numeraire/probability when asset pays dividends

So I was looking at Margrabe's formula for exchange call options in the book 'Mathematical Methods for Financial Markets' (Jeanblanc, Chesney, Yor), and I was having trouble justifying their change of ...
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1answer
62 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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0answers
32 views

Filtrations and the different “kinds” of pre-knowledge

I am searching for a reference I think I saw in a book by either Shreve or Oskendahl. I am struggling with a theoretical question. As I recall how it was posed, the idea of no prior information (or ...
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1answer
312 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} ...