Questions tagged [itos-lemma]

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20
votes
2answers
27k 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 ...
14
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3answers
9k 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\...
12
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 ...
12
votes
2answers
3k 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 = ...
8
votes
3answers
1k 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
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
4k 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
3k 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 ...
7
votes
4answers
10k 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 ...
7
votes
1answer
484 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
166 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
288 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 ...
6
votes
1answer
319 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
426 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 ...
6
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0answers
111 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,...
5
votes
1answer
345 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?
5
votes
1answer
369 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
230 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
399 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
499 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
294 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
115 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
88 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 ...
4
votes
1answer
167 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 ...
4
votes
1answer
167 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 ...
3
votes
4answers
11k views

Difference between ito process, brownian motion and random walk

Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out. 1.Ito process 2....
3
votes
4answers
603 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
1answer
105 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, ...
3
votes
2answers
289 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 $Z_s=f(t-s,X_s)$ ...
3
votes
2answers
139 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 ...
3
votes
1answer
121 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 ...
3
votes
1answer
437 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
2answers
103 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 ...
3
votes
1answer
595 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. ...
3
votes
1answer
488 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $ ...
3
votes
2answers
164 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 $...
3
votes
1answer
216 views

On an application of Ito's lemma

Assume that instantaneous returns are generated by the continuous time martingale: $$dp_t = \sigma_t dW_t$$ where $W_t$ denotes a standard Weiner process and One day returns are denoted by $r_{t+1} =...
3
votes
1answer
219 views

Integration of stochastic total derivative

Super basic question. I think I am doing this correctly, but just want a sanity check. Say I have a stochastic process $r(t)$. Say I have an equation $$d(e^{\beta (t-s)}r(s))=\dots$$ where the $...
3
votes
1answer
117 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 ...
3
votes
1answer
112 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 ...
3
votes
1answer
393 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 ...
3
votes
1answer
340 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
3
votes
1answer
166 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 = \...
3
votes
1answer
217 views

Derivation using Ito's Lemma of price process

Define $q(t)$ as the log price minus a linear trend $$ q(t) = \ln P(t) - \mu t $$ Assume the log price process = Equation 1: $$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$ Can you show that the ...
3
votes
1answer
66 views

Show that Z(t)/Z(0) is a positive mean-1 martingale

We look at a standard no dividends Black-Scholes model and here we have a process Z, which is defined by: Z(t)=(S(t)/H)^p , where H is a positive constant and p=1-2r/sigma^2 I am now asked to show ...
3
votes
0answers
107 views

Dixit & Pindyck (1993) Chapter 4, equation 13

Starting with the Bellman equation for the optimal stopping problem: $$F(x,t)=max\{\Omega(x,t), \pi(x,t)+(1+\rho dt)^{-1} E[F(x+dx, t+dt)|x]\}$$ In the continuation region where the second term is the ...
2
votes
1answer
267 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 {...
2
votes
1answer
2k views

How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?

The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given: $$S_t=S_0\,e^{\...
2
votes
1answer
175 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 ...