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Questions tagged [itos-lemma]

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3
votes
1answer
218 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} =...
1
vote
1answer
430 views

What are the dynamics of the reverse of this FX process?

Assuming the dynamics of the exchange rate between two currencies at time $t$ is given by: $$ dX_t=\Delta r X_t dt+ σ X_t dW_t$$ Is the FX Reverse process $\frac{1}{X_t}$ a brownian motion? How can ...
0
votes
1answer
238 views

Simulating a GBM with martingale condition - Ito process moving downwards

I want to correctly simulate a $\mathcal{Q}$ - martingale $S$, which is a geometric Brownian motion and an exponential of a process $X$, \begin{equation} X_t = X_0 + \mu t + \sigma B_t = X_{t-\Delta t}...
7
votes
1answer
292 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 ...
2
votes
2answers
180 views

Markov Pricing kernel

I'm reading about Markov pricing kernels in the lecture notes of a course I'm following, but I have a big doubt on an application of Ito's lemma. The setting is the following: We define the pricing ...
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 ...
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
108 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 ...
0
votes
2answers
100 views

Shreve book II Question 4.6 Error?

I'm working through Shreve II, and on question 4.6, you are asked to compute $d(S_t^p)$ where $S_t$ = $S_0e^{\sigma W_t + (\alpha - \frac{1}{2}\sigma^2)t}$ I get the answer $pS_t^p[\sigma dW_t + (\...
1
vote
0answers
148 views

In what kind of stochastic process Ito's lemma is adopted?

I have been told that Ito's lemma serves as the stochastic calculus counterpart of the chain rule. And yet again my tutor mentioned it is not used for all stochastic processes. Is this statement true?...
3
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^{\...
12
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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 ...
1
vote
1answer
394 views

How to derive equivalent martingale measure using Ito's Lemma

Can someone explain how to get equation 27.14 below? I understand the first usage of Ito's Lemma to get $d(\ln f-\ln g)$ but I do not understand how to use Ito's Lemma to go from $d(\ln \frac{f}{g})$ ...
3
votes
1answer
227 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 $...
5
votes
1answer
413 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}...
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\...
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}-...
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 ...