Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
223
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How to express the process of number of stock (nt) in a portfolio using ito's lemma
We have a regular self-financing portfolio $W_t$:
$$dW_t = n_t dS_t + (W_t − n_t S_t) r dt$$
Where $W_t$ is total wealth, $n_t$ is amount of stock, $S_t$ is stock price, $r$ is the risk-free rate.
And ...
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Deriving an Analytical Expression for Standard Deviation of Log Returns
I am looking to find an expression for the standard deviation log returns of a stock price process.
I have a stock price which follows the following dynamics:
$dY(t) = Y(t)(r(t)dt + η(t)dW(t))$
Here,...
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Pricing various classes of derivatives and replicating them
Consider the following three derivative styles and assume zero dividends for simplicity.
The "american style", "european style", and "infinite" style:
$$L_{A}(S,K,t,T)=f(...
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Volatility of the product of two correlated asset following a log normal distribution [duplicate]
I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the ...
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How calculate expectation and variation of stochastic integral Based on Heston model?
I was calculated Heston volatility model. But I think it is wrong.
$dS_t = \mu dt + \sqrt V_t dW_t^s$
$dV_t = k(\theta - V_t)dt + \sigma \sqrt V_t dW_t^v$.
$dW^s_t dW^v_t = \rho dt$
take integral to ...
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Is the time derivative of asset returns expressible as an SDE?
Consider the following SDE for $(S_t)_{t\geq 0}$ under $\mathbb{Q}$,
\begin{equation}
\mathop{dS_t}=S_t\left(r\mathop{dt}+\sigma(t,S_t)\mathop{dW_t}\right),
\end{equation}
which (in Langevin form) may ...
- 128
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The solution of SDE after Itô lemma for diffusion process [closed]
Consider the one-dimensional diffusion process $dX_t = \mu_tdt + \sigma_tdB_t$ and function $f : \mathbb{R} \to \mathbb{R}$, which is twice differentiable. Here we have another SDE by using Itô lemma ...
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Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
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Analytical expression for SDE
I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
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Deriving the stochastic process for a dividend-yielding stock (under Black-Scholes assumptions)
In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$
$$
S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,}
$$ M. ...
- 346
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Integral of brownian motion wrt. time over [t;T]
From the post Integral of Brownian motion w.r.t. time we have an argument for
$$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$
However, how does this generalise for the interval $[t;T]$? I.e. ...
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Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
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Value of trading strategy
A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process.
The problem is to find the probability of the ...
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Show that the solution to a SDE is strong
I have the following SDE
\begin{equation}
dX_t = - \frac{1}{1+t}X_t dt + \frac{1}{1+t}dB_t
\end{equation}
that has the solution:
\begin{equation}
\begin{aligned}
X_t = \frac{X_0 + B_t}{1+t} = \frac{...
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Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$
which is ...
- 221
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128
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Ito's lemma for option pricing with Levy-alpha stable drift
Consider
$$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$
such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
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103
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Stochastic process as integral over window function
Consider the following stochastic integral of a deterministic function $f(t,s)$ with respect to the Wiener process $W_s$:
$$\int_0^\infty f(t,s) d W_s$$
My questions are:
Is such an integral ...
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Difficulty with stochastic calculus problem
I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is:
Consider a stock whose price differential is
$$
dS(t) = r(t) S(...
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Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
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Why this stochastic integral is calculated with Riemann integral
This picture is from Neftci's textbook, 'An Introduction to the Mathematics of Financial Derivatives, Third Edition'
What makes me uncomfortable is equation [10.61] In above picture.
In this equation,$...
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103
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Performance of dollar cost averaging
If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested?
For example, modelling a stock as GBM without contributions, $ ...
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203
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Clarification on Paul Wilmott's derivation of Ito's Lemma
I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
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Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$
Suppose a stock follows the stochastic differential equation
$$dS=\frac{P-S}{\omega}dt+SdW_t,$$
such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
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Volatility and drift of the instantaneous forward rate under risk neutral measure using the zero coupon bond
I have question about this problem. I believe I have derived $f(t,T)$ correctly using the zero-coupon bond. But I am unsure about how to go forward with the question and how to use the second part.
...
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Parametric Stochastic Integral
I need help.
Defining the parametric stochastic integral
$$
F_t = \int_t^T\xi(t,s)g(s)ds
$$
$\\\\$
with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
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Covariance of the product of log normal process and normal procces
I tried to compute the following covariance :
$$Cov(e^{\int_{t}^{T}W^1_sds},\int_{t}^{t+1}W^2_sds)$$
where $W^1_t$ and $W^2_t$ are Brownian motions such that $dW_t^1dW_t^2=\rho dt $
My idea was to ...
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Question on Ito's lemma involving $\mathrm{d}W(t)$
I am new to Ito-calculus, so please forgive me if the question is stupid.
Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
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Confused by derivation of variance swap payoff
I'm trying to follow
https://en.wikipedia.org/wiki/Variance_swap#Pricing_and_valuation
where it seems to me that they're just subtracting a simple return:
$$ R_t = \frac{\mathrm{d}S_t}{S_t} = \mu \...
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Ito's lemma $f(t,W_t^2)$
Let $f$ be a function of $t$ and $W_t^2$.
a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration.
b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
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Ito multiplication
Let $\{N_t|0<t\leqslant T \}$ and $\{M_t|0<t\leqslant T \}$ be two Poisson processes with intensities $\lambda_n, \lambda_m>0$, respectively.
Based on the implicit results of Corollaries 1 ...
user57062
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Ito's lemma results in negative volatility processes
I struggle with the interpretation of a process I derive from Ito's Lemma. Let's say I have function f(S,t) which is twice differentiable wrt S.
I thus can apply Ito's Lemma to get $df(S,t)$. So far ...
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Ito Lemma for Poisson Process
I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question.
Let $g_t$ be a $\mathcal{F_t}$-...
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Solving an SDE using Ito's Lemma
Suppose that
$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$
with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process
There is also ...
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MGF of Generalised Itô Integral
The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$.
Consider $W_t$ ...
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Drift Term in Black-Scholes Model Martingale
How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof).
For instance if Im looking at:
$$dS_{t} = \mu S_{t} + \...
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Multidimentional Black Scholes Formula
I need to write the Black-Scholes formula for option $V = (S_1, S_2, t)$, where:
$$d S_1 = \mu_1 S_1 dt + \sigma_1 S_1 d W_1,$$
$$ d S_2 = \mu_2 S_2 dt + \sigma_2 S_2 d W_2.$$
We know that $W_1$ and $...
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Derivation of option pricing PIDE: Why does the drift need to be zero?
I started studying PIDE methods for option pricing and am struggling to understand or find the necessary theory that shows why the PIDE is obtained by the condition that the drift term has to be zero.
...
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How is the formula of Quadratic Variation of Brownian Motion derived? [closed]
This is a follow up on this question on quant SE:
The question mentions for a Brownian motion :
$X_t = X_0 + \int_0^t\mu ds + \int_0^t\sigma dW_t $
, the quadratic variation is calculated as
$dX_t ...
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Covariance between integral of brownian motion and brownian motion
Let
$$
I = \int_0^1W_tdt,
$$
where $W_t$ is a Brownian motion.
From Integral of Brownian motion w.r.t. time we have that
$$
\mathbb{E}[I]=0,
$$
by Fubini's theorem. And that
$$
\mathbb{V}\text{ar}[I] =...
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Justification for substituting "Itô differentials"
I'm reading Shreve's Stochastic Calculus for Finance, Volume II. In it, he uses the stochastic differential notation. For example, he may write
$$\mathrm{d}X(t) = \sigma(t)\mathrm{d}W(t)+\alpha(t)\...
- 437
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Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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341
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Ito calculus is Gaussian (using method of characteristic function)
Let $h$ be a deterministic function and define $X_{t}=\int_{0}^{t} h(s) d W_{s} .$ Show that
$$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
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103
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Mutual variation of Brownian motions
Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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What does it mean to "compute" an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
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votes
2
answers
465
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Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
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1
answer
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Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
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votes
1
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206
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Derivative of Stochastic Integral
I am trying to take the derivative of the following stochastic integral,
$$d\left(\int g(S_t) dS_t \right),$$
where $dS(t) = \sigma S(t) dW_t$ and $g(.)$ is some (smooth) deterministic function. My ...
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votes
2
answers
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Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles
I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
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1
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655
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How can I learn stochastic process & stochastic calculus in two weeks? [closed]
I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
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2
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Show that $\int_0^T(T-t)dW_t=\int_0^TW_t \ dt$
I want to show that $$\int_0^T(T-t)dW_t=\int_0^TW_t \ dt \tag1$$
By Ito's lemma we can write $$d((T-t)W_t)=(T-t)dW_t-W_t \ dt.\tag{2}$$
Now If I integrate this expression and use that $W_0=0$ I should ...
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