Questions tagged [itos-lemma]

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25
votes
2answers
38k views

Worked examples of applying Ito's lemma

In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
16
votes
2answers
4k views

Solution of Merton's Jump-Diffusion SDE

In many textbooks and also in the original Merton's paper the solution of the SDE $$ dS_t = S_t\,\mu\,dt+S_t\,\sigma\,dW_t+S_{t^-}\,d\left(\sum_{j=1}^{N_t}V_j-1\right) $$ is written as $$ S_t = ...
15
votes
3answers
11k views

What is Ito's lemma used for in quantitative finance?

Further to my question asked here: prior post and which left some points unanswered, I have reformulated the question as follows: What is Ito's lemma used for in quantitative finance? and when is it ...
14
votes
5answers
2k views

Monte carlo methods for vanilla european options and Ito's lemma.

I understand that by applying Ito's lemma to the following SDE $$dX=\mu\,X\,dt+\sigma\,X\,dW$$ one obtains a solution to the above SDE which is as follows: $${X}\left( t\right) =\mathrm{X}\left( 0\...
13
votes
2answers
2k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
11
votes
4answers
13k views

How to use Itô's formula to deduce that a stochastic process is a martingale?

I'm working through different books about financial mathematics and solving some problems I get stuck. Suppose you define an arbitrary stochastic process, for example $ X_t := W_t^8-8t $ where $ W_t ...
9
votes
2answers
368 views

conditional expectation of stochastic integral

let $M_t$ be the following stochastic integral $$ M_t = \int_0^t \sigma_s dW_s $$ where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
9
votes
3answers
2k views

Demonstration of Ito's correction term/lemma in binomial tree

I am preparing an undergraduate QuantFinance lecture. I want to demonstrate the ideas of Ito's correction term and Ito's lemma in the most accessible manner. My idea is to take the "working horse" of ...
8
votes
2answers
4k views

Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
8
votes
1answer
1k views

How to perform basic integrations with the Ito integral?

From the text book Quantitative Finance for Physicists: An Introduction (Academic Press Advanced Finance) I have this excercise: Prove that $$ \int_{t_1}^{t_2}W(s)^ndW(s)=\frac{1}{n+1}[W(t_2)^{n+1}-...
8
votes
1answer
6k views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in \mathbb{R}^{...
7
votes
2answers
2k 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 ...
7
votes
1answer
656 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=...
7
votes
1answer
178 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 $...
7
votes
1answer
451 views

Is this application of Ito's lemma correct?

Suppose that $S$ follows a geometric brownian motion $$dS=S(\mu dt+\sigma dB).$$ It is well understood that $$S_{T}=S_{0}exp((\mu-\dfrac{\sigma^{2}}{2})T+\sigma B_{T}).$$ Method 1 (I have no ...
7
votes
1answer
572 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
7
votes
0answers
153 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,...
6
votes
2answers
473 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 ...
6
votes
1answer
232 views

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 ...
6
votes
1answer
384 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 $\mathbb{E}[Y(t_0)]=0$?
6
votes
1answer
490 views

How to tackle this exercise about Ito's formula?

In the following exercise, I can't get started on question 2) as I am not sure what to do when there is an integral inside: Could you help me out?
6
votes
2answers
125 views

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 ...
6
votes
1answer
217 views

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 ...
5
votes
2answers
859 views

Clarification on Deriving Ito's Lemma

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
5
votes
1answer
508 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, & {...
5
votes
3answers
143 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_{...
5
votes
1answer
826 views

Square of arithmetic brownian motion process

We have an arithmetic Brownian motion process $X_t$ that follows $dX_t=\mu dt + \sigma dZ_t$ and we define the asset price $S_t=X_t^2$ and we are asked to find the stochastic differential equation ...
5
votes
1answer
122 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 ...
5
votes
1answer
129 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
323 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 ...
5
votes
1answer
756 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}\...
5
votes
1answer
202 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 ...
5
votes
1answer
572 views

Ito's Lemma - Integrand depends on upper limit of integration

A problem I came across while practicing using Ito's Lemma had a process with an integral whose integrand depends on the upper limit of integration (the goal is to find $dZ_{t}$): $Z_{t}=\int_{0}^{t}...
4
votes
1answer
519 views

Deriving the solution for European call option in the Heston Model

I'm deriving the solution for European call option in the Heston Model. I follow the original paper by Heston and Fabrice Douglas Rouah's derivations in his book The Heston Model and Its Extensions in ...
4
votes
1answer
240 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, ...
4
votes
2answers
628 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
333 views

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 $$\mathbb{E} \exp \left(i u X_{t}\right)=\exp \left(-\frac{u^{2}}{2} \int_{0}^{t} h^{2}(s) d s\right),$$ ...
4
votes
2answers
207 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(...
4
votes
1answer
547 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}
4
votes
2answers
171 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 ...
4
votes
2answers
226 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 ...
4
votes
1answer
1k 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. ...
4
votes
1answer
586 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
528 views

clarification to log-stock price formula

Having financial market with safe rate r and risky asset S with dynamics under physical measure P $$\frac{dS_t}{S_t}=\mu dt +\sigma dW_t$$ what is the log-stock price? Using Ito formula it is ...
4
votes
1answer
109 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 $dY_t=B_tdt+t\cdot dB_t$ ...
4
votes
2answers
229 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 $$...
4
votes
1answer
131 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
1answer
202 views

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(...
4
votes
1answer
132 views

Differential of time over Browninan motion

I know that $\frac{dW_t}{dt}$, with $W_t$ a brownian motion, does not exist. However, does $\frac{dt}{dW_t}$ exists? Or does it even make sense? I am trying to calculate the quotient of two ...
4
votes
1answer
163 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....

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