Questions tagged [itos-lemma]
The itos-lemma tag has no usage guidance.
142
questions
2
votes
0answers
22 views
Ito formula for $Y_t=tB_t$
someone can help me to solve this problem:
$B_t$ is a Standard Brownian Motion.
Let $Y_t=tB_t$.
Using Ito formula, find drift and volatility of $Y_t$.
The result I found is $dY_t=B_tdt+t\cdot dB_t$ ...
1
vote
1answer
82 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
1
vote
0answers
38 views
Application Itô's Lemma: Forward to Spot process
I am working on the following equation (I want to apply Ito's lemma on it):
and I know that:
and also
and
My problem is that I want the dynamic of F(S,T) without S because I need first to ...
0
votes
1answer
41 views
Compute dZ(t) : Ito's formula/lemma
We need to find dZ(t). I know I have to use Ito's formula. But I am confused because in the Ito's formula we have f(y,t) is a twice differentiable function with two variables
But here Z(t) = 1/(2+x(t)...
0
votes
0answers
52 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
0
votes
0answers
22 views
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 - ...
4
votes
1answer
251 views
Pricing call option using risk-neutral martingale approach with squared stock price boundary?
I have to use the risk-neutral martingale 5 step approach under BS pricing framework to price the following call option at time 0:
$$X = \begin{cases}1, &{if} &S_T^2\geq K,\\0, & {...
3
votes
1answer
134 views
Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift
Given the stochastic differential equation:
$$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$
for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply Itô's Lemma to obtain:
...
5
votes
0answers
63 views
Relating two equations in a jump-diffusion process
I am trying to understand an argument involving the pricing kernel $\xi_t$ in the context of a simple jump diffusion model for the price of an asset $S_t$:
\begin{align}
\xi_t = \exp \left[ -\theta ...
3
votes
2answers
137 views
Partial derivative of Ito integral without product rule
I'm thinking about the problem of deriving the stochastic differential of an integral with both time and state part of the integrand but not in a way that you can easily factor it out - for example I ...
2
votes
1answer
103 views
How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion
As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
2
votes
0answers
112 views
Itô’s formula and Wiener process
The Wikipedia page on the formula https://en.wikipedia.org/wiki/It%C3%B4%27s_lemma and some textbooks I have looked at say we must assume that the relevant time-dependent function is over an Itô ...
6
votes
2answers
306 views
Variance of a time integral with respect to a Brownian Motion function
Let process
$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$
where $W_s$ is standard Brownian motion. My question are the following:
We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
4
votes
2answers
92 views
Volatility of Exchange Option
I got a question and its partial solution, and have some doubts about the volatility of its geometric Brownian motion process:
Question:
How would you price an exchange call option that pays $max(S_{...
0
votes
0answers
44 views
How to proof the formula to be martingale under ITO process?
How can implies that is a martingale when using the defaultable bond price?
4
votes
1answer
111 views
How to determine components of Affine Term Structure for an Ohrnstein-Uhlenbeck process?
I wonder how I can determine the components $A(t,T)$ and $B(t,T)$ for the zero-coupon bond price process $p(t,T)=e^{A(t,T)-r(t)B(t,T)}$? The components are defined in the following link: https://en....
0
votes
0answers
30 views
Confirm If Risk-Neutral Measure is Unique in My Following Case
I'm reading a book that discusses about derivatives pricing and have some doubts about a particular problem and really appreciate your advice:
Question:
Assume a non-dividend paying stock follows a ...
1
vote
0answers
90 views
How to determine exchange rate dynamics in currency derivatives
I need some guidance regarding exchange rate dynamics in currency derivatives.
Following three dynamics are defined below,
$\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
1
vote
1answer
48 views
How to determine the no arbitrage price of following claim? (change of numeraire)
How do I determine the no arbitrage price for claims such as $min(S_1(T),S_2(T))$ or $max(S_1(T),S_2(T))$? We can consider a standard Black Scholes model. Hence $S_i(T)=S_i(t)e^{(r-\sigma_i^2/2)(T-t)+\...
0
votes
1answer
74 views
Determining the No Arbitrage price of max[B(T), S(T)]
Following is given,
$dB(t)=rB(t)dt$
$dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$
where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
3
votes
2answers
172 views
Stochastic Calculus problem with three processes? (Itô calculus)
Can someone help me solve this following Itô Calculus problem?
Let $Z(t):= [B(t)*X(t)]/S(t)$
We have the following dynamics of B(t), X(t) and S(t):
$dS(t)=\alpha S(t)dt+\sigma S(t)dW(t)$
$dB(t)=rB(...
5
votes
1answer
65 views
Three proofs regarding brownian motions and martingales
1. Let $(B_t)_{t \geq 0}$ and $(W_t)_{t \geq 0}$ be two standard Brownian motions and let $X_t := B_t W_t$. Is $(X_t)_{t \geq 0}$ a martingale?
The easiest way to proceed seems to be to apply Ito's ...
3
votes
1answer
114 views
Computing Itô differential of conditional expectation process (Heston SDE)
Going through this article
on Heston's model, where the variance evolves following the SDE
\begin{equation}
\label{sd1}
d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
3
votes
1answer
182 views
How to calculate the mean and variance of this Ito integral?
I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process.
$$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
2
votes
1answer
68 views
Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
3
votes
2answers
148 views
Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$
I am wondering about the probability distribution of the stochastic process
$$X_t=\int_0^t \frac{u} {t} dW_{u}$$
I thought of using the Kolmogorov equation but after converting this into An SDE
$$...
0
votes
0answers
48 views
How does this follow from Ito's formula?
Let $S_t$ be a geometric brownian motion and let $M_t = \sup_{0 \le u \le t} S_t$ be the maximum process.
Define $X_t = \frac{M_t}{S_t}$.
A book I am reading states the following:
How does this ...
5
votes
2answers
341 views
More questions about integral of Brownian Motion w.r.t time
A similar question have been posted earlier but one part has remained unanswered. Let us define:
$$X_t = \int_0^t W_s ds,$$
where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
2
votes
1answer
101 views
Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?
In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma:
Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying
$$dX_t = \mu(X_t,t)dt + \sigma(...
2
votes
1answer
103 views
Stochastic Processes (Applying Ito's Lemma on Ho-Lee Model )
I seek a basic form (SDE) to understand the Ho-Lee model.
I already understand the models from Vasicek, Merton and Cox-Ingereoll-Ross, etc.. For example,
\begin{align*}
dX_t &= -1/2 \alpha X_t ...
1
vote
1answer
64 views
How to derive the expression for the forward rate?
The following RN dynamics of a ZCB maturing at time is given:
$$\frac{dZ(t,T)}{Z(t,T)} = r_tdt + \sigma_Z(t,T)dX_t$$
and the forward rate is given:
$$f(t,T,T+\delta) = \frac{ln(Z(t,T)) - ln(Z(t,T,...
1
vote
1answer
113 views
Integration and expectation of geometric Brownian motion
Let the stock price S follows the geometric brownian motion:
$$dS=\mu Sdt+\sigma Sdz$$
$$\frac{dS}S=\mu dt+\sigma dz$$
where $dz$ is a wiener process.
Naively integrating the second equation above ...
2
votes
1answer
92 views
Zero-coupon bond pricing equation derivation
I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation?
The equation is:
$\frac{\partial B}{\partial t} + \...
3
votes
1answer
154 views
List: Behavioural characteristics of key Ito processes used in finance
My hope from this question is to become a repository of the behavioural characteristics, use-cases and interesting features of the key Ito processes used in quantitative finance - examples being GBM, ...
2
votes
0answers
98 views
Ito's lemma for special case
Assume a HJM framework with the same Brownian motion driving the dynamics for every tenor.
$$
df(t,T) = \alpha(t, T)dt + \sigma(t,T) dw_t \,,
$$
with $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$....
3
votes
1answer
160 views
What is the easiest way to learn Option pricing with PDE?
I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
1
vote
1answer
115 views
Brownian Motions theorems
I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0.
How can I prove/demonstrate this theorem?
Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
2
votes
1answer
192 views
Ito formula (lemma) problem
I am trying to solve this problem
Consider the following one-dim. stochastic process
$$dX_t = b_t dt + \sigma_t dW_t$$
where $W$ is a one-dim. Brownian motion. The above SDE is well-defined.
...
1
vote
1answer
86 views
Stochastic solution (mean, variance) to lognormal drift and normal volatility
I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its
a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model
I ...
-1
votes
1answer
77 views
Different Forms of Geometric Brownian Motion [closed]
If the stock price S follows the geometric brownian motion:
$$dS=\mu Sdt+\sigma Sdz$$
$$\frac{dS}S=\mu dt+\sigma dz$$
Where $dz=\epsilon\sqrt{dt}$ is a wiener process.
Integrating this to get $S_T$ ...
4
votes
1answer
489 views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
3
votes
4answers
725 views
Log of square of Geometric Brownian Motion
Which of the two calculations below, is wrong?
Why?
$dF = \sigma F dW$
First:
$dF^2 = (F^2)' dF + \frac{1}{2}(F^2)''dF.dF$
$dF^2 = 2F dF + dF.dF$
$dF^2 = 2 \sigma F^2 dW + \sigma^2 F^2 dt$
$\...
3
votes
0answers
59 views
Ito Diffusion with Change of Measure
Let $(X_t)$ be an Ito diffusion with speed $(V_t)$, under a probability measure P. Could there exist a change of measure to a probability measure Q, with Q ~ P, under which $(X_t)$ is an Ito diffusion ...
3
votes
1answer
142 views
Expectation in a stochastic differential equation
I'm new to stochastic calculus, I want to find the mean of $X_2$ with $X_t = \exp(W_t)$, with $W_t$ a Wiener process.
I used Ito's Lemma is arrive at the SDE:
\begin{align}
d(X_t) = \frac{1}{2}X_t dt ...
2
votes
0answers
160 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
5
votes
1answer
212 views
Ito's Lemma for this problem
I'm attempting to prove a lemma from a paper, in the context of optimal contracts.
$r,\rho,\gamma,\alpha,\sigma$ are all known constants.
$dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
3
votes
1answer
120 views
HJM model Baxter Rennie: differentiating the discounted asset price using Ito
From Baxter and Rennie Page 145:
$Z(t,T) = exp(\int_{0}^{t}\Sigma(s,T)dW_s - \int_{0}^{T}f(o,u)du - \int_{0}^{t}\int_{s}^{T}\alpha(s,u)duds)$
where $\Sigma(t,T) = \int_{t}^{T}\sigma(t,u)du$
How ...
7
votes
0answers
124 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: $$M_s = U(s,...
4
votes
1answer
117 views
Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
4
votes
2answers
417 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 $Z_s=f(t-s,X_s)$ ...
