Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
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Views on Functional Ito Calculus [closed]
Setting up a thread to see your views and opinions on the applications of Functional Ito Calculus.
Would like to get an overview intuition of it before delving deep into the mathematics.
Bruno Dupire ...
-1
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1answer
61 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
86 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
157 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 :
$$dC = \...
-2
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2answers
61 views
1
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0answers
58 views
Pricing caplet with Bachelier (normal dynamic) using forward measure
I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution.
Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing :
$C_t(T,T+d)$ ...
0
votes
0answers
81 views
Dynamics of an option on a future
I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE
$dV_s=\theta_s dW_s+r_s V_s ds$.
I tried of course with ...
2
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0answers
68 views
Basic question on Ito integrals
$Let \space X(t) =\begin{cases}
2, \qquad\text{if} \space 0\le t \le 1 \\
3, \qquad\text{if} \space 1 < t \le 3 \\
-5, \qquad\text{if}\space 3 < t \le 4
\end{cases}
$
or in one forumala $...
1
vote
0answers
143 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
2
votes
1answer
106 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 ...
0
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0answers
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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
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1answer
78 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 $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$
$$F_t(T)= \exp\...
0
votes
1answer
101 views
Self finance conditions - proof check
Find expressions for the process $\psi=(\psi(t),\ 0\leq t\leq T)$ , so the portfolio $(\phi,\ \psi)$ is self-financing when:
(1) $\phi(t)= \int_{0}^{t}S_{s}ds $
(2) $\phi(t)=S_{t}$
where $\phi(t)$ ...
3
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2answers
126 views
Show that the two solutions of the SDE are equivalent
I have a process:
$$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$
where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$.
I want to show that the solution of this ...
0
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0answers
25 views
Value of stock based contract Ito Lemma
Assuming a stock with alpha 0.14, delta 0.01 and volatility of 0.48, how do I derive the price process of a derivative with value V = S^(1/1)e^(0.4)*(1-t) [Equation 1], by using Ito lemma (equation 2)
...
0
votes
1answer
189 views
exercise on multivariate Ito's lemma + jumps (Poisson)
Given the two jump-diffusions:
\begin{equation}
\begin{aligned}
dX_{1,t} &= a_1 dt + b_1 dW_t + c_1 dN_t(\lambda) \\
dX_{2,t} &= a_2 dt + b_2 dW'_t + c_2 dN_t(\lambda) \\
corr(dW,dW') &= \...
1
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1answer
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Mark Joshi, Chapter 5 Problem 2 of The concepts and practice of mathematical finance
If $$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$$ where $V$ is constant. How unique is $f$...
0
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0answers
154 views
Change measure and derivative pricing in Heston model
Consider the Heston-Model
$$\begin{cases}
dS_t=\mu S_t dt+ \sqrt{v_t} S_tdB_t^1 \\
dv_t=k(\theta - v_t)dt+\eta \sqrt{v_t}dB_t^2 \\
\end{cases}
$$
where $B_1,B_2$ are correlated Brownian motions with ...
2
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0answers
58 views
How to calculate the product of forward rates with different reset times using Ito's lemma?
I am curious about a calculation I saw in this question.
Specifically in this equation:
\begin{align*}
&\ L(T_s, T_p, T_e) L(T_s, T_s, T_e) \\
=&\ L(t_0, T_p, T_e) L(t_0, T_s, T_e) e^{-\...
4
votes
1answer
458 views
Ito's Lemma: Multiplication Rule
I have a conceptual question about Ito's lemma, in particular, the multiplication.
Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
5
votes
1answer
159 views
Ito representation unique up to indistinguishability? Proof?
Given an Ito-process $X(t)$, $t\in[0,T]$
$$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$
with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$.
It is now often claimed that this ...
1
vote
1answer
224 views
Intuition behind Ln transformation of stock price when applying Ito lemma [closed]
I am able to replicate steps and arrive to the option price using Black Scholes framework. Here however I am more interested to understand, at least intuitively, why the ln transformation of price ...
2
votes
1answer
185 views
Quantile normal and lognormal
Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows:
\begin{equation}
\Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...
0
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0answers
79 views
Forward Kolmogorov initial distribution
According to this Wiki article Kolmogorov Backward Equations (Diffusion)
Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state $x$ of the ...
0
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0answers
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First order recursive equation for discrete modelling of stock price
I'm looking for a general recursive discrete model for stock price and in the process I want to compare some of the forms described as under:
1.) I found this one in a paper by Bertsimas & Lo ...
-3
votes
1answer
374 views
On the application of Itos lemma to Geometric Brownian motion [closed]
I recently read this from a book:
The canonical SDE in financial math, the geometric Brownian motion,
${{d{S_t}} \over {{S_t}}} = \mu dt + \sigma d{W_t}$ has solution
$${S_t} = {S_0}{e^{(\mu -...
0
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0answers
118 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
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1answer
173 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
66 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
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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
75 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):
$$ \mathrm{d}\vec{r}(t) = \vec{u}(\vec{r}(t))\mathrm{d}t + \...
2
votes
1answer
229 views
How to show that $E\left[ \int_0^t \sigma(s) e^{iuX(s)} dW(s)\right] = 0$?
Let $\sigma(t)$ be a given deterministic function of time and define the
process $X_t$ by
$$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
400 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
161 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
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1answer
153 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
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1answer
192 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
243 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 ...
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votes
1answer
561 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
442 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}$$
...
4
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1answer
337 views
Chain rule for Ito's Lemma
The CIR short rate model is
$$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}$$
where the continuously ...
1
vote
0answers
256 views
Stochastic Leibniz rule
We have the following single-factor HJM model
$$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$$
The discounted T bond is then
\begin{align}
Z(...
1
vote
1answer
101 views
Simple HJM model, differentiating the bond price
We have the following simple HJM model
$$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
633 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
317 views
Baxter & Rennie HJM: differentiating Ito integral
From Baxter and Rennie, page 138:
$$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$
$$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$
$$dZ_t=...
1
vote
1answer
398 views
Ito's Lemma, differentiating an integral with Brownian motion
In How were these SDE derived? I don't understand one part of Gordon's answer, specifically:
$$\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
111 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 ...
3
votes
1answer
182 views
Is this a poorly written example, or could volatility in fact be negative?
I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
0
votes
1answer
111 views
Is there a better, more rigorous explanation for why this partial derivative is 0 using Ito's Lemma?
I encountered the following slide in a lecture on Ito's Lemma.
The lecturer explained that $$\frac{\partial V}{\partial t} = 0$$ because the first two derivatives on the slide already took into ...
5
votes
1answer
261 views
How to compute the expectation of integral of this random function?
Let $W_t$ be a standard wiener process and
$$Y_t=\int_{0}^{t}\frac{W_s}{(1+W_s^2)^2}ds$$
If $W(t_0)=\sqrt{3}$, then how can we compute $\mathbb{E}[Y(t_0)]$?
Is $\mathbb{E}[Y(t_0)]=0$?
