Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
5
votes
0answers
56 views
Random variable minus Integral of Ito Generator is a Martingale under what conditions?
I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
4
votes
1answer
97 views
Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
2answers
169 views
Application of Ito's lemma
Let $X_t$ be some stochastic process driven by wiener process ($W_t)$ so it can be expressed as:
$$dX_t=(...)dt+(...)dW_t$$
Let $f(t,x)$ be some $C^2$ function. Define the process $Z_s=f(t-s,X_s)$ ...
4
votes
1answer
80 views
How to express a process using Itos formula
Let $F(t,x)$ be the solution to the PDE
$$
F_t(t,x)=aF_x(t,x)+\frac{1}{2}F_{xx}(t,x),t>0
$$ $$F(0,x)=g(x)$$ for some function $g$.
Let $X_t$ be a process defined by
$$dx_t=aX(t)dt+dW(t)$$
Now ...
5
votes
1answer
70 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
3
votes
2answers
89 views
Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
-1
votes
1answer
63 views
Differential product Correlated processes
I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now:
We have the following two stochastic processes:
$dX_t= \mu_t dt +\sigma_t dW_t$...
2
votes
1answer
97 views
Differential of integrating factor $d(e^{at}r_t)$ in Vasicek model
I am attempting to solve the Vasicek model SDE (using Wikipedia parametrisation):
$$
dr_t = a(b-r_t)dt + \sigma dW_t
$$
Every solution is proceeding to multiply both sides of the equation by the ...
3
votes
1answer
159 views
Why is Ito applied this way?
Given the price of a call option :
$$C = \mathbb{E}\left[ D_{0,T} (s-K)1_{s>K} |\mathcal{F_0}\right] $$
with $D_{0,T}=e^{-\int_0^Tr(u)du}$
I read somewhere that applying Itô gives :
$$dC = \...
-2
votes
2answers
63 views
2
votes
0answers
71 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
0
votes
0answers
83 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
5
votes
1answer
131 views
Basic question on Ito integrals
$Let \space X(t) =\begin{cases}
2, \qquad\text{if} \space 0\le t \le 1 \\
3, \qquad\text{if} \space 1 < t \le 3 \\
-5, \qquad\text{if}\space 3 < t \le 4
\end{cases}
$
or in one forumala $...
2
votes
0answers
185 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
2
votes
1answer
142 views
Black Scholes in the case of dividends
Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
-1
votes
0answers
47 views
Pricing kernel dynamics in a JDSV model
I have the following model
\begin{align}
d M_t & = r M_t dt \\
d S_t & = S_t [\alpha dt + \sqrt{V_t} d B_t + J d N_t] \\
d V_t & = k(\theta - V_t) dt + \eta\sqrt{V_t} \left(\rho d B_t +...
1
vote
0answers
63 views
$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
2
votes
1answer
79 views
How to derive the dynamic of the log forward price?
I have the following Schwartz model:
$$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$
$$X_t=\ln S_t$$
$$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$
with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$
$$F_t(T)= \exp\...
0
votes
1answer
106 views
Self finance conditions - proof check
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ ...
3
votes
2answers
130 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
0
votes
0answers
25 views
Value of stock based contract Ito Lemma
Assuming a stock with alpha 0.14, delta 0.01 and volatility of 0.48, how do I derive the price process of a derivative with value V = S^(1/1)e^(0.4)*(1-t) [Equation 1], by using Ito lemma (equation 2)
...
0
votes
1answer
223 views
exercise on multivariate Ito's lemma + jumps (Poisson)
Given the two jump-diffusions:
\begin{equation}
\begin{aligned}
dX_{1,t} &= a_1 dt + b_1 dW_t + c_1 dN_t(\lambda) \\
dX_{2,t} &= a_2 dt + b_2 dW'_t + c_2 dN_t(\lambda) \\
corr(dW,dW') &= \...
1
vote
1answer
115 views
Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
If $$dX_t = \mu(t,X_t)dt + \sigma(X_t)dW_t$$
with $\sigma$ positive, show there exists a function $f$ such that $$d\left(f(X_t)\right) = v(t,X_t)dt + V dW_t$$ where $V$ is constant. How unique is $f$...
2
votes
0answers
60 views
How to calculate the product of forward rates with different reset times using Ito's lemma?
I am curious about a calculation I saw in this question.
Specifically in this equation:
\begin{align*}
&\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\
=&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\...
4
votes
1answer
494 views
Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
5
votes
1answer
160 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
1
vote
1answer
244 views
Intuition behind Ln transformation of stock price when applying Ito lemma [closed]
I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price ...
2
votes
1answer
201 views
Quantile normal and lognormal
Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows:
\begin{equation}
\Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
0
votes
0answers
87 views
Forward Kolmogorov initial distribution
According to this Wiki article Kolmogorov Backward Equations (Diffusion)
Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state $x$ of the ...
-3
votes
1answer
464 views
On the application of Itos lemma to Geometric Brownian motion [closed]
I recently read this from a book:
The canonical SDE in financial math, the geometric Brownian motion,
${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution
$${S_t} = {S_0}{e^{(\mu -...
0
votes
0answers
129 views
Stochastic Continuously Dividend paying stock
I am a beginner and I got confused on the concept of continuously paying dividend. Let say the process of dividend payment evolve as
$$ dD_t = \mu D_tdt + \sigma D_tdZ_t$$
where $Z_t$ is a Wiener ...
0
votes
1answer
183 views
Self-Financing Portfolio
Why when we are using self-financing portfolios to replicate some external payoff we do not consider the quadratic variation of the portfolio weights? Say, in Black-Scholes world, when we are using $...
1
vote
1answer
69 views
Is it possible to approach finding the risk premium of this derivative using Ito's Lemma?
I understand the author's intended solution to the below problem, but I thought I would see if I could solve this using first principles and Ito's Lemma instead for practice.
Let $V(S(t), t) = e^{rt}\...
-1
votes
1answer
1k 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 $...
2
votes
0answers
76 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):
$$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
2
votes
1answer
233 views
How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?
Let $\sigma(t)$ be a given deterministic function of time and define the
process $X_t$ by
$$X(t) = \int_0^t \sigma(s)dW(s)$$
I want to show $$E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$$...
-1
votes
1answer
422 views
Bond price and its process
Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process
$dx=a(x_0-x)dt + sxdz$
where $a, x_0$ and $s$ ...
4
votes
2answers
162 views
Moment Ito's Process Proof
I have a following stochastic integral - related problem that I have difficulty to solve:
Given
\begin{equation}
dX_t = -\alpha X_tdt+\sigma\sqrt{X_t}dW_t
\end{equation}
and the second moment of $...
-1
votes
1answer
166 views
Why does the partial derivative, $X_t$, of an ABM $X(t)$ not involve standard Brownian motion $Z(t)$, even though $Z(t)$ varies with $t$?
Consider the arithmetic Brownian motion $X(t) = \alpha t + \sigma Z(t)$ and evaluating $dX(t)$ using Ito's lemma.
We have $\frac{\partial X}{\partial t} = \alpha$, which does not involve $Z(t)$, even ...
0
votes
1answer
214 views
How to define the $f$ function to apply Ito's lemma?
\begin{equation}
Z(t) = \exp (a W(t))
\end{equation}
I am asked to find $dZ$. I am pretty sure it can be done using Ito's lemma. But in all my textbook (Bjork) examples Ito's lemma is giving from a $...
1
vote
1answer
257 views
Stochastic differential equation of a Brownian Motion
I have two questions about Ito's Lemma with respect to calculating SDEs. The examples are simple enough, but I haven't found an answer yet.
Take $W_t$ as a standard Brownian motion and $g(s)$ as some ...
-2
votes
1answer
632 views
How is the Wiener integral $\int{WdW}$ calculated?
I want to calculate $\int ^t _0 W_tdW_t$
I know that the reasoning is the following:
Let $x(t)=W(t)$ with $a=0$ and $b=1$ in the definition of an Ito Process, and $f(t,x)=x^2$.
Then, applying Ito'...
5
votes
1answer
451 views
Application of Ito's Lemma, finding the condition for the martingale
The Vasicek short rate model is
$$dr_t=\kappa(\theta-r_t)dt+\sigma dW_t$$
Define the processes $x_t$ and $f(x,t)$
$$x_t=\frac{r_t}{\kappa}(1-e^{-\kappa(T-t)})+\int_0^tr_sds$$
$$f(x,t)=e^{a(T-t)-x_t}$$
...
4
votes
1answer
351 views
Chain rule for Ito's Lemma
The CIR short rate model is
$$dr_t=k(\theta-r_t)dt+\sigma\sqrt{r_t}dW_t$$
under the risk-neutral measure.
The bond price is of the form
$$P(t,T)=A(t,T)e^{-B(t,T)r_t}$$
where the continuously ...
1
vote
0answers
278 views
Stochastic Leibniz rule
We have the following single-factor HJM model
$$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$
$$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$
The discounted T bond is then
\begin{align}
Z(...
1
vote
1answer
103 views
Simple HJM model, differentiating the bond price
We have the following simple HJM model
$$f(t,T)=f(0,T)+\int_0^t\alpha(s,T)ds+\sigma W_t$$
$$r_t=f(0,t)+\int_0^t\alpha(s,t)ds+\sigma W_t$$
$$P(t,T)=\exp-\bigg(\int_t^Tf(0,u)du+\int_0^t\int_t^T\alpha(s,...
1
vote
1answer
714 views
Partial derivative of an integral
Suppose I have a model for the short rate $r$ as ($W(t)$ is standard Brownian motion)
$r(t) = c+ \int_0^t \sigma (s) ^2 (t-s) ds+ \int_0^t \sigma (s) dW(s)$
I then want to find the dynamics of $r$, ...
7
votes
1answer
326 views
Baxter & Rennie HJM: differentiating Ito integral
From Baxter and Rennie, page 138:
$$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$
$$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$
$$dZ_t=...
1
vote
1answer
458 views
Ito's Lemma, differentiating an integral with Brownian motion
In How were these SDE derived? I don't understand one part of Gordon's answer, specifically:
$$\ln S_t=\ln F_{0,t}-\frac{\sigma^2}{4\lambda}(1-e^{-2\lambda t})+\sigma e^{-\lambda t}\int_0^t e^{\...
1
vote
1answer
119 views
Piecewise Ito formula
Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions.
My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only ...
